REPRESENTATION THEORY An Electronic Journal of the American Mathematical Society Volume 2, Pages 106–223 (April 10, 1998) S 1088-4165(98)00011-9

FINITE QUATERNIONIC MATRIX GROUPS

GABRIELE NEBE

Abstract. Let D be a definite quaternion algebra such that its center has degree d over ℚ. A subgroup G of GL_{n}(D) is absolutely irreducible if the ℚ-algebra spanned by the matrices in G is D^{n×n}. The finite absolutely irreducible subgroups of GL_{n}(D) are classified for nd ≤ 10 by constructing representatives of the conjugacy classes of the maximal finite ones. Methods to construct the groups and to deal with the quaternion algebras are developed. The investigation of the invariant rational lattices yields quaternionic structures for many interesting lattices.

Table of contents.

1. Introduction.

2. Definitions and first properties.

3. The Schur subgroup of the Brauer group.

4. Algorithms for quaternion algebras.

5. Notation for the finite matrix groups.

6. The a.i.m.f. subgroups of GL_{1}(Q).

7. Normal subgroups of primitive groups.

8. The normal p-subgroups of primitive groups and their automorphism groups.

9. The candidates for quasi-semi-simple normal subgroups.

10. Some building blocks.

11. Special dimensions.

12. The a.i.m.f. subgroups of GL_{2}(Q).

13. The a.i.m.f. subgroups of GL_{3}(Q).

14. The a.i.m.f. subgroups of GL_{4}(Q).

15. The a.i.m.f. subgroups of GL_{5}(Q).

16. The a.i.m.f. subgroups of GL_{6}(Q).

17. The a.i.m.f. subgroups of GL_{7}(Q).

18. The a.i.m.f. subgroups of GL_{8}(Q).

19. The a.i.m.f. subgroups of GL_{9}(Q).

20. The a.i.m.f. subgroups of GL_{10}(Q).

21. Appendix: The new r.i.m.f. subgroups in dimension 32, 36, and 40.

1. Introduction

The rational group algebra of any finite group is a semisimple algebra, hence a direct sum of matrix rings over division algebras. Whereas for any n ∈ ℕ there

\begin{matrix}c \\ ◯ \end{matrix}1998 American Mathematical Society

Received by the editors December 23, 1996 and, in revised form, February 3, 1998 and February 16, 1998.

1991 Mathematics Subject Classification. Primary 20C10, 11E39, 11R52.

exists a finite group G such that ℚ^{n×n} is a direct summand of ℚG (in other words GL_{n}(ℚ) has a finite absolutely irreducible subgroup) this is not true for an arbitrary division algebra ≠ ℚ. Clearly the centers of the occurring division algebras are generated by the character values of the corresponding character of G and hence finite abelian extensions of ℚ. If the center is real, then the involution on ℚG defined by g ↦ g^{−1} for all g ∈ G preserves the corresponding direct summand D^{n×n} and therefore induces an involution on it. From this one deduces the theorem of Brauer and Speiser, which says that if the center K := Z(D) is a totally real number field, then D is either K or a quaternion algebra over K.

For fixed n and fixed degree d := [K : ℚ] of the real field K over ℚ the unit group of the direct summand of ℚG embeds into GL_{dn}(ℚ) if D = K is abelian and into GL_{4dn}(ℚ) if D is a quaternion algebra over K. The image of G under this homomorphism is a finite subgroup of GL_{dn}(ℚ) resp. GL_{4dn}(ℚ) with enveloping algebra D^{n×n}. Since for given m ∈ ℕ the group GL_{m}(ℚ) has finitely many conjugacy classes of finite subgroups. This shows that for fixed n and d, there are only finitely many possibilities for D.

This paper deals with the case where D is a definite quaternion algebra over K. All D are determined for which D^{n×n} is a direct summand of ℚG for a finite group G; in other words, G has an absolutely irreducible representation into GL_{n}(D), if n · [K : ℚ] ≤ 10. We derive much finer information on the unit group GL_{n}(D) of D^{n×n} by determining all of its absolutely irreducible (cf. Definition 2.1) maximal finite (a.i.m.f.), subgroups. The classification results are given in the form of tables containing representatives for the primitive a.i.m.f. subgroups of GL_{n}(D) and some information on the invariant lattices (cf. Table 6.3, Theorem 12.1, Table 12.7, and the Theorems 12.15, 12.17, 12.19, 13.1, 13.3, 13.5, 14.1, 14.14, 15.1, 15.3, 16.1, 17.1, 18.1, 19.1, and 20.1). The conjugacy classes of the a.i.m.f. subgroups are interrelated via common absolutely irreducible subgroups (cf. Definition 2.12) and we determine the resulting simplicial complexes for n ≤ 7, (n, d) ≠ (4, 2).

Quaternionic matrix groups have already been studied by various authors. For instance in [Ami 55] the finite subgroups of GL_{1}(D) are classified, [HaS 85] treats the quasisimple finite subgroups of GL_{2}(D), and A.M. Cohen determines the finite quaternionic reflection groups in [Coh 80]. Quite a few of these reflection groups are a.i.m.f. subgroups (cf. Remark 5.2). The last article is somehow closer to the present paper, since Cohen describes the corresponding root systems. But none of the authors treat the subject from the arithmetic point of view so they do not look at the G-invariant lattices for the various maximal orders in D.

Any subgroup G of GL_{n}(D) may be considered as a subgroup of GL_{4dn}(ℚ) via the regular representation of D. The rational irreducible maximal finite subgroups of GL_{m}(ℚ) are classified for m ≤ 31 (cf. [PlN 95], [NeP 95], [Neb 95], [Neb 96], [Neb 96a]). As a consequence of this paper one obtains certain maximal finite subgroups of GL_{m}(ℚ) which contain an a.i.m.f. group. So the results give a partial classification of the rational irreducible maximal finite subgroups of GL_{m}(ℚ) for the new degrees m=32, 36, and 40 (see Appendix).

Finite subgroups of GL_{m}(ℚ) act on Euclidean lattices. In particular the maximal finite groups are automorphism groups of distinguished lattices. The action of an a.i.m.f. subgroup G of GL_{n}(D) ≤ GL_{4dn}(ℚ) on such a lattice L defines a Hermitian structure on L as a lattice over its endomorphism ring End_{G}(L), which is an order in the commuting algebra C_{ℚ^{m×m}}(G) ≅ D. Only those lattices L where End_{G}(L) is a maximal order in D are investigated. This yields Hermitian structures for many nice lattices. For example for the Leech lattice, the unique even unimodular lattice of dimension 24 without roots, we find, in addition to the two well known structures as Hermitian lattice described in [Tit 80], nine other structures over a maximal order of a definite quaternion algebra D preserved by an a.i.m.f. subgroup of GL_{n}(D).

There is a mysterious connection between large class numbers of number fields and the existence of nice lattices. For example the Leech lattice occurs as an invariant lattice of the group SL_{2}(23) due to the fact that the class number of ℤ[\frac{1+\sqrt{-23}}{2}] is 3. The occurence of large prime divisors of the determinants of invariant lattices of maximal finite groups also has an explanation using class groups. In GL_{16}(ℚ) there are two irreducible maximal finite subgroups fixing no lattice of which the determinant only involves primes dividing the group order. The same phenomenon happens in GL_{32}(ℚ) where there are at least four such primitive groups. These four groups contain a.i.m.f. subgroups of GL_{2}(D) where D is the quaternion algebra with center ℚ[\sqrt{3}, \sqrt{5}] ramified only at the four infinite places. The narrow class group of the center of D, which is isomorphic to the group of stable classes of left ideals of a maximal order in D (cf. [Rei 75]), is of order 2 and the norm of any integral generator has a prime divisor ≥ 11. Related to this the determinants of the integral lattices of the six maximal finite groups have prime divisors ≥ 11.

The article is organized as follows: Chapter 2 contains the fundamental definitions and generalizations of some important properties of rational matrix groups to matrix groups over quaternion algebras. The most important notion is the one of imprimitivity reducing the determination of a.i.m.f. groups to the one of primitive ones. In the next chapter known restrictions on the quaternion algebras that can be Morita equivalent to a direct summand of a group algebra of a finite group are used to introduce a notation for these quaternion algebras. Chapter 4 derives methods to compute representatives of the conjugacy classes of maximal orders in a definite quaternion algebra D and describes the results for the occurring D by expanding the mass formulas. Chapter 5 introduces some notation used for finite matrix groups. As an application of the classification of finite subgroups of GL_{2}(ℂ) and a theorem of Brauer the a.i.m.f. subgroups of GL_{1}(D) for arbitrary quaternion division algebras D are determined in Chapter 6. The invariant lattices are only determined if the degree of the center of D over ℚ is ≤ 5 since the class number of D rapidly increases afterwards.

In this paper one does not know the quaternion algebra D in advance. So, before one can use arithmetic structures and calculate the a.i.m.f. group G as an automorphism group of a lattice, one has to build up a fairly large subgroup of G to get enough restrictions on D. For this purpose, methods to conclude from the existence of a small normal subgroup N in G, the existence of a (much) bigger one, the generalized Bravais group of N (cf. Definition 7.1), are developed further. For some groups N this generalized Bravais group splits as a tensor factor and reduces the determination of G to the one of C_{G}(N) which is a maximal finite subgroup in the unit group of the commuting algebra of N. If the enveloping algebra of N is a central simple K-algebra, the general case is not much harder.

The possible normal p-subgroups of primitive a.i.m.f. groups may be derived from a theorem of P. Hall and are essentially extraspecial groups. The investigation of the automorphism groups of the relevant p-groups leads to a determination of the generalized Bravais groups of these groups in Chapter 8. The next chapter contains a table of the occurring quasisimple groups to fix the notation for the irreducible characters and to give the information that is used from the classification of finite simple groups and their character tables in [CCNPW 85].

When building up the primitive maximal finite subgroups of GL_{n}(D) by normal subgroups one needs not only the maximal finite matrix groups in smaller dimension but the maximal pairs of finite groups together with their normalizers. Some of these “building blocks” are classified in Chapter 10. In Chapter 11 four infinite series of a.i.m.f. groups which come from representations of the groups SL_{2}(p) for primes p are presented. The last nine chapters deal with the determination of the a.i.m.f. groups of GL_{n}(D) for definite quaternion algebras D with n[Z(D) : ℚ] ≤ 10. There is one chapter for each dimension n = 2, …, 10.

The general strategy is as follows: Let G be a primitive a.i.m.f. subgroup of GL_{n}(D) for some n ∈ ℕ and a d-dimensional ℚ-division algebra D. Then the order of G is bounded in terms of nd (cf. Proposition 2.16). One has only finitely many possibilities for the maximal nilpotent normal subgroup P := ∏_{p||G|} O_{p}(G) (see Table 8.7). The centralizer C_{G}(P) is an extension of a direct product of quasisimple groups Q := C_{G}(P)^{(∞)} by a subgroup of the outer automorphism group of Q. The possibilities for Q are deduced from the classification of finite simple groups and their character tables in [CCNPW 85] (cf. Table 9.1). So one has a finite list of possible normal subgroups QP ⊴ G with G/QP ≤ Out(QP). The methods developed in Chapter 7 now allow one to conclude the existence of a usually much larger normal subgroup B := B^{∘}(QP) in G. The possible extensions G of B by outer automorphisms of QP not induced by elements of B can now be handled case by case.

In an appendix the invariants of the lattices of the new maximal finite subgroups of GL_{32}(ℚ), GL_{36}(ℚ), and GL_{40}(ℚ) are displayed in the form of tables.

The computer calculations were mainly done by stand alone C-programs (for solving systems of linear equations, calculating sublattices invariant under an order in ℚ^{n×n} with the algorithm described in [PlH 84], calculating automorphism groups of positive definite lattices as described in [PlS 97], …) developed at the Lehrstuhl B für Mathematik of the RWTH Aachen (Germany). These algorithms are or will be also available in MAGMA (cf. [MAGMA]). The investigation of the isomorphism type of the matrix groups was done with the help of the two group theory packages GAP (cf. [GAP 94]) and MAGMA. Invariant Hermitian forms for the primitive a.i.m.f. subgroups of GL_{n}(D) can be obtained from the author’s home page, via http://www-math.math.rwth-aachen.de/∼LBFM/.

The work for this paper was a research project during a one year DFG-fellowship at the University of Bordeaux. I want to thank both organizations. In particular I express my gratitude to J. Martinet who encouraged me to treat quaternion algebras and towards the institute of applied mathematics of the University of Bordeaux for allowing me to use one of their computers.

2. Definitions and first properties

In this paper maximal finite subgroups of the unit group GL_{n}(D) of D^{n×n} for totally definite quaternion algebras D over totally real number fields will be determined.

Since the representation theoretical methods generalize to arbitrary division algebras D, let D be a division algebra whose center K is a finite extension of ℚ.

The module V := D^{1×n} is a right module for D^{n×n}. Its endomorphisms are given by left multiplication with elements of D. For computations it is convenient also to let the endomorphisms act from the right. Then End_{D^{n×n}}(V) ≅ D^{op} which we identify with D in the case of quaternion algebras.

The following definition can also be found in [ShW 86].

Definition 2.1. Let G be a finite group and Δ : G → GL_{n}(D) be a representation of G.

(i) Let L be a subring of K. The enveloping L-algebra \overline{LΔ(G)} is defined as

\overline{LΔ(G)} := \{ ∑_{g∈Δ(G)} l_{g}g | l_{g} ∈ L\} ⊆ D^{n×n}.

(ii) Δ is called absolutely irreducible if the enveloping ℚ-algebra \overline{Δ(G)} := \overline{ℚΔ(G)}

of Δ(G) is D^{n×n}. (iii) Δ is called centrally irreducible if the enveloping K-algebra \overline{KΔ(G)} is D^{n×n}. (iv) Δ is called irreducible if the commuting algebra C_{D^{n×n}}(Δ(G)) is a division

algebra.

(v) A subgroup G ≤ GL_{n}(D) is called irreducible (resp. centrally irreducible, absolutely irreducible), if its natural representation id : G → GL_{n}(D) is irreducible (resp. centrally irreducible, absolutely irreducible).

Being only interested in those groups G, to which the quaternion algebra D is really attached, only the absolutely irreducible maximal finite (a.i.m.f.) subgroups of GL_{n}(D) will be determined. The irreducible maximal finite subgroups G of GL_{n}(D) are absolutely irreducible in their enveloping ℚ-algebra \overline{G}, which is a matrix ring over some division algebra \overline{G} ≅ D^{′m×m} with m^{2}dim_{ℚ}(D′) dividing n^{2}dim_{ℚ}(D). The reducible maximal finite subgroups of GL_{n}(D) can be built up from the irreducible maximal finite subgroups of GL_{l}(D) for l < m.

As in the case D = ℚ, the notion of primitivity gives an important reduction in the determination of the maximal finite subgroups.

Definition 2.2. (cf. [Lor 71] (1.3)) Let G ≤ GL_{n}(D) be an irreducible finite group. Consider V := D^{1×n} as D-G-bimodule. G is called imprimitive, if there exists a decomposition V = V_{1} ⊕ … ⊕ V_{s} of V as a nontrivial direct sum of D-left modules such that G permutes the V_{i} (i.e. for all g ∈ G, for all 1 ≤ i ≤ s, ∃1 ≤ j ≤ s such that V_{i}g ⊆ V_{j}). If G is not imprimitive, G is called primitive.

If G is an imprimitive group and V = V_{1} ⊕ … ⊕ V_{s} a nontrivial G-stable decomposition of V as in the definition, the natural representation of G is induced up from the natural representation Δ_{1} of the subgroup U := Stab_{G}(V_{1}) on V_{1}. If G is absolutely irreducible, then Δ_{1} : U → End_{D}(V_{1}) is also absolutely irreducible. Especially the imprimitive a.i.m.f. subgroups of GL_{n}(D), being maximal finite, are wreath products of primitive a.i.m.f. subgroups of GL_{d}(D) with the full symmetric group S\frac{n}{d} of degree \frac{n}{d} for divisors d of n.

As for D = ℚ, the primitive groups have the following frequently used property:

Remark 2.3. Let G ≤ GL_{n}(D) be a primitive finite group and N ⊴ G be a normal subgroup. Then the enveloping K-algebra \overline{KN} ⊆ D^{n×n} is simple.

Proof. Assume that \overline{KN} is not simple. Since \overline{KN} is semisimple there exists a decomposition 1 = e_{1} + … + e_{s} of 1 ∈ \overline{KN} into centrally primitive idempotents e_{i} ∈ \overline{KN} (1 ≤ i ≤ s). The group G acts by conjugation on N, hence on \overline{KN} and therefore on the set of centrally primitive idempotents in \overline{KN}. So the decomposition V = V e_{1}⊕…⊕V e_{s} is stable under the action of G. By primitivity of G this implies s = 1.

Corollary 2.4. Let G ≤ GL_{n}(D) be a primitive finite subgroup and N ⊴ G be a normal subgroup.

(i) If N is a p-group, then (p − 1) · p^{α} divides dim_{ℚ}(K) · n for some α ≥ 0. (ii) If N is abelian, then N is cyclic.

Notation 2.5. Let G be a primitive subgroup of GL_{n}(D) and N ⊴ G a normal subgroup of G. By Remark 2.3, the restriction of the natural character of G to N is a multiple of a K-irreducible character χ, where K := Z(D). By the theorem of Skolem-Noether the knowledge of χ is sufficient to identify the conjugacy class of N in GL_{n}(D). This will be expressed by the phrase G contains N with character χ.

Invariant Hermitian lattices. For the rest of this chapter assume that D is a totally definite quaternion algebra, and let ^{∗} : D → D be the canonical involution of D such that xx^{∗} ∈ K = Z(D) for all x ∈ D and such that ^{∗} induces the identity on the center K. (cf. [Scha 85, (8.11.2)]) We extend ^{∗} to D^{n×n} by applying the involution to the entries of the matrices. Then g ↦ (g^{∗})^{t} where g^{t} denotes the transposed matrix of g is an involution on the algebra D^{n×n}.

Then the maximal finite subgroups of GL_{n}(D) can be described as full automorphism groups of totally positive definite Hermitian lattices as follows:

Definition and Lemma 2.6. Let G be a finite subgroup of GL_{n}(D) and V := D^{1×n} the natural G-right-module and let 𝔐 be an order in D = End_{D^{n×n}}(V).

(i) An 𝔐-lattice L ≤ V is a finitely generated projective 𝔐-left module with ℚL = V.

(ii) The set

Z_{𝔐}(G) := {L ≤ V | L is a 𝔐-lattice and Lg ⊆ L}

of G-invariant 𝔐-lattices in V is nonempty. (iii) The K vector space

F(G) := {F ∈ D^{n×n} | F^{t} = F^{∗} and gF (g^{∗})^{t} = F for all g ∈ G}

of G-invariant Hermitian forms contains a totally positive definite form, i.e.

F_{>0}(G) := {F ∈ F(G) | ϵ(F)is positive definite

for all embeddings ϵ : K ↪ ℝ} ≠ ∅.

(iv) Let L be an 𝔐-lattice in V and F ∈ D^{n×n} a totally positive definite Hermitian form. The automorphism group

Aut(L, F) := {g ∈ GL_{n}(D) | Lg ⊆ L and gF (g^{∗})^{t} = F }

of L with respect to F is a finite group.

(v) The a.i.m.f supergroups of G are of the form Aut(L, F) for some L ∈ Z_{𝔐}(G) and F ∈ F_{>0}(G).

Proof. (ii) Let (b_{1}, …, b_{n}) be a D-basis of V. Then

L := {∑_{i=1}^{n} m_{i}b_{i}g_{i} | m_{i} ∈ 𝔐, g_{i} ∈ G} ∈ Z_{𝔐}(G).

(iii) Choose any totally positive definite Hermitian form F ∈ D^{n×n}. Define F_{0} :=

∑_{g∈G} gF (g^{∗})^{t}. Then F_{0} ∈ F_{>0}(G) is totally positive definite. (iv) Fix an embedding ϵ : K → ℝ and let m := max{ϵ(vF (v^{∗})^{t}) | v ∈ S}, where S

is a finite subset of L generating L. Then the set {x ∈ L | ϵ(xF (x^{∗})^{t}) ≤ m} is a finite set containing the images of the elements of S under the automorphisms g ∈ Aut(L, F). Since g is uniquely determined by these images, one has only finitely many possibilities for g.

(v) Follows from (ii)-(iv).

In view of 2.6 (v), one may calculate all a.i.m.f. supergroups of a finite subgroup G ≤ GL_{n}(D) as automorphism groups of G-invariant lattices.

The centralizer C_{GL_{n}(D)}(G) of G in GL_{n}(D) acts on Z_{𝔐}(G). Two lattices are called isomorphic if they lie in the same orbit under this action. Clearly a system of representatives of isomorphism classes of lattices in Z_{𝔐}(G) suffices to get all a.i.m.f. supergroups. So the theorem of Jordan and Zassenhaus says that one may always find a finite critical set of invariant lattices in the sense of the following definition.

Definition 2.7. A set of lattices S ⊆ Z_{𝔐}(G) is called critical (resp. normal critical) if for all finite groups H with G ≤ H ≤ GL_{n}(D) (resp. G ⊴ H ≤ GL_{n}(D)) there is an L ∈ S and some F ∈ F_{>0}(G) such that H ≤ Aut(L, F). If S = {L} consists of one lattice, L itself is called (normal) critical.

Definition 2.8. Let L ∈ Z_{𝔐}(G) and F ∈ F_{>0}(G).

(i) The Hermitian dual lattice is defined as

L^{∗} := {v ∈ D^{1×n} | vF l^{t} ∈ 𝔐 for all l ∈ L}.

(ii) If L is integral (i.e. L^{∗} ⊇ L), then its Hermitian determinant is det(L) := |L^{∗}/L|.

Remark 2.9. For all L ∈ Z_{𝔐}(G) and F ∈ F_{>0}(G) the Hermitian dual lattice L^{∗} is also in Z_{𝔐}(G) (cf. [Neb 98, Lemma 1.1]).

The rational maximal finite supergroups. Recall that D is a totally definite quaternion algebra with center K and d := [K : ℚ]. Via the regular representation of D, one may embed GL_{n}(D) into GL_{4dn}(ℚ). Therefore it makes sense to ask for the rational maximal finite supergroups G ≤ H ≤ GL_{4dn}(ℚ) of an a.i.m.f. subgroup G of GL_{n}(D). The relation of the lattices is given in the following

Definition 2.10 (cf. [Scha 85, p. 348]). Let 𝔐 be an order in D and L an 𝔐lattice. Let F ∈ D^{n×n} be a totally positive definite Hermitian form. The corresponding Euclidean ℤ-lattice L is the set L (considered as ℤ-module) together with the trace form tr(F) : (v, w) ↦ tr(vF (w^{∗})^{t}) where tr is the reduced trace of D over ℚ.

Remark 2.11. (i) Since tr(x) = tr(x^{∗}), the trace form tr(F) of a Hermitian form F is a symmetric ℚ-bilinear form. If F is totally positive definite, then tr(F) is positive definite.

(ii) Let G ≤ GL_{n}(D) be absolutely irreducible. Since D is totally definite the ℚ-vector space of G-invariant quadratic forms on ℚ^{4dn} is {tr(F) | F ∈ F(G)} = {tr(aF_{0}) | a ∈ K} for any F_{0} ∈ F_{>0}(G). As in Remark 2.6 one gets that the rational maximal finite supergroups of G are of the form Aut(L, tr(F)) := {g ∈ GL_{4dn}(ℚ) | Lg = L, gtr(F)g^{t} = tr(F)} for some F ∈ F_{>0}(G) and L ∈ Z_{𝔐}(G), where 𝔐 := End_{G}(L) is an order in D.

(iii) As the G-invariant Hermitian forms give rise to embeddings of G into an orthogonal group over K, one may also consider the invariant skew-Hermitian forms to get embeddings of G into the symplectic group over K. For an a.i.m.f. subgroup G of GL_{n}(D) the K vector space F(G) is of dimension one, whereas the K vector space of invariant skew-Hermitian forms is D_{0}F(G), where D_{0} denotes the quaternions of trace 0 in D, is of dimension 3 over K. Therefore the embedding of G into the symplectic group is not unique.

Common absolutely irreducible subgroups. Having found the a.i.m.f. subgroups of GL_{n}(D), one may interrelate them via common absolutely irreducible subgroups in the sense of the following definition.

Definition 2.12. The simplicial complex M_{n}^{irr}(D) of a.i.m.f. subgroups of GL_{n}(D) has the GL_{n}(D)-conjugacy classes of a.i.m.f. groups of degree n as vertices. The s + 1 vertices P_{0}, …, P_{s} form an s-simplex, if there exist representatives G_{i} ∈ P_{i} and an absolutely irreducible subgroup H ≤ GL_{n}(D) with H ≤ G_{i} for i = 0, …, s.

This definition is a straightforward generalization of the definition of M_{n}^{irr}(ℚ) for rational irreducible matrix groups (cf. [Ple 91]). One might think of generalizations of this definition to common uniform subgroups U (i.e. dim_{K}(F(U)) = 1) of a.i.m.f. groups in GL_{n}(D) and GL_{n}(D′) for (different) quaternion algebras D and D′ with the same center K. A second possibility to interrelate a.i.m.f. subgroups of GL_{n}(D) for different quaternion algebras D (or even simplices in M_{n}^{irr}(D)) is described in Remark 12.11.

In this paper we determine the simplicial complexes M_{n}^{irr}(D) for n ≤ 7 with (n, [Z(D) : ℚ]) ≠ (4, 2).

As for the simplicial complexes of rational matrix groups the D-isometry class of the invariant Hermitian forms distinguishes the different components of M_{n}^{irr}(D). By [Scha 85, Theorem 10.1.7] two Hermitian forms are isometric if and only if their trace forms (cf. Definition 2.10) are isometric quadratic forms over the center K = Z(D). Hence in our situation all totally positive definite Hermitian forms of a given dimension are isometric.

To build up the a.i.m.f. groups, but also to find their common absolutely irreducible subgroups, it is helpful to know divisors of the group order.

Lemma 2.13. Let U ≤ GL_{n}(D) be an absolutely irreducible subgroup, 𝔐 a maximal order in D and L ∈ Z_{𝔐}(U). Let p ∈ ℤ be a prime.

(i) If p ramifies in D, then p divides the order of U. (ii) Let F be a U-invariant Hermitian form on L. If p divides |AL^{∗}/L| for all

fractional ideals A of K such that AL^{∗} ⊇ L, then p divides the order of U.

Proof. (i) If p ramifies in D , then p divides the discriminant of the maximal orders in D^{n×n}. The order \overline{ℤU} is contained in some maximal order and hence its discriminant is also divisible by p. Since \overline{ℤU} is an epimorphic image of the group ring ℤU, p divides the order of U.

(ii) Since 𝔐 is a maximal order, by [Neb 98, Lemma 1.1] the Hermitian dual lattice L^{∗} is also a 𝔐 − U-lattice. Let R be the ring of integers in K and ℘ be a prime ideal containing p. Assume that p | |̸U|. Then by (i) ℘ does not ramify in D and therefore the two-sided ideals of R_{℘} ⊗_{R} 𝔐 are of the form R_{℘} ⊗_{R} A𝔐 for fractional ideals A of R. If for all fractional ideals A of R the completion R_{℘} ⊗_{R} AL^{∗} ≠ R_{℘} ⊗_{R} L, then the R_{℘}-order R_{℘} ⊗_{R} 𝔐U is not a maximal order and therefore p divides |U| which is a contradiction.

The next lemma may be proved similarly to Lemma (II.7) of [NeP 95]:

Lemma 2.14. Let N ⊴ G ≤ GL_{n}(D) be a normal subgroup of G with |G/N| =: s. Then s · dim_{ℚ}(\overline{ℚN}) ≥ dim_{ℚ}\overline{ℚG}.

Proof. If G = ⋃_{i=1}^{s} Ng_{i} and (b_{1}, …, b_{m}) is a ℚ-basis of \overline{ℚN}, then the elements b_{j}g_{i} (1 ≤ i ≤ s, 1 ≤ j ≤ m) generate \overline{ℚG}.

For normal subgroups of index 2 in an absolutely irreducible subgroup G of GL_{n}(D) one may now strengthen Lemma 2.13:

Lemma 2.15. Let N ⊴ G be a normal subgroup of index two in an absolutely irreducible subgroup G of GL_{n}(D). If p is a prime ramifying in D, then p divides the discriminant of the enveloping ℤ-order \overline{ℤN} of N.

Proof. If N is already absolutely irreducible, the lemma follows from Lemma 2.13. So assume that N is not absolutely irreducible. Let g ∈ G − N. Then \overline{ℤG} contains the order O := \overline{ℤN} ⊕ \overline{ℤN}g of finite index. The discriminant of O is disc(\overline{ℤN})^{2} the square of the discriminant of \overline{ℤN}.

By the formula in [Schu 05] the prime divisors of the order of a finite group G may be bounded in terms of the character degree and the (degree of) the character field of an irreducible faithful character of G.

Proposition 2.16. Let χ be a faithful irreducible rational character of a finite group G with χ(1) = n. Then the order of G divides

M_{n} := ∏_{p≤n+1} p^{⌊n/(p−1)⌋+⌊n/(p(p−1))⌋+⌊n/(p^{2}(p−1))⌋+...}

where the product runs over all primes p ≤ n + 1.

In view of Lemma 2.13 (i), one now has only finitely many candidates for quaternion algebras D such that GL_{n}(D) has a finite absolutely irreducible subgroup, if one bounds n and the degree of the center of D over ℚ.

But there remain too many candidates for D to be dealt with separately. So the main strategy to find the primitive maximal finite absolutely irreducible subgroups G ≤ GL_{n}(D) will be to build them up using normal subgroups.

The following lemma can be shown to hold as in [Neb 96, Lemma 1.13]:

Lemma 2.17. Let N ⊴ G ≤ GL_{n}(D) be a normal subgroup of G with |G/N| = 2. Assume that \overline{ℚN} and \overline{ℚG} = D^{n×n} are simple algebras with centers K resp. K^{+}, where K is complex and K^{+} is the maximal totally real subfield of K. Then the isoclinic group is not a subgroup of GL_{n}(D).

Immediately from the theorem of Brauer and Witt one gets the following lemma:

Lemma 2.18. Let U ≤ G, χ an irreducible character of G and χ_{1} an irreducible constituent of χ_{|U}. Assume that the character fields of χ and χ_{1} are equal. If (χ_{|U}, χ_{1}) is odd, then the Schur index of χ is 2 at exactly those primes where χ_{1} has Schur index 2.

3. The Schur subgroup of the Brauer group

Standard references for this section are [Yam 74] and [Rei 75].

Let K be a number field and Br(K) denote the Brauer group of K. If G is a finite group, then by Maschke’s theorem, the algebra KG is a finite dimensional semisimple K-algebra, hence KG = ⨁ A_{i} decomposes into a direct sum of simple K-algebras A_{i} ≅ D_{i}^{n_{i}×n_{i}}, which are full matrix rings over K-division algebras D_{i}. The Schur subgroup S(K) of Br(K) consists of the classes [D] where D is a central simple K-division algebra for which there is an n ∈ ℕ and a finite group G such that D^{n×n} is a ring direct summand of KG.

If D^{n×n} is a ring direct summand of the rational group algebra ℚG and K contains the center Z(D), then [K ⊗_{Z(D)} D] ∈ S(K), so the algebra ℚG contains all information about the Schur subgroups of the Brauer group of algebraic number fields.

Definition 3.1. Let D be a ℚ-division algebra. Define

μ(D) := min{n ∈ ℕ | D^{n×n} is a ring direct summand of ℚG

for some finite group G} ∈ ℕ ∪ {∞}.

If μ(D) < ∞, then the center K := Z(D) is the character field of some character of a finite group, hence an abelian extension of ℚ.

Moreover, by a theorem of Benard and Schacher (cf. [Yam 74], Theorem 6.1), D has uniformly distributed invariants, which means in particular, that the Schur index of the completions D ⊗ K_{℘}, does not depend on the prime ℘ of K but only on the rational prime ℘ ∩ ℚ contained in it.

In this paper, we only treat the case, where K is a (totally) real number field. In this case, the theorem of Brauer and Speiser says, that D is a quaternion algebra, i.e. all local Schur indices are 1 or 2. So D is uniquely determined by the set of the rational primes that are contained in primes of K that ramify in D.

Notation 3.2. Let μ(D) < ∞ and assume that K := Z(D) = ℚ[α] is a (totally) real number field. Let r(D) := {p_{1}, …, p_{k}} ⊆ ℕ ∪ {∞} be the set of those rational primes that are contained in a prime of K that ramifies in D. Then D is denoted by Q_{α,p_{1},... ,p_{k}}.

If K = ℚ or K = ℚ[\sqrt{d}] is a real quadratic field, then Theorems 7.2, 7.8, and 7.14 of [Yam 74] characterize the set of all central simple K-division algebras D with μ(D) < ∞.

The results of this paper in particular give information on μ(D) for quaternion algebras D with totally real center and [Z(D) : ℚ] · μ(D) ≤ 10 (cf. Table 4.1). It turns out that in these small dimensions, the p_{i} are either inert or ramified in Z(D).

It is not true, that for all n > μ(D) there is a finite group G such that D^{n×n} is a ring direct summand of ℚG. However, taking wreath products (or tensor products with absolutely irreducible subgroups of GL_{d}(ℚ)) one shows that this holds for all multiples n = d · μ(D) of μ(D). It would be interesting to know, if the ideal generated by the n for which there is a finite group G such that D^{n×n} is a ring direct summand of ℚG in general is ℤ.

4. Algorithms for quaternion algebras

Let D be a totally definite quaternion algebra over K = Z(D). Let G be an a.i.m.f. subgroup of GL_{n}(D). To find some distinguished integral lattices on which G acts, we embed GL_{n}(D) (and hence G) into GL_{4dn}(ℚ). In the tables we will give rational irreducible maximal finite (r.i.m.f.) subgroups of GL_{4dn}(ℚ) containing G and fixing a G-lattice L for which End_{G}(L) ⊆ D is a maximal order. The set of isomorphism classes of such G-lattices is the union of the sets of isomorphism classes of 𝔐G-lattices where 𝔐 runs through a system of representatives of conjugacy classes of maximal orders in D.

To check completeness we use the well known mass formulas developed by M. Eichler [Eic 38] (cf. [Vig 80]).

Let h be the class number of K, D the discriminant of D over K, and 𝔐 any maximal order in D. Let (I_{i})_{1≤i≤s} be a system of representatives of left ideal classes of 𝔐, 𝔐_{i} := {x ∈ D | I_{i}x ⊆ I_{i}} the right order of I_{i} and ω_{i} := [𝔐_{i}^{∗} : R^{∗}] the index of the unit group of R, the ring of integers in K, in the unit group of 𝔐_{i}. Then one has:

∑_{i=1}^{s} ω_{i}^{−1} = 2^{1−d} · |ζ_{K}(−1)| · h · ∏_{℘|D}(n(℘) − 1)

where the product is taken over all primes ℘ of R dividing the discriminant D of D and n denotes the norm of K over ℚ.

If 𝔐_{i} and 𝔐_{j} are conjugate in D, one may choose a new representative for the class of I_{j} to achieve that 𝔐_{i} = 𝔐_{j}. Then I_{i}^{−1}I_{j} is a 2-sided 𝔐_{i}-ideal. Moreover the 𝔐-left ideals I_{i} and I_{j} are equivalent, if and only if I_{i}^{−1}I_{j} is principal.

So if one reorders the 𝔐_{i} such that the first t orders 𝔐_{1}, …, 𝔐_{t} form a system of representatives of conjugacy classes of maximal orders in D and H_{i} is the number of isomorphism classes of 2-sided ideals of 𝔐_{i} (1 ≤ i ≤ t), then

∑_{i=1}^{s} ω_{i}^{−1} = ∑_{i=1}^{t} ω_{i}^{−1}H_{i}.

The occurring quaternion algebras have the additional property of having uniformly distributed invariants (cf. Chapter 3). Therefore the Galois group Gal(K/ℚ) acts on D:

Choose a K-basis (1 =: b_{1}, b_{2}, b_{3}, b_{4}) of D. An element σ ∈ Gal(K/ℚ) defines an automorphism σ of the ℚ-algebra D by σ(∑ a_{i}b_{i}) := ∑ σ(a_{i})b_{i}. By the theorem of Skolem and Noether the class σInn(D) of the automorphism σ does not depend on the chosen basis. Therefore one gets a well defined action of Gal(K/ℚ) on the set of conjugacy classes of maximal orders in D. This action preserves ω_{i} and H_{i}.

Let ω_{i}^{1} := \frac{1}{2}|\{x ∈ 𝔐_{i} | xx^{∗} = 1\}| be the index of ^{±}1 in the group of units in 𝔐_{i} of norm 1, and ω_{i}^{ns} := N(𝔐_{i}^{∗})/(R^{∗})^{2}. Then ω_{i} = ω_{i}^{1} · ω_{i}^{ns}.

If n_{i} denotes the length of the orbits of the class of 𝔐_{i} under Gal(K/ℚ) one gets Table 4.1 below.

In the first column the degree d := [K : ℚ] is given, in the second one the name of the quaternion algebra D as explained in Notation 3.2. The third column contains the relevant dimensions n and in the last column, the mass formula of D is expanded. Here the sum is taken over a system of representatives of the orbits of Gal(K/ℚ) on the conjugacy classes of maximal orders in D.

Table 4.1. The totally definite quaternion algebras Q with d := [Z(Q) : ℚ] ≤ 5 for which there is an n ≤ \frac{10}{d} such that GL_{n}(Q) has a finite absolutely irreducible subgroup:

d  D  n  ∑ n_{i}(ω_{i}^{1} · ω_{i}^{ns})^{−1} · H_{i}
1  Q_{∞,2}  1 … 10  12^{−1}
Q_{∞,3}  1 … 10  6^{−1}
Q_{∞,5}  2, 4, 6, 8, 10  3^{−1}
Q_{∞,2,3,5}  8  3^{−1} + 3^{−1}
Q_{∞,7}  3, 4, 6, 8, 9, 10  2^{−1}
Q_{∞,11}  5, 6, 10  2^{−1} + 3^{−1}
Q_{∞,13}  6  1
Q_{∞,17}  8  1 + 3^{−1}
Q_{∞,19}  9, 10  1 + 2^{−1}
2  Q_{\sqrt{2},∞}  1 … 5  24^{−1}
Q_{\sqrt{2},∞,2,3}  2, 4  1
Q_{\sqrt{2},∞,2,5}  4  3^{−1}
Q_{\sqrt{3},∞}  1 … 5  (12 · 2)^{−1} + (12 · 2)^{−1}
Q_{\sqrt{5},∞}  1 … 5  60^{−1}
Q_{\sqrt{5},∞,2,3}  4  5^{−1} · 2
Q_{\sqrt{5},∞,2,5}  2, 4  5^{−1}
Q_{\sqrt{5},∞,5,3}  2, 4  5^{−1} + 3^{−1}
Q_{\sqrt{6},∞}  2, 4  (12 · 2)^{−1} + (6 · 2)^{−1} + (4 · 2)^{−1}
Q_{\sqrt{7},∞}  3, 4  (4 · 2)^{−1} + (3 · 2)^{−1} + (12 · 2)^{−1}
Q_{\sqrt{10},∞}  4  3^{−1} + 2^{−1} + 12^{−1} + 4^{−1}
Q_{\sqrt{11},∞}  5  12^{−1} + 2^{−1}
Q_{\sqrt{13},∞}  3  12^{−1}
Q_{\sqrt{15},∞}  4  3^{−1} + (1 · 2)^{−1} + (2 · 2)^{−1} + 6^{−1}+
(3 · 2)^{−1} + 2^{−1} + 12^{−1} + (2 · 2)^{−1}
Q_{\sqrt{17},∞}  4  6^{−1}
Q_{\sqrt{21},∞}  3, 4  12^{−1} + 6^{−1}
Q_{\sqrt{33},∞}  5  6^{−1} + 3^{−1}
3  Q_{θ_{7},∞,7}  1 … 3  14^{−1}
Q_{θ_{7},∞,2}  2  12^{−1}
Q_{θ_{7},∞,3}  2  6^{−1} + 7^{−1}
Q_{θ_{9},∞,3}  1 … 3  18^{−1}
Q_{θ_{9},∞,2}  2  12^{−1} + 9^{−1}
Q_{ω_{13},∞,13}  2  1
Q_{ω_{19},∞,19}  3  2^{−1} + 1 + 3 · 1
4  Q_{θ_{15},∞}  1, 2  (30 · 2)^{−1} + 60^{−1}
Q_{θ_{16},∞}  1, 2  16^{−1} + 24^{−1}
Q_{θ_{20},∞}  1, 2  (20 · 2)^{−1} + (12 · 2)^{−1} + 60^{−1}
Q_{θ_{24},∞}  1, 2  (24 · 2)^{−1} + (8 · 2)^{−1} + 24^{−1}
Q_{η_{17},∞}  2  6^{−1} + 2 · 12^{−1}
Q_{\sqrt{2}+\sqrt{5},∞}  2  24^{−1} + 60^{−1}
Q_{\sqrt{2}+\sqrt{5},∞,2,5}  2  5^{−1} + 2 · 1 · 2
Q_{η_{40},∞}  2  (10 · 2)^{−1} + 60^{−1} + 5^{−1}
+(2 · 2)^{−1} + (12 · 2)^{−1} + (4 · 2)^{−1}
Q_{\sqrt{3}+\sqrt{5},∞}  2  60^{−1} + (12 · 2)^{−1} + (12 · 2)^{−1} + (5 · 2)^{−1}
Q_{η_{48},∞}  2  (6 · 2)^{−1} + (2 · 2)^{−1} + 2 · 3^{−1} + 24^{−1}

+(8 · 2)^{−1} + 2 · (1 · 2)^{−1} + (4 · 2)^{−1}

+(1 · 2)^{−1} + (8 · 2)^{−1} + (2 · 2)^{−1}
5  Q_{θ_{11},∞,11}  1, 2  22^{−1} + 3^{−1}
Q_{θ_{11},∞,2}  2  12^{−1} + 1^{−1} + 11^{−1}
Q_{θ_{11},∞,3}  2  6^{−1} + 1^{−1} · 2 + 5 · 1^{−1} + 1^{−1} · 2
Q_{σ_{25},∞,5}  2  3^{−1} + 5 · 3^{−1} · 2 + 5 · 1^{−1} · 2 + 5 · 1^{−1} + 5 · 1^{−1}

For the algebraic numbers the following notation is used:

Notation 4.2. As usual ζ_{m} denotes a primitive m-th root of unity in ℂ and \sqrt{m} a square root of m. Moreover θ_{m} := ζ_{m} + ζ_{m}^{−1} denotes a generator of the maximal totally real subfield of the m-th cyclotomic field. ω_{m} (resp. η_{m}, σ_{m}) denote generators of a subfield K of ℚ[ζ_{m}] with Gal(K/ℚ) ≅ C_{3} (resp. C_{4}, C_{5}).

The algorithmic problems in evaluating these formulas are:

a) determine the ideals I_{j}.

b) decide whether two maximal orders are conjugate in D.

c) determine the length of the orbit of 𝔐 under the Galois group Gal(K/ℚ).

d) determine ω_{i}^{−1}H_{i}.

Problem a) is the major difficulty here. There is of course the well known geometric approach to this question using the Minkowski bound on the norm of a representative of the ideal classes. From the arithmetic point of view one may apply two different strategies to find the ideals I_{j}:

There is a coarser equivalence relation than conjugacy namely the stable isomorphism cf. [Rei 75, (35.5)]. The theorem of Eichler [Rei 75, (34.9)] says that the reduced norm is an isomorphism of the group of stable isomorphism classes of 𝔐-left ideals onto the narrow class group of the center K. This gives estimates for the norms of the ideals I_{j}.

A second arithmetic strategy is to look for (commutative, nonfull) suborders O of D. The number of the maximal orders 𝔐_{i} containing O as a pure submodule can be calculated using the formula [Vig 80, (5.12)].

Example. Let D := Q_{\sqrt{3}+\sqrt{5},∞}. Then the narrow class group of K = ℚ[\sqrt{3} + \sqrt{5}] has order 2 and is generated by a prime ideal dividing 11. So there are two stable isomorphism classes of 𝔐-ideals one containing the ideal classes of I_{1}, I_{2}, and I_{3}, the other one contains that of of I_{4} (in the notation of Table 4.1). The second strategy applied to O = ℤ[ζ_{5}, \sqrt{3}] gives that there are two orders 𝔐_{i} containing a fifth root of unity, because the class number of O is 2 (and again a prime ideal dividing 11 generates the class group).

The problems b), c), and d) can be dealt with using the normform of D:

Let D be a definite quaternion algebra over K and N be its reduced norm which is a quadratic form with associated bilinear form ⟨x, y⟩ = tr(xy^{∗}) where tr is the reduced trace and ^{∗} the canonical involution of D. The special orthogonal group SO(D, N) := {φ : D → D | N(φ(x)) = N(x) for all x ∈ D, det(φ) = 1} is the group of all proper isometries of D with respect to the quadratic form N. The following proposition is surely well known (cf. [Vig 80, Théorème 3.3]) (cf. also [DuV 64] for a geometric interpretation of the quaternionic conjugation).

Proposition 4.3. With the notation above one has

SO(D, N) = {x ↦ a_{1}xa_{2}^{−1} | a_{i} ∈ D^{∗}, N(a_{1}) = N(a_{2})}

is induced by left multiplication with elements of D of norm 1 and conjugation with elements of D^{∗}.

Proof. Clearly the mapping x ↦ a_{1}xa_{2}^{−1} with a_{i} ∈ D^{∗} and N(a_{1}) = N(a_{2}) is a proper isometry of the K-vector space (D, N).

To see the converse inclusion let D = ⟨1, i, j, ij = k = −ji⟩_{K} with i^{2} = a and j^{2} = b and φ : D → D be an isometry of determinant 1 with respect to N. Then N(φ(1)) = 1 and after left multiplication by φ(1)^{−1} we may assume that φ(1) = 1. Let b_{2} := φ(i), b_{3} := φ(j), and b_{4} := φ(k). Then tr(b_{i}1) = 0 and hence b_{i}^{∗} = −b_{i} for all i = 2, 3, 4, and b_{2}^{2} = a, b_{3}^{2} = b, b_{4}^{2} = −ab. Moreover tr(b_{i}b_{j}^{∗}) = 0 = −tr(b_{i}b_{j}) and hence b_{i}b_{j} = −b_{j}b_{i} for all 2 ≤ i ≠ j ≤ 4. Thus (b_{2}b_{3})b_{4} = b_{4}(b_{2}b_{3}) and therefore b_{4} ∈ Kb_{2}b_{3} is an element of trace 0 in the field generated by b_{2}b_{3}. Since b_{4}^{2} = (b_{2}b_{3})^{2}, this implies that b_{4} = ^{±}b_{2}b_{3}. If b_{4} = b_{2}b_{3}, then φ is an K-algebra automorphism of D and hence induced by conjugation with an element of D^{∗} and we are done. In this case φ is of determinant 1. Hence if b_{4} = −b_{2}b_{3}, the mapping φ has determinant −1, which is a contradiction.

Corollary 4.4. Let 𝔐_{i} (i = 1, 2) be two orders in D. Then 𝔐_{1} is conjugate to 𝔐_{2} if and only if the lattices (𝔐_{1}, N) and (𝔐_{2}, N) are properly isometric.

Proof. Clearly if the two orders are conjugate the lattices are properly isometric, so we show the converse: let φ : 𝔐_{1} → 𝔐_{2} be a proper isometry with respect to N. By the proposition there are elements a_{1}, a_{2} ∈ D^{∗} with N(a_{1}) = N(a_{2}) such that a_{1}𝔐_{1}a_{2}^{−1} = 𝔐_{2}. Since 1 ∈ 𝔐_{1} this implies that a_{1}a_{2}^{−1} is an element of norm 1 in 𝔐_{2} and hence 𝔐_{2} = a_{1}a_{2}^{−1}a_{2}𝔐_{1}a_{2}^{−1} = a_{2}𝔐_{1}a_{2}^{−1} is conjugate to 𝔐_{1}.

Since ^{∗} is the identity on the subspace K and the negative identity on the 3dimensional subspace 1^{⊥} consisting of the elements of D with trace 0, one easily sees that ^{∗} is an improper isometry (of determinant -1) of (D, N). Thus, if one of the orders 𝔐_{1} or 𝔐_{2} is stable under ^{∗}, one may omit the word “properly” in the corollary above. Note that this holds particularly for maximal orders.

Corollary 4.5. Let 𝔐 be an order in D. The group of proper isometries of the lattice (𝔐, N) is induced by the transformations of the form b ↦ axbx^{−1}, where a ∈ 𝔐 is an element of norm 1 and x ∈ N_{D^{∗}}(𝔐) normalizes 𝔐.

By Corollary 4.5 the order of the isometry group |Aut(𝔐_{i}, N)| = ω_{i}^{1} · ω_{i} · 2^{s} · 2 · 2 · H_{i}^{−1}, where s is the number of finite primes of K that ramify in D. Now 2ω_{i}^{1} is simply the number of shortest vectors of the lattice (𝔐_{i}, N) and can easily be calculated. Hence ω_{i}^{−1}H_{i} = |Aut(𝔐_{i}, N)|^{−1} · 2^{s+2} · ω_{i}^{1}.

Corollary 4.6. Let 𝔐 be an order in D and σ ∈ Gal(K/ℚ). Then 𝔐 is conjugate to σ(𝔐), if and only if the R-lattices (𝔐, N) and (𝔐, σ ∘ N) are isometric.

Proof. The corollary follows from Corollary 4.4 and the fact that tr(σ(x)σ(y^{∗})) = σ(tr(xy^{∗})).

5. Notation for the finite matrix groups

The notation for the absolutely irreducible maximal finite (a.i.m.f.) subgroups of GL_{n}(D) is similar to the one for the rational irreducible maximal finite (r.i.m.f.) subgroups of GL_{n}(ℚ).

If D = Q_{α,p_{1},... ,p_{s}}, then the (conjugacy class of an) a.i.m.f. group G ≤ GL_{n}(D) is denoted by _{α,p_{1},... ,p_{s}}[G]_{n}.

The automorphism groups of root lattices are usually denoted by the name of the corresponding root system A_{n}, …, E_{8}, F_{4}.

For the quasisimple groups the notation in [CCNPW 85] is used with the exception that the alternating group of degree m is denoted by Alt_{m}.

A maximal finite matrix group always contains the negative unit matrix. If G is a matrix group, then ^{±}G denotes the group generated by G and the negative unit matrix.

The symbols M_{p+1,i} and A_{p−1}^{(j)} (i, \frac{p+1}{2j} ∈ \{2, 3, 4, 6\} with 2i | p − 1) denote (automorphism groups of) lattices of P SL_{2}(p) as described in Chapter V of [PlN 95].

Let G ≤ GL_{n}(D) and H ≤ GL_{m}(D′) be two irreducible finite matrix groups. Let A be a subalgebra of C_{D^{n×n}}(G) such that A^{op} is isomorphic to a subalgebra of C_{D^{′m×m}}(H), such that D^{n×n} ⊗_{A} D^{′m×m} ≅ D′′^{l×l} is again simple. Then tensoring the natural representations of G and H yields a representation of the direct product G × H. The corresponding matrix group G ⊗_{A} H is a subgroup of GL_{l}(D′) and isomorphic to a central product of G and H. If A ≅ ℚ[α] is a field or A ≅ Q_{α} is a quaternion algebra, the matrix group G ⊗_{A} H is abbreviated as G⊗_{α} H and if A = ℚ as G ⊗ H.

Already the groups C_{5} ≅ G ≤ GL_{1}(Q[ζ_{5}]) and S̃_{3} ≅ H ≤ GL_{1}(Q_{∞,3}) show that this tensor notation needs to be extended. Though ℚ[ζ_{5}] ⊗ Q_{∞,3} ≅ ℚ[ζ_{5}]^{2×2} the maximal common subalgebra of the two algebras ℚ[ζ_{5}] and Q_{∞,3} is ℚ. We use the symbol G ⊗\sqrt{5}′ H to denote the corresponding subgroup of GL_{2}(ℚ[ζ_{5}]).

To describe quite frequently occurring extensions of tensor products of matrix groups of index 2 as in Theorem 7.11, we use the notation introduced in [PlN 95,

Proposition (II.4)]: The symbols C ⊗_{α}^{2(p)} N, C ⊃ × ⊏_{α}^{2(p)} N, and C ⊠_{α}^{2(p)} N denote primitive

matrix groups G that are extensions of the tensor product of the two matrix groups N and C by an automorphism x with x^{2} ∈ C⊗_{α} N. Since N (as well as C = C_{G}(N)) is a normal subgroup of G, one may write the elements of G as tensor products as

in [CuR 81, Theorem (11.17)]. Let x = y ⊗ z. In the first case G = C ⊗_{α}^{2(p)} N, y ∈ C̄,

z ∈ N̄, Ḡ = C̄⊗_{α} N̄ and p ∈ Z(Ḡ) is the norm of y which is also the norm of z (cf.

Definition 7.10). If G = C ⊃ × ⊏_{α}^{2(p)} N, then y ∉ C̄ but still z ∈ N̄ and p is the norm of

z. In the last case x induces nontrivial automorphisms on both centers Z(N̄) and Z(C̄).

If p = 1, it is omitted. Also the symbol × and α is omitted if either N or C is contained in A.

Remark 5.1. As the referee pointed out, one should like to compare the classification of maximal finite subgroups of GL_{n}(D) with Aschbacher’s classification of subgroups of the finite classical groups in [Asc 84].

In Aschbacher’s classification the groups in C_{1} (reducible groups), C_{2} (imprimitive groups), C_{3}, and C_{5} reduce to smaller situations, the same is true here. But cases 3 and 5 are harder to deal with, since Galois groups of abelian extension fields are not necessarily cyclic. One may not always extend the cocylces to overgroups, as one sees from the maximal finite subgroups [D_{120}.2]_{16,i} i = 1, 2 of GL_{16}(ℚ).

The types C_{6} and C_{8} (extraspecial resp. simple groups) are dealt with in Chapter 8 and 9 of this paper.

The main difficulties are the tensor products (types C_{4} and C_{7} of Aschbacher’s classification). These difficulties lead to Chapter 10, where a first approach is made to classify the possible tensor factors. These factors are not necessarily maximal finite subgroups of the unit group of a smaller algebra as the following example shows.

Consider the symmetry group D_{8} of a square. This is an imprimitive maximal finite subgroup of GL_{2}(ℚ). It admits an outer automorphism α ∈ N_{GL_{2}(ℚ)}(D_{8}) satisfying α^{2} = 2I_{2}. Similarly the matrix group L_{2}(7) ≤ Aut(A_{6}) admits an additional outer automorphism β ∈ N_{GL_{6}(ℚ)}(L_{2}(7)) with β^{2} = 2I_{6}. Hence α ⊗ β also

normalizes the tensor product D_{8} ⊗ L_{2}(7) ≤ GL_{12}(ℚ) The group D_{8} ⊗^{2(2)} L_{2}(7) = ⟨D_{8} ⊗ L_{2}(7), \frac{1}{2}α ⊗ β⟩ is a maximal finite subgroup of GL_{12}(ℚ) though the group ±L_{2}(7) is not maximal finite in GL_{6}(ℚ).

The general phenomenon may be described by the groups Glide(N) of gliding automorphisms as defined in Definition 7.3.

Example 5.2. The quaternionic reflection groups of Table III in [Coh 80] that are a.i.m.f. groups are

\begin{aligned}O_{2} & = _{∞,3}[SL_{2}(9)]_{2}, O_{3} = _{\sqrt{3},∞,∞}[2.S_{6}]_{2}, \\ P_{2} & = _{∞,2}[(D_{8} ⊗ Q_{8}).Alt_{5}]_{2}, P_{3} = _{\sqrt{2},∞,∞}[(D_{8} ⊗ Q_{8}).S_{5}]_{2}, \\ Q & = _{∞,3}[^{±}U_{3}(3)]_{3}, R = _{\sqrt{5},∞,∞}[2.J_{2}]_{3}, \\ S_{3} & = _{∞,2}[2_{−}^{1+6}.O_{6}^{−}(2)]_{4}, \\ T & = _{\sqrt{5},∞}[(SL_{2}(5) ∘ SL_{2}(5)⊗ \sqrt{5} SL_{2}(5)) : S_{3}]_{4}, and U = _{∞,2}[^{±}U_{5}(2)]_{5}. \end{aligned}

6. The a.i.m.f. subgroups of GL_{1}(Q)

Let Q be a totally definite quaternion algebra over its center K and assume that K is a (totally real) number field. Let G ≤ GL_{1}(Q) be a finite subgroup such that \overline{ℚG} = Q. The classification of finite subgroups of P GL_{2}(ℂ) in [Bli 17] shows that G/Z(G) is either abelian, a dihedral group or one of the 3 exceptional groups Alt_{3}, S_{4} or Alt_{5}. Using this classification one gets the following:

Theorem 6.1. Let G ≤ GL_{1}(Q) be an a.i.m.f. subgroup of GL_{1}(Q), where Q is a totally definite quaternion algebra over a totally real number field K = Z(Q).

Then K is the maximal totally real subfield of a cyclotomic field.

If [K : ℚ] ≤ 2, then G is one of _{∞,2}[SL_{2}(3)]_{1}, _{∞,3}[S̃_{3}]_{1}, _{\sqrt{2},∞}[S̃_{4}]_{1}, _{\sqrt{3},∞}[Q_{24}]_{1}, or _{\sqrt{5},∞}[SL_{2}(5)]_{1}.

If [K : ℚ] ≥ 3, let m be even such that K = ℚ[θ_{m}] ≤ ℚ[ζ_{m}]. Then G = Q_{2m} = C_{m}.C_{2} ≤ GL_{1}(Q) is a generalized quaternion group, a nonsplit extension of a cyclic group of order m by a C_{2}. If \frac{m}{2} = p^{α} is a power of the prime p ≡ 3 (mod 4), then Q = Q_{θ_{m},∞,p} is ramified at the place over p. In all other cases, the quaternion algebra Q is only ramified at the infinite places of K.

Proof. The possible groups G may immediately be obtained from [Bli 17]. So we only have to compute the local Schur indices of the groups Q_{2m}. To this purpose let p be a prime and 2 < r be a divisor of m. Then the restriction χ′ of the natural character χ of Q_{2m} to the subgroup Q_{4r} remains irreducible. By the theorem of Brauer [Bra 51] (cf. also [Lor 71]) the Schur index of χ and the one of χ′ over ℚ_{p}[θ_{m}] are equal. If r is a prime such that p does not divide 2r, then by Lemma 2.13 the p-adic Schur index of χ′ is 1. This is also true if p = 2 and r are odd, since then the Sylow 2-subgroup of the cyclic subgroup of index 2 in Q_{4r} is C_{2} ([Lor 71, p. 98]). So if \frac{m}{2} is not a prime power, the quaternion algebra Q is not ramified at any finite prime.

If \frac{m}{2} is a power of some prime l ≡ 1 (mod 4), then ℚ_{l} contains a fourth root of unity. Hence the l-adic Schur index of χ′ is 1. (This follows also from the parity of the number of ramified primes, since [ℚ[θ_{l}] : ℚ] is even.) If m is a power of 2, then m ≥ 16. Since [ℚ_{2}[θ_{m}] : ℚ_{2}] is even, ℚ_{2}[θ_{m}] splits ℚ_{2} ⊗ Q_{∞,2}. Choosing r = 4 in the consideration above, yields that the 2-adic Schur index of χ′ = χ_{|Q_{8}} over ℚ_{2}[θ_{m}] is 1. If \frac{m}{2} = p^{α} is a power of a prime p ≡ 3 (mod 4). Choosing r = p, the Schur index of χ′ over ℚ_{p}, hence also the one over the character field ℚ_{p}[χ′] = ℚ_{p}[θ_{p}] is 2. But now [ℚ_{p}[θ_{m}] : ℚ_{p}[θ_{p}]] = p^{α−1} is odd, hence the character field ℚ_{p}[χ] does not split the quaternion algebra ℚ_{p} ⊗Q_{θ_{p},∞,p}. Therefore Q is ramified at the place over p, which again also follows from the fact that [ℚ[θ_{m}] : ℚ] is odd in this case.

Remark 6.2. Let Q be an indefinite quaternion algebra with totally real center K. If GL_{1}(Q) has an absolutely irreducible finite subgroup G, then Q = K^{2×2}.

Table 6.3. The a.i.m.f. subgroups of GL_{1}(Q), where Q is a definite quaternion algebra such that [Z(Q) : ℚ] ≤ 5:

lattice L  |Aut(L)| r.i.m.f. supergroups
∞,2[SL_{2}(3)]_{1}  2^{3} · 3 F_{4}
∞,3[S̃_{3}]_{1}  2^{2} · 3 A_{2}^{2}
\sqrt{2},∞[S̃_{4}]_{1}  2^{4} · 3 E_{8}, F_{4}^{2}
\sqrt{3},∞[Q_{24}]_{1}  2^{3} · 3 A_{2}^{4}, F_{4}^{2}
\sqrt{5},∞[SL_{2}(5)]_{1} 2^{3} · 3 · 5 E_{8}, [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8}
θ_{15},∞[Q_{60}]_{1}  2^{2} · 3 · 5 (A_{2} ⊗ A_{4})^{2}, [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8}^{2}, E_{8}^{2}

A_{2} ⊗ [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8}, A_{2} ⊗ E_{8},

[SL_{2}(5) ∘ SL_{2}(5) : 2 ⊠_{\sqrt{5}}^{2} D_{10}]_{16}

θ_{20},∞[Q_{40}]_{1}  2^{3} · 5 A_{4}^{4}, [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8}^{2}, E_{8}^{2}

[SL_{2}(5) ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+4}.Alt_{5}]_{16}

[(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8}^{2}, E_{8}^{2},

[SL_{2}(5) ∘ SL_{2}(5) : 2 ⊠_{\sqrt{5}}^{2} D_{10}]_{16}

θ_{24},∞[Q_{48}]_{1}  2^{4} · 3 A_{2}^{8}, E_{8}^{2}, [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8}^{2},

E_{8}^{2}, [SL_{2}(5) ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+4}.Alt_{5}]_{16}

[L_{2}(11) ⊗^{2(3)} D_{12}]_{20} [L_{2}(11) ⊃ × ⊏^{2(3)}D_{12}]_{20}

A_{2} ⊗ F_{4}, E_{8} θ_{7},∞,7[Q_{28}]_{1}  2^{2} · 7 (A_{6})^{2}, (A_{6}^{(2)})^{2} θ_{9},∞,3[Q_{36}]_{1}  2^{2} · 3^{2} A_{2}^{6}, E_{6}^{2} θ_{16},∞[Q_{32}]_{1}  2^{5}  (B_{16})

F_{4}^{4}, E_{8}^{2} A_{4} ⊗ F_{4}, F_{4} ˜⊗F_{4}, (A_{2} ⊗ F_{4})^{2}, F_{4}^{4} (F_{4}^{4})

(A_{2} ⊗ F_{4})^{2}, F_{4} ˜⊗F_{4}, A_{2} ⊗ E_{8}, θ_{11},∞,11[Q_{44}]_{1} 2^{2} · 11 (A_{10})^{2}, (A_{10}^{(2)})^{2}, (A_{10}^{(3)})^{2}

Proof of Remark 6.2. Let G ≤ GL_{1}(Q) be a finite absolutely irreducible subgroup and p be a finite prime ramified in Q. As in the proof of Theorem 6.1 one concludes that G = ^{±}C_{p^{α}}.C_{2}, K = ℚ[θ_{p^{α}}] and the finite primes of K ramified in Q divide p. Since all infinite primes of K are not ramified in Q and p is totally ramified in K, this contradicts the fact that the number of primes in K that ramify in Q is even.

If Q is a definite quaternion algebra and the degree of the center of Q over ℚ is ≤ 5, the a.i.m.f. subgroups of GL_{1}(Q) and their r.i.m.f. supergroups are given in Table 6.3. The first column gives a name for the a.i.m.f. group G also indicating the quaternion algebra Q. This entry is followed by the order of G. In the last column the r.i.m.f. supergroups fixing a G-lattice with maximal order as endomorphism ring are given. If there is no such group, at least one r.i.m.f. supergroup of G is specified in brackets. If there is more than one conjugacy class of maximal orders 𝔐 in Q, they are listed in the next lines separated by lines in the third column in the same order as they are displayed in Table 4.1.

7. Normal subgroups of primitive groups

Throughout this chapter let K denote an abelian number field with complex conjugation and R the maximal order in K. Then R is a Dedekind ring.

Let N ⊴G be a normal subgroup of the primitive group G in GL_{n}(K). As proved in Chapter 2 (cf. 2.3) the enveloping K-algebra of N is a simple algebra.

Generalizing the notion of a generalized Bravais group (as defined in Definition (II.9) of [NeP 95] for K = ℚ) to arbitrary number fields K, one may often conclude the existence of a larger normal subgroup B_{K}^{∘}(N) in G if G is maximal finite.

For this purpose recall the radical idealizer process (cf. [BeZ 85]): Let Λ be an R-order in a simple K-algebra A. The arithmetic (right) radical AR_{r}(Λ) of Λ is defined as the intersection of all those maximal right ideals of Λ which contain the discriminant ideal of Λ. The arithmetic radical is a full R-module in A. Its (right) idealizer Id_{r}(AR_{r}(Λ)), which is defined as the set of all elements a ∈ A, such that AR_{r}(Λ)a ⊆ AR_{r}(Λ), is again an R-order in A containing Λ. The repeated application of (Id_{r} ∘ AR_{r}) is called the radical idealizer process. It constructs a finite ascending chain of R-orders in A. The maximal element of this chain (Id_{r} ∘ AR_{r})^{∞}(Λ) is necessarily a hereditary order in A (cf. [Rei 75] pp. 356-358).

Definition 7.1. Let N ≤ GL_{n}(K) be a finite group and F an N invariant Hermitian form on K^{n}. Assume that the algebra A := \overline{KN} is simple. Then the natural A-module K^{n} decomposes into a direct sum of l copies of an irreducible A-module V. Let Λ := \overline{RN}, be the R-order generated by the matrices in N and Λ_{0} := (Id_{r} ∘ AR_{r})^{∞}(Λ) be the hereditary order in A obtained by applying the radical idealizer process to the R-order Λ. Let L_{1}, …, L_{s} ⊆ V be representatives of the isomorphism classes of the irreducible Λ_{0}-lattices in V. Then the generalized Bravais group of N is defined as

B_{K}^{∘}(N) := \{g ∈ \overline{KN} | L_{i}g = L_{i}, 1 ≤ i ≤ s, gF ḡ^{t} = F \}.

If K = ℚ, the group B_{ℚ}^{∘}(N) is also denoted by B^{∘}(N).

Note that the definition of B_{K}^{∘}(N) does not depend on the choice of F ∈ F_{>0}(N), since the elements in \overline{KN} commute with all F′F^{−1} for F′ ∈ F(N).

As for K = ℚ in [NeP 95, Proposition (II.10)] one proves:

Proposition 7.2. Let k be a subfield of K.

(i) If X is a finite subgroup of the unit group (\overline{kN})^{∗} of \overline{kN} with N ⊴ X, then X ≤ B_{k}^{∘}(N).

(ii) If D is a central K-division algebra of index s and G is a primitive a.i.m.f. subgroup of GL_{n}(D) ≤ GL_{s^{2}n}(K) with N ⊴ G, then N ⊴ B_{k}^{∘}(N) =: B ⊴ G. In particular B = G∩\overline{kN} is the unique maximal finite subgroup of the normalizer of N in (\overline{kN})^{∗}. Moreover the centralizer of N in G is C_{G}(B) = C_{G}(N).

Example. The generalization of the definition to arbitrary number fields K provides stronger restrictions on the possible normal subgroups of G. For example, for K := ℚ[\sqrt{2}] and N = Q_{8} one has B_{K}^{∘}(N) = S̃_{4} whereas B_{ℚ}^{∘}(N) = SL_{2}(3).

In the situation of Proposition 7.2 (i), the a.i.m.f. group G contains the normal subgroup NC_{G}(N). The quotient group G/NC_{G}(N) embeds into the outer automorphism group Out(N) of N. The image of G/C_{G}(N) in Aut(N) contains the group of automorphisms that are induced by B_{K}^{∘}(N) and is contained in the subgroup Aut_{stab}(N) of Aut(N) consisting of those automorphisms of N, that stabilize the irreducible constituent χ of the natural character of N.

Definition 7.3. Let N ≤ GL_{n}(K) be a finite subgroup of GL_{n}(K), such that the enveloping algebra \overline{KN} is a simple K-algebra. Let χ be an absolutely irreducible character occurring in the natural character of N.

(i) The automorphism group Aut(N) acts on the set of irreducible characters of N. Aut_{stab}(N) := Stab_{Aut(N)}(χ) is called the group of stable automorphisms of the matrix group N.

(ii) N is called primitively saturated over K, if N ⊴ B_{K}^{∘}(N) and all stable auto-

morphisms of N are induced by conjugation with elements of B_{K}^{∘}(N). (iii) The factor group Glide(N, χ, K) := Glide_{K}(N) := Aut_{stab}(N)/κ(B_{K}^{∘}(N)) of

Aut_{stab}(N) modulo the group of automorphisms induced by conjugation with elements of B_{K}^{∘}(N) is called the group of gliding automorphisms of the matrix group N. We set Glide(N) := Glide_{ℚ}(N).

Remark 7.4. Let N, A, Λ_{0} be as in Definition 7.1. Since the elements of Aut_{stab}(N) define Z(A)-algebra automorphisms of A, the theorem of Skolem and Noether says that there are elements in A^{∗} inducing these automorphisms on A. Since these elements normalize N, they also normalize Λ_{0} and therefore act as inclusion preserving bijections on the set of Λ_{0}-lattices. The subgroup induced by conjugation with elements in B_{K}^{∘}(N) is the kernel of this action and therefore a normal subgroup of Aut_{stab}(N). Hence Glide_{K}(N) is well defined.

For a maximal ideal ℘ of R let A_{℘} and Λ_{℘} denote the completions of A and Λ_{0} at ℘. Since Λ_{℘} is hereditary the Λ_{℘}-lattices in a simple A_{℘}-module are linearly ordered by inclusion. Since B_{K}^{∘}(N) stabilizes all Λ_{℘}-lattices one gets an action of Glide_{K}(N) on this chain of Λ_{℘}-lattices (shifting up or down).

Proposition 7.5. In the situation of Proposition 7.2 (ii), assume that N is primitively saturated over K and that the center of \overline{KN} is K. If B := B_{K}^{∘}(N) and C := C_{G}(N), then G = BC.

Proof. Since G is primitive, the automorphisms of N that are induced by g ∈ G stabilize χ. Hence there is a b ∈ B such that bg ∈ C.

The automorphism groups of the indecomposable root lattices A_{n} (n ≥ 4), E_{6}, E_{7}, and E_{8} provide examples for primitively saturated groups.

Corollary 7.6. Let D be a ℚ-division algebra and G a primitive irreducible maximal finite subgroup of GL_{n}(D). Assume that G contains a normal subgroup N isomorphic to either Alt_{n} (n ≥ 5), U_{4}(2) = Aut(E_{6})′, S_{6}(2) = Aut(E_{7})′, or 2.O_{8}^{+}(2) = Aut(E_{8})′ (where the corresponding irreducible constituent χ of the natural character of N is of degree n−1, 6, 7, respectively 8). Then G = B^{∘}(N)⊗C_{G}(N).

Proof. In all cases Aut_{stab}(N) is already induced by conjugation with elements of B^{∘}(N).

Corollary 7.7. Let Q be a definite quaternion algebra with center K and G a primitive a.i.m.f. subgroup of GL_{n}(Q). Then G has no normal subgroup N isomorphic to M_{11}, 2.M_{12}, or 2.M_{22} where the restriction of the natural character of G to N is a multiple of the sum of the two Galois conjugate complex characters of degree 10.

Proof. Since the whole outer automorphism group of N is already induced by conjugation with elements in B^{∘}(N), the group G is of the form G = B^{∘}(N)C_{G}(N). In particular the character field of the natural character of G is complex. Therefore G is not an absolutely irreducible subgroup of GL_{n}(Q).

The following theorem is a version of a well known theorem of Clifford (cf. [CuR 81, Theorem (11.20)]), which is usually only formulated for algebraically closed fields.

Theorem 7.8. Let G ≤ GL_{n}(K) be a finite group, N ⊴ G a normal subgroup such that the enveloping K-algebra A := \overline{KN} is central simple. Let C := C\overline{KG}(A) be the commuting algebra of A in \overline{KG}. Then the natural representation Δ : G → GL_{n}(K) is a tensor product Δ = Δ_{1} ⊗ Δ_{2} of projective representations Δ_{1} : G → A^{∗} and Δ_{2} : G → C^{∗}.

Proof. Let g ∈ G. Since N ⊴ G, conjugation with g induces a K-algebra automorphism of A. By the theorem of Skolem and Noether, there is an a ∈ A^{∗}, such that ag =: b ∈ C. Hence g = a ⊗ b ∈ A ⊗ C = \overline{KG}.

In the situation of Theorem 7.8, Δ_{1}(G) is a (not necessarily finite) subgroup of the normalizer of N in the unit group of its enveloping algebra N_{A^{∗}}(N). If one additionally assumes that G is (primitive and) maximal finite and chooses Δ_{1} and Δ_{2} appropriately, then B := B_{K}^{∘}(N) = Ker(Δ_{2}) is the unique maximal finite subgroup of N_{A^{∗}}(N).

Lemma 7.9. Let N ≤ GL_{n}(K) be a finite matrix group such that the algebra \overline{KN} is simple with center Z. Let ¯ also denote the complex conjugation on the abelian number field Z and let Z^{+} be the maximal totally real subfield of Z. Let α ∈ Glide_{K}(N). Then there is a ∈ \overline{KN}^{∗} such that a representative of α is induced by conjugation with a. Moreover there is q ∈ Z^{+} such that aF ā^{tr} = qF for all F ∈ F(N). The element q is a totally positive element of Z^{+} unique up to multiplication with elements of the group {zz̄ | z ∈ Z}.

Proof. Let F ∈ F_{>0}(N) be an N-invariant K-Hermitian form. Then the matrix aF ā^{t} is again N-invariant, because a normalizes N. Hence there is a q ∈ C := C_{K^{n×n}}(N), such that aF ā^{t} = qF. Every element x ∈ C may be written as a sum of a symmetric and a skew symmetric element (with respect to F), i.e. x = x^{+} + x^{−} with x^{+}, x^{−} ∈ C and x^{+}F = F \overline{(x^{+})}^{t} and x^{−}F = −F \overline{(x^{−})}^{t}. Then clearly q = aF ā^{t}F^{−1} is symmetric. Since a commutes with x^{+} and x^{−} one has x^{+}q = ax^{+}F ā^{t}F^{−1} = qx^{+} and x^{−}q = qx^{−}. Hence q ∈ Z lies in the center of C. If F′ ∈ F(N) is another N-invariant Hermitian form, then there is c ∈ C such that F′ = cF. Then aF′ā^{t} = acF ā^{t} = caF ā^{t} = cqF = qF′. Since a is unique up to multiplication with elements of Z and F z^{t}F^{−1} = z̄ for all z ∈ Z, q is unique up to norms (resp. up to squares if Z = Z^{+}) of elements in Z. Moreover if F is totally positive definite, then aF ā^{t} = qF is also totally positive definite, whence q is totally positive.

Definition 7.10. The element q in the lemma above is called the norm of α.

If A is a central simple algebra over a totally real field K, then Δ_{1}(G)/(K^{∗}B_{K}^{∘}(N)) is of exponent ≤ 2, as shown in the next theorem. This is somehow an explanation for the fact that the constructions given in Proposition (II.4) of [PlN 95] suffice to describe all r.i.m.f. groups in dimension ≤ 31.

Theorem 7.11. Let K be a real abelian number field and N ≤ GL_{n}(K) a finite matrix group such that the enveloping algebra \overline{KN} is simple with center K. Assume that N ⊴ B_{K}^{∘}(N) =: B. Then Glide_{K}(N) is of exponent 1 or 2.

Proof. Let α ∈ Aut_{stab}(N). Since \overline{KN} is central simple, there is an a ∈ (\overline{KN})^{∗}, such that α is induced by conjugation with a. Let F be an N-invariant K-quadratic form. By Lemma 7.9 there is q ∈ K = Z(\overline{KN}) such that aF a^{t} = qF. Therefore a^{2}q^{−1}F (a^{2}q^{−1})^{t} = F. Since the automorphism α has finite order, there is an m ∈ ℕ such that (a^{2}q^{−1})^{m} ∈ Z(\overline{KN}) = K. One calculates F = (a^{2}q^{−1})^{m}F ((a^{2}q^{−1})^{m})^{t} = (a^{2}q^{−1})^{2m}F. Hence (a^{2}q^{−1})^{2m} = 1 and (a^{2}q^{−1}) is an element of finite order in (\overline{KN})^{∗} normalizing N. By Proposition 7.2 a^{2}q^{−1} ∈ B.

If Glide_{K}(N) is of order 2 and \overline{KN} is a central simple K-algebra, the primitive a.i.m.f. groups G with normal subgroup N contain a subgroup of index 1 or 2 which is a tensor product B ⊗ C_{G}(N). If α ∈ Aut_{stab}(N) − κ(B) and q are as in the proof above, we call N nearly tensor decomposing with parameter q.

Note that Theorem 7.11 is false if one omits the assumption that K is real. One counterexample is provided by the faithful character of degree 144 of the group 3.U_{3}(5) (cf. [CCNPW 85]). The smallest counterexample I know is N ≅ C_{3} × (C_{7} : C_{3}). Let N := ⟨z, x, y | z^{3}, x^{7}, y^{3}, x^{y} = x^{2}⟩. Then N has an automorphism s of order 3, with z^{s} = z, x^{s} = x, y^{s} = yz. N has a faithful representation into GL_{3}(K) where K := ℚ[\sqrt{−3}, \sqrt{−7}]. The corresponding character χ extends to ±N : ⟨s⟩ but the character value of xs involves further irrationalities. So the order of Glide_{K}(N) is divisible by 3.

Corollary 7.12. With the notation of the proof of Theorem 7.11, the element α ∈ Glide_{K}(N) is uniquely determined by the class of q in K^{∗}/(K^{∗})^{2}.

Proof. Let α, β ∈ Aut_{stab}(N) be induced by conjugation with a resp. b ∈ (\overline{KN})^{∗} such that aF a^{t} = qF and bF b^{t} = r^{2}qF, with q, r ∈ K^{∗}. Replacing b by br^{−1} we assume that r = 1. Then ab^{−1}F (ab^{−1})^{t} = F. As in the proof above, the matrix ab^{−1} ∈ \overline{KN} is an element of finite order normalizing N and hence ab^{−1} ∈ B_{K}^{∘}(N).

8. The normal p-subgroups of primitive groups and their automorphism groups

In this chapter we calculate the generalized Bravais groups and outer automorphism groups of the relevant p-groups N which are candidates for normal psubgroups of a primitive a.i.m.f. group G. Since all abelian characteristic subgroups of N are cyclic (Corollary 2.4), these groups are classified by P. Hall:

Theorem 8.1 (cf. [Hup 67], p. 357). Let N be a p-group, such that all abelian characteristic subgroups of N are cyclic.

If p > 2, then N is a central product of a cyclic group and an extraspecial group of exponent p.

If p = 2, then N is a central product of an extraspecial 2-group with a cyclic dihedral, generalized quaternion, or quasidihedral 2-group.

If K is an abelian number field, then all of these groups have a (up to automorphism) unique K-irreducible faithful representation. The corresponding matrix group is called an admissible p-group over K.

The automorphism groups of the extraspecial groups are well known (cf. [Win 72]). For these groups one finds:

Proposition 8.2. Let n ∈ ℕ, p be a prime, and N = p_{+}^{1+2n} or N = 2_{−}^{1+2(n−1)} if p = 2. If N ⊴ B^{∘}(N) =: B, then B is as follows:

If p > 2, then B = ^{±}p_{+}^{1+2n}.Sp_{2n}(p).

If N = 2_{+}^{1+2n}, then B = 2_{+}^{1+2n}.O_{2n}^{+}(2).

If N = 2_{−}^{1+2(n−1)}, then B = 2_{−}^{1+2(n−1)}.O_{2(n−1)}^{−}(2).

Proof. Case p > 2: By [Win 72] the subgroup of the outer automorphism group of the extraspecial p-group p_{+}^{1+2n} = N of exponent p which centralizes the center C_{p} of N is the symplectic group Sp_{2n}(p). Hence if N ⊴ B, then B ≤ ^{±}N.Sp_{2n}(p). In [Wal 62], Wall constructs a lattice of dimension (p − 1) · p^{n} on which ^{±}N.Sp_{2n}(p) acts. Therefore ^{±}N.Sp_{2n}(p) ≤ B.

Case p = 2: If p = 2, the proposition can be checked directly for n ≤ 3. So assume n ≥ 4. Let ϵ be + or −. Then by [Win 72] the outer automorphism group of 2_{ϵ}^{1+2n} is the orthogonal group SO_{2n}^{ϵ}(2) = GO_{2n}^{ϵ}(2). It contains a subgroup O_{2n}^{ϵ}(2) of index 2 (cf. [CCNPW 85, p. xii]).

By [Wal 62] the group B_{1} := 2_{+}^{1+2n}.O_{2n}^{+}(2) is the full automorphism group of a lattice of dimension 2^{n}. Now N = D_{8} ⊗ … ⊗ D_{8} is conjugate in GL_{2^{n}}(ℚ) to the tensor product of n copies of D_{8}. If D_{8} ∈ GL_{2}(ℚ) is given in the monomial representation, then α := ( \begin{matrix}1 & 1 \\ 1 & −1 \end{matrix} ) normalizes D_{8} and satisfies α^{2} = 2I_{2}. Hence the element a := α ⊗ I_{2} ⊗ … ⊗ I_{2} normalizes N and hence B. Since a^{2} = 2 is not a square in ℚ^{∗}, there is no element of finite order in GL_{2^{n}}(ℚ) inducing the same automorphism on N as a. Therefore ⟨B_{1}, a⟩ ≅ N.SO_{2n}^{+}(2) is the full holomorph of N and B_{1} = B the unique maximal finite subgroup of the normalizer N_{GL_{2^{n}} (ℚ)}(N).

If ϵ = −, then N = 2_{−}^{1+2(n−1)} = D_{8} ⊗ … ⊗ D_{8} ⊗ Q_{8} is the centralizer of a subgroup Q_{8} ≤ 2_{+}^{1+2n}. One finds a subgroup of index 2 in the holomorph of N as a centralizer of the subgroup Q_{8} in 2_{+}^{1+2n}.O_{2n}^{+}(2). Since a normalizes N and lies in the enveloping ℚ-algebra of N the proposition follows.

Lemma 8.3. Let N := 2_{+}^{1+2n}YC_{4} ≅ 2_{−}^{1+2n}YC_{4}. Then the outer automorphism group Out(N) is isomorphic to O_{2n+1}(2) × C_{2}.

Proof. The mapping q : N/N′ → N′, xN′ ↦ x^{2} is a well defined nondegenerate quadratic form on N/N′ ≅ 𝔽_{2}^{2n+1}. The inner automorphisms induce the identity on N/N′ and q is Out(N)-invariant. Since every isometry of (N/N′, q) can be extended to an automorphism of N, one gets an epimorphism Aut(N) → Out(N) → O_{2n+1}(2). The kernel H consists of all automorphisms of N inducing the identity on N/N′. Now N = Q_{8}Y … YQ_{8}YC_{4} = ⟨A_{1}, B_{1}⟩Y … Y⟨A_{n}, B_{n}⟩Y⟨A⟩ where ⟨A_{i}, B_{i}⟩ ≅ Q_{8} and A_{i} and B_{i} commute with A_{j} and B_{j} for i ≠ j. Let α ∈ H. If α(A_{i}) = A^{2}A_{i} for some i, we multiply α with the inner automorphism κ_{B_{i}} induced by conjugation with B_{i} to achieve α(A_{i}) = A_{i} for all i. Analogously for B_{i}. After this α is either the identity or α = α_{0} where α_{0}(A_{i}) = A_{i}, α_{0}(B_{i}) = B_{i} for all i, and α_{0}(A) = A^{3}. Hence H/Inn(N) = (⟨α_{0}⟩Inn(N))/Inn(N) ⊴ Out(N) is a normal subgroup of order 2 of Out(N). Since O_{2n+1}(2) ≅ Aut(N)/H ≅ C_{Out(N)}(Z(N)) ⊴ Out(N) one has Out(N) ≅ O_{2n+1}(2) × ⟨α_{0}⟩.

Corollary 8.4. Let m, n ∈ ℕ, m > 1, p be a prime, and N = p_{+}^{1+2n}YC_{p^{m}}. If N ⊴ B^{∘}(N) =: B, then B = ^{±}N.Sp_{2n}(p).

Moreover Out(N) = Sp_{2n}(p) × Aut(C_{p^{m}}).

Proof. If p > 2, then C_{p^{m}} = Z(N) and p_{+}^{1+2n} = Ω_{1}(N) are characteristic subgroups of N. Since the elements in B centralize the center of N, the first statement follows from Proposition 8.2. For p = 2, the groups C_{2^{m}} = Z(N) and V := 2^{1+2n}YC_{4} = Ω_{2}(N) are characteristic subgroups of N. The holomorph of V can be constructed as the centralizer of an element of order 4 in 2_{+}^{1+2(n+1)} in 2_{+}^{1+2(n+1)}.O_{2(n+1)}^{+}(2). Hence by Lemma 8.3 the subgroup of the automorphism group of N centralizing the center of N is induced by B. Since O_{2n+1}(2) ≅ Sp_{2n}(2) (cf. e.g. [Tay 92, Theorem 11.9]), the first statement follows.

The outer automorphism group Out(N) contains a normal subgroup κ(B) = C_{Out(N)}(Z(N)) ≅ B/^{±}N, the image of B in Out(N). The automorphisms of Z(N) may be extended to outer automorphisms of N, hence Out(N)/κ(B) ≅ Aut(Z(N)) is isomorphic to the automorphism group of C_{p^{m}} = Z(N). The kernel of the epimorphism Out(N) → Sp_{2n}(p) constructed above is a normal complement of κ(B) in Out(N) which shows that Out(N) = κ(B) × Aut(C_{p}^{m}).

Lemma 8.5. Let m > 3. The outer automorphism groups of the dihedral, quasidihedral, or generalized quaternion groups U are:

Out(D_{2^{m}}) ≅ Out(Q_{2^{m}}) ≅ C_{2} × C_{2^{m−2}} and Out(QD_{2^{m}}) ≅ C_{2^{m−2}}.

Proof. In all three cases U has a unique subgroup V isomorphic to C_{2^{m−1}} of index 2 which is therefore characteristic in U. U/V induces a subgroup of order 2 of the automorphism group of V. Since Aut(V) is abelian Out(U) has an epimorphic image C_{2^{m−2}} with kernel H consisting of those outer automorphisms that induce the identity on V modulo inner automorphisms of U.

Let D_{2^{m}} = ⟨x, y | x^{2^{m−1}}, y^{2}, (xy)^{2}⟩. The elements of order 2 in D_{2^{m}} − V are x^{i}y with 1 ≤ i ≤ 2^{m−1}. Since y^{x} = x^{−2}y these form 2 orbits under the group of inner automorphisms of D_{2^{m}}. Via the absolutely irreducible faithful representation of degree 2 the group D_{2^{m}} can be viewed as a subset G of the algebra ℚ[θ_{2^{m−1}}]^{2×2}. If x, y denote the corresponding elements of G, the element (1 − x) normalizes G since (1 − x)^{−1}y(1 − x) = (1 − x^{−1})(1 − x)^{−1}y = −x^{2^{m−1}−1}y. Hence the outer automorphism induced by (1 − x) generates H.

Analogous considerations hold for the generalized quaternion group Q_{2^{m}}.

The elements of the group QD_{2^{m}} = ⟨x, y | x^{2^{m−1}}, y^{2}, x^{y} = x^{2^{m−2}−1}⟩ that are not in V again form two conjugacy classes. But now one of them consists of elements of order 4, the other one of elements of order 2. Hence here H = 1.

Corollary 8.6. Let m, n ∈ ℕ, m > 3, and U be one of D_{2^{m}}, Q_{2^{m}}, or QD_{2^{m}}. If N := 2_{+}^{1+2n}YU is a normal subgroup of an a.i.m.f. matrix group G, then B := B^{∘}(N) = N.O_{2n+1}(2) and Out(N) is isomorphic to O_{2n+1}(2) × C_{2} × C_{2^{m−2}} if U = D_{2^{m}} or Q_{2^{m}}, and Out(N) ≅ O_{2n+1}(2) × C_{2^{m−2}} if U = QD_{2^{m}}.

Proof. Since m > 3, U has a unique subgroup V isomorphic to C_{2^{m−1}} of index 2. V is the center of the subgroup 2_{+}^{1+2n}YV =: W of N generated by the elements of order 2^{m−1}. Therefore V and W are characteristic subgroups of N and hence normal in G. Thus with Corollary 8.4 W.O_{2n+1}(2) ≅ B^{∘}(W) = G∩\overline{ℚW} is a normal subgroup of G (and therefore also of B). In particular, these automorphisms do extend to automorphisms of N.

If g ∈ N − W, then g induces the Galois automorphism of the center of \overline{ℚW} over the center of \overline{ℚN}. Hence N_{(\overline{ℚN})^{∗}}(W) = ⟨N_{(\overline{ℚW})^{∗}}(W), g⟩ with the theorem of Skolem and Noether. Hence B = N.O_{2n+1}(2) by Corollary 8.4.

By the same corollary the full automorphism group of W is C_{2} × C_{2^{m−2}} × O_{2n+1}(2). These automorphisms extend to automorphisms of N. Since a subgroup C_{2} of Out(W) is induced by conjugation with elements of U, Out(N) has an epimorphic image C_{2^{m−2}} ×O_{2n+1}(2). Let H be the kernel and 1 ≠ x̄ ∈ H. Then one may choose the representative x of x̄ modulo the group of inner automorphisms of N such that x centralizes W (and N/W ≅ C_{2}). Then x maps C := C_{N}(U) =≅ 2_{+}^{1+2n} and hence U = C_{N}(C) into themselves. Hence H is a subgroup of the outer automorphism group of U. Since all automorphisms of U can be extended to N, the group H is isomorphic to Out(U) and the corollary follows from Corollary 8.5.

The results of this section are summarized in the following table.

Table 8.7. Let D be a finite dimensional ℚ-division algebra and G be a primitive a.i.m.f. group in GL_{d}(D). Then O_{p}(G) is one of the following groups:

\begin{matrix}N & B^{∘}(N) & N̄ & Glide(N) \\ p_{+}^{1+2n}, p > 2 & ^{±}N.Sp_{2n}(p) & ℚ[ζ_{p}]^{p^{n}×p^{n}} & 1 \\ 2_{+}^{1+2n} & N.O_{2n}^{+}(2) & ℚ^{2^{n}×2^{n}} & C_{2} (2) \\ 2_{−}^{1+2n} & N.O_{2n}^{−}(2) & Q_{∞,2}^{2^{n−1}×2^{n−1}} & C_{2} (2) \\ C_{p^{m}} & ^{±}N & ℚ[ζ_{p^{m}}] & 1 \\ p_{+}^{1+2n}YC_{p^{m}}, m > 1 & ^{±}N.Sp_{2n}(p) & ℚ[ζ_{p^{m}}]^{p^{n}×p^{n}} & 1 \\ 2_{+}^{1+2n}YD_{2^{m}}, m > 3 & ^{±}N.Sp_{2n}(2) & ℚ[θ_{2^{m−1}}]^{2^{n+1}×2^{n+1}} & C_{2} (2 − θ_{2^{m−1}}) \\ 2_{+}^{1+2n}YQ_{2^{m}}, m > 3 & ^{±}N.Sp_{2n}(2) & Q_{θ_{2^{m−1}} ,∞}^{2^{n}×2^{n}} & C_{2} (2 − θ_{2^{m−1}}) \\ 2_{+}^{1+2n}YQD_{2^{m}}, m > 3 & ^{±}N.Sp_{2n}(2) & ℚ[ζ_{2^{m−1}} − ζ_{2^{m−1}}^{−1}]^{2^{n+1}×2^{n+1}} & 1 \end{matrix}

The first column contains the isomorphism type of the admissible p-group N. The information in the second column is only proved under the assumption that N is a normal subgroup of its generalized Bravais group over ℚ (cf. Definition 7.1), which is necessarily the case if N is a normal subgroup of a primitive a.i.m.f. subgroup. Under this assumption, the second column contains the generalized Bravais group of N. The third column gives the enveloping ℚ-algebra \overline{N} of N and the last column contains the factor group Glide(N) of the subgroup of the automorphism group of N that acts trivially on the center Z of \overline{N}. For all a.i.m.f. subgroups G containing N as a normal subgroup the quotient G/(B^{∘}(N)C_{G}(N)) is a subgroup of Glide(N).Gal(Z/ℚ). If |Glide(N)| = 2, a norm (cf. Definition 7.10) of a nontrivial element in Glide(N) is given in brackets.

Definition 8.8. Let N ⊴ G. Then N is called self-centralizing, if C_{G}(N) ≤ N.

Proposition 8.9. Let D be a definite quaternion algebra with center K. Let G be a primitive a.i.m.f. subgroup of GL_{n}(D) and O_{2}(G) a self-centralizing normal subgroup. Then G = B_{K}^{∘}(O_{2}(G)), n = 2^{m−1} is a power of 2, and O_{2}(G) is centrally irreducible. Moreover one of the following three possibilities occurs:

(i) K = ℚ, O_{2}(G) = 2_{−}^{1+2m}, and G = 2_{−}^{1+2m}.O_{2m}^{−}(2). (ii) K = ℚ[\sqrt{2}] and G is one of 2_{−}^{1+2m}.GO_{2m}^{−}(2) or (2_{+}^{1+2(m−1)} ⊗ Q_{16}).O_{2m−1}(2). (iii) K = ℚ[θ_{2^{s}}] with s > 3 and G = (2_{+}^{1+2(m−1)} ⊗ Q_{2^{s+1}}).O_{2m−1}(2).

Proof. By Theorem 8.1 the group O_{2}(G) is a central product of an extraspecial 2group with a cyclic, dihedral, quasidihedral, or generalized quaternion group. Since O_{2}(G) is self-centralizing, G/O_{2}(G) is a subgroup of Out(O_{2}(G)) with O_{2}(G/O_{2}(G)) = 1. Hence by Corollary 8.6 and Lemma 8.3 either G = B^{∘}(O_{2}(G)) or O_{2}(G) = 2_{−}^{1+2n}, K = ℚ[\sqrt{2}] and G = B_{K}^{∘}(O_{2}(G)) = 2_{−}^{1+2m}.GO_{2m}^{−}(2).

Corollary 8.10. Let N be an admissible p-group over K and p^{a} := |Z(N)| the order of the center of N. If N is not an extraspecial 2-group or K contains ℚ[\sqrt{2}], then Glide_{K}(N) = 1 and Aut(N)/Aut_{stab}(N) is isomorphic to the Galois group Gal(K[ζ_{p^{a}}]/K).

Lemma 8.11. Let G ≤ GL_{n}(D) be a primitive a.i.m.f. group such that F it(G) := ∏_{p||G|} O_{p}(G) is a self-centralizing normal subgroup. Then F it(G) is irreducible.

Proof. If O_{2}(G) is not an extraspecial 2-group, the Lemma follows from Corollary 8.10 and Lemma 2.14.

So assume that O_{2}(G) is an extraspecial 2-group and let B := B_{K}^{∘}(O_{2}(G)). Then N := C_{G}(B)B is a normal subgroup of index 1 or 2 in G.

Let Z := Z(\overline{KF it(G)}) be the center of the enveloping K-algebra of F it(G), z := [Z : K], f := dim_{K}(\overline{KF it(G)}) = m^{2}z, and g := dim_{K}(\overline{KG}) = 4n^{2}. By Corollary 8.10, the center of \overline{KN} is a subfield K ⊆ Z(\overline{KN}) ⊆ Z, say of degree x over K.

Assume that F it(G) is reducible. Then mz < 2n and by Lemma 2.14 and Corollary 8.10 dim_{K}(\overline{KN}) = (mz)^{2}x^{−1} ≤ (mz)^{2} ≤ n^{2}. With Lemma 2.14, this contradicts the absolute irreducibility of G.

The next lemma is useful to exclude cyclic normal subgroups (cf. also Lemma 11.2).

Lemma 8.12. Let G be a primitive a.i.m.f. group and 3 < p ≡ 3 (mod 4) be a prime. If O_{p}(G) = C_{p}, then N := C_{p} : C_{\frac{p−1}{2}} is not a normal subgroup of G.

Proof. Assume that N ⊴ G. Since G is primitive, the enveloping algebra of N is \overline{ℚN} = ℚ[\sqrt{-p}]^{\frac{p−1}{2} × \frac{p−1}{2}} . If C := C_{G}(N), then G/CN embeds into C_{2}, the outer automorphism group of N. Now N is a subgroup of M := L_{2}(p) ⊆ \overline{ℚN}. Since NḠ^{∗}(M) > NḠ^{∗}(N), the group G also normalizes M and hence ⟨G, M⟩ is a proper supergroup of G.

Remark 8.13. If p ≡ 1 (mod 4), then the outer automorphism group of C_{p} : C_{\frac{p−1}{2}} =: N ≤ GL_{\frac{p−1}{2}} (ℚ[\sqrt{p}]) is C_{2} × C_{2}, where the additional automorphism is

induced by conjugation with (1 − ζ) ∈ \overline{ℚN} = ℚ[\sqrt{p}]^{\frac{p−1}{2} × \frac{p−1}{2}} , where ζ generates the normal p-subgroup of N.

9. The candidates for quasi-semi-simple normal subgroups

In this chapter we list the information used from the classification of finite simple groups and their character tables as given in [CCNPW 85]. Let G be a primitive a.i.m.f. subgroup of GL_{n}(D). Then the minimal normal subgroups of the centralizer in G of the Fitting group of G are central products of isomorphic quasisimple groups. Such groups are called quasi-semi-simple. The candidates for the quasi-semi-simple normal subgroups of G may be derived from the following table:

Table 9.1. Table of the quasisimple matrix groups admitting a homogenous representation into Q^{n×n} for a totally definite quaternion algebra Q with center of degree d over ℚ and d · n ≤ 10:

group N B^{∘}(N) character of N

\overline{ℚN}

Glide(N)
Alt_{5}  ^{±}Alt_{5}  χ_{3a} + χ_{3b} ℚ[\sqrt{5}]^{3×3}  1
Alt_{5}  ^{±}S_{5}  χ_{4}  ℚ^{4×4}  1
Alt_{5}  ^{±}S_{6}  χ_{5}  ℚ^{5×5}  -
SL_{2}(5)  SL_{2}(5) 2(χ_{2a} + χ_{2b}) Q_{\sqrt{5},∞,∞}  1
SL_{2}(5)  SL_{2}(9)  2χ_{4}  Q_{∞,3}^{2×2}  -
SL_{2}(5)  SL_{2}(5)  2χ_{6}  Q_{∞,2}^{3×3}  C_{2} (2)
L_{2}(7)  ^{±}L_{2}(7)  χ_{3a} + χ_{3b} ℚ[\sqrt{-7}]^{3×3}  1
L_{2}(7)  ^{±}L_{2}(7)  χ_{6}  ℚ^{6×6}  C_{2} (2)
L_{2}(7)  ^{±}S_{6}(2)  χ_{7}  ℚ^{7×7}  -
L_{2}(7)  ^{±}L_{2}(7) : 2  χ_{8}  ℚ^{8×8}  1

group N B^{∘}(N) character of N

\overline{ℚN}

Glide(N)
SL_{2}(7)  SL_{2}(7)  χ_{4a} + χ_{4b}  ℚ[\sqrt{-7}]^{4×4}  1
SL_{2}(7)  SL_{2}(7)  2(χ_{6a} + χ_{6b})  Q_{\sqrt{2},∞,∞}^{3×3}  C_{2} (2 + \sqrt{2})
SL_{2}(7)  SL_{2}(7)  2χ_{8}  Q_{∞,3}^{4×4}  C_{2} (3)
Alt_{6}  ^{±}S_{6}  χ_{5a} resp. χ_{5b}  ℚ^{5×5}  1
Alt_{6}  ^{±}S_{10}  χ_{9}  ℚ^{9×9}  -
Alt_{6}  ^{±}S_{6}  χ_{10}  ℚ^{10×10}  C_{2} (2)
SL_{2}(9)  SL_{2}(9) 2χ_{4a} resp. 2χ_{4b}  Q_{∞,3}^{2×2}  C_{2} (3)
SL_{2}(9)  SL_{2}(9)  2(χ_{8a} + χ_{8b})  Q_{\sqrt{5},∞,∞}^{4×4}  C_{2} (5 + 2\sqrt{5})
SL_{2}(9)  SL_{2}(9)  2(χ_{10a} + χ_{10b})  Q_{\sqrt{2},∞,∞}^{5×5}  C_{2} (2 + \sqrt{2})
3.Alt_{6}  ^{±}3.Alt_{6}  \begin{matrix}χ_{3a} + χ′_{3a} \\ +χ_{3b} + χ′_{3b} \end{matrix}  ℚ[\sqrt{5}, \sqrt{-3}]^{3×3}  1
3.Alt_{6}  ^{±}3.Alt_{6}  χ_{6} + χ′_{6}  ℚ[\sqrt{-3}]^{6×6}  C_{2} (2)
3.Alt_{6}  ^{±}3.M_{10}  χ_{9} + χ′_{9}  ℚ[\sqrt{-3}]^{9×9}  1
L_{2}(8)  ^{±}S_{6}(2)  χ_{7}  ℚ^{7×7}  -
L_{2}(8)  2.O_{8}^{+}(2).2  χ_{8}  ℚ^{8×8}  -
L_{2}(11)  ^{±}L_{2}(11)  χ_{5a} + χ_{5b}  ℚ[\sqrt{-11}]^{5×5}  1
L_{2}(11) ^{±}L_{2}(11) : 2  χ_{10a}  ℚ^{10×10}  1
L_{2}(11)  ^{±}L_{2}(11)  χ_{10b}  ℚ^{10×10}  C_{2} (3)
SL_{2}(11) SL_{2}(11)  χ_{6a} + χ_{6b}  ℚ[\sqrt{-11}]^{6×6}  1
SL_{2}(11) SL_{2}(11)  2χ_{10}  Q_{∞,2}^{5×5}  C_{2} (2)
SL_{2}(11) SL_{2}(11) 2(χ_{10a} + χ_{10b})  Q_{\sqrt{3},∞,∞}^{5×5}  C_{2} (2)
L_{2}(13)  ^{±}L_{2}(13)  χ_{7a} + χ_{7b}  ℚ[\sqrt{13}]^{7×7}  1
SL_{2}(13) SL_{2}(13)  2(χ_{6a} + χ_{6b})  Q_{\sqrt{13},∞,∞}^{3×3}  1
SL_{2}(13) SL_{2}(13)  2χ_{14}  Q_{∞,2}^{7×7}  C_{2} (2)
SL_{2}(17) SL_{2}(17)  2(χ_{8a} + χ_{8b})  Q_{\sqrt{17},∞,∞}^{4×4}  1
SL_{2}(17) SL_{2}(17)  2χ_{16}  Q_{∞,3}^{8×8}  C_{2} (3)
Alt_{7}  ^{±}S_{7}  χ_{6}  ℚ^{6×6}  1
Alt_{7}  ^{±}Alt_{7}  χ_{10a} + χ_{10b}  ℚ[\sqrt{-7}]^{10×10}  1
2.Alt_{7}  2.Alt_{7}  χ_{4a} + χ_{4b}  ℚ[\sqrt{-7}]^{4×4}  1
2.Alt_{7}  2.Alt_{7}  2χ_{20a}  Q_{∞,3}^{10×10}  C_{2} (3)
2.Alt_{7}  2.Alt_{7}  2χ_{20b}  Q_{∞,3}^{10×10}  C_{2} (6)
3.Alt_{7}  6.U_{4}(3).2  χ_{6} + χ′_{6}  ℚ[\sqrt{-3}]^{6×6}  -
L_{2}(19)  ^{±}L_{2}(19)  χ_{9a} + χ_{9b}  ℚ[\sqrt{-19}]^{9×9}  1
SL_{2}(19) SL_{2}(19)  χ_{10a} + χ_{10b}  ℚ[\sqrt{-19}]^{10×10}  1
SL_{2}(19) SL_{2}(19)  2χ_{18}  Q_{∞,2}^{9×9}  C_{2} (2)
SL_{2}(19) SL_{2}(19)  2χ_{20}  Q_{∞,3}^{10×10}  C_{2} (3)

group N B^{∘}(N) character of N

\overline{ℚN}

Glide(N)
U_{3}(3)  ^{±}U_{3}(3)  2χ_{6}  Q_{∞,3}^{3×3}  C_{2} (3)
U_{3}(3)  ^{±}S_{6}(2)  χ_{7}  ℚ^{7×7}  -
U_{3}(3)  U_{3}(3) ∘ C_{4}  χ_{7a} + χ_{7b}  ℚ[\sqrt{-1}]^{7×7}  1
SL_{2}(25)  SL_{2}(25)  2χ_{12}  Q_{∞,5}^{6×6}  C_{2} (5)
M_{11}  ^{±}S_{11}  χ_{10a}  ℚ^{10×10}  -
M_{11}  ^{±}M_{11}  χ_{10b} + χ_{10c} ℚ[\sqrt{-2}]^{10×10}  1
Alt_{8}  ^{±}S_{6}(2)  χ_{7}  ℚ^{7×7}  -
2.Alt_{8}  2.O_{8}^{+}(2).2  χ_{8}  ℚ^{8×8}  -
2.L_{3}(4) 2.L_{3}(4) : 2_{2} χ_{10a} + χ_{10b} ℚ[\sqrt{-7}]^{10×10}  1
6.L_{3}(4)  6.L_{3}(4)  χ_{6} + χ′_{6}  ℚ[\sqrt{-3}]^{6×6}  C_{2} (2)
U_{4}(2)  ^{±}U_{4}(2) ∘ C_{3} χ_{5a} + χ_{5b}  ℚ[\sqrt{-3}]^{5×5}  1
U_{4}(2)  ^{±}U_{4}(2) : 2  χ_{6}  ℚ^{6×6}  1
U_{4}(2)  ^{±}U_{4}(2) ∘ C_{3} χ_{10a} + χ_{10b} ℚ[\sqrt{-3}]^{10×10}  1
2.U_{4}(2) 2.U_{4}(2) ∘ C_{3} χ_{4a} + χ_{4b}  ℚ[\sqrt{-3}]^{4×4}  1
2.U_{4}(2)  2.U_{4}(2)  2χ_{20}  Q_{∞,2}^{10×10}  C_{2} (2)
U_{3}(4)  2.G_{2}(4)  2χ_{12}  Q_{∞,2}^{6×6}  -
2.M_{12}  2.M_{12} : 2  χ_{10a} + χ_{10b} ℚ[\sqrt{-2}]^{10×10}  1
U_{3}(5)  ^{±}U_{3}(5) : 3  2χ_{20}  Q_{∞,5}^{10×10}  C_{2} (5)
Alt_{9}  2.O_{8}^{+}(2).2  χ_{8}  ℚ^{8×8}  -
2.Alt_{9}  2.O_{8}^{+}(2).2 χ_{8a} resp. χ_{8b}  ℚ^{8×8}  -
2.M_{22}  2.M_{22} : 2  χ_{10a} + χ_{10b} ℚ[\sqrt{-7}]^{10×10}  1
2.J_{2}  2.J_{2}  2(χ_{6a} + χ_{6b}) Q_{\sqrt{5},∞,∞}^{3×3}  1
2.J_{2}  2.J_{2}  2χ_{14}  Q_{∞,2}^{7×7}  C_{2} (2)
S_{6}(2)  ^{±}S_{6}(2)  χ_{7}  ℚ^{7×7}  1
2.S_{6}(2)  2.O_{8}^{+}(2).2  χ_{8}  ℚ^{8×8}  -
Alt_{10}  ^{±}S_{10}  χ_{9}  ℚ^{9×9}  1
2.U_{4}(3)  2.U_{4}(3).4  2χ_{20}  Q_{∞,3}^{10×10}  C_{2} (3)
6.U_{4}(3)  6.U_{4}(3).2  χ_{6} + χ′_{6}  ℚ[\sqrt{-3}]^{6×6}  1
U_{5}(2)  ^{±}U_{5}(2)  2χ_{10}  Q_{∞,2}^{5×5}  C_{2} (2)
Alt_{11}  ^{±}S_{11}  χ_{10}  ℚ^{10×10}  1
2.O_{8}^{+}(2) 2.O_{8}^{+}(2).2  χ_{8}  ℚ^{8×8}  1
2.G_{2}(4)  2.G_{2}(4)  2χ_{12}  Q_{∞,2}^{6×6}  C_{2} (2)

The first column contains the quasisimple matrix group N, the second column its generalized Bravais group over ℚ (cf. Definition 7.1) followed by the character χ of a ℚ-irreducible constituent of the natural \overline{ℚN}-module and the enveloping algebra \overline{ℚN}. If G is a primitive maximal finite group in GL_{n}(D) with normal subgroup N, then G has a normal subgroup B^{∘}(N)C_{G}(N) such that the factor group G/(B^{∘}(N)C_{G}(N)) embeds into Glide(N).Gal(ℚ[χ]/ℚ) (cf. Definition 7.3). Especially if |Glide(N)| = 2, a norm of a nontrivial element of this group (cf. Definition 7.10) is given in brackets.

10. Some building blocks

By Chapter 7 we may build up the primitive maximal finite matrix groups using normal subgroups that satisfy a certain maximality condition.

Let D be a definite quaternion algebra with center K and N = B^{∘}(N) be a normal subgroup of a primitive a.i.m.f. subgroup G of GL_{n}(D). Assume that \overline{KN} is a central simple K-algebra and let G = (Δ_{1} ⊗ Δ_{2})(G) be as in Theorem 7.8. Since K is totally real, Theorem 7.11 says that G contains the normal subgroup U := C_{G}(N) = ker(Δ_{1}) with (Δ_{2}(G)K^{∗})/(Δ_{2}(U)K^{∗}) ≅ (Δ_{1}(G)K^{∗})/(Δ_{1}(N)K^{∗}) of exponent 1 or 2. Choose g_{i} ∈ Δ_{2}(G) such that (g_{1}, …, g_{s}) maps onto a basis of (Δ_{2}(G)K^{∗})/(Δ_{2}(U)K^{∗}). Then there are q_{i} ∈ K^{∗} such that q_{i}^{−1}g_{i}^{2} ∈ U.

Lemma 10.1. In the situation above, the pair (U, S = {g_{1}, …, g_{s}}) satisfies the following maximality condition: For all finite supergroups V ≥ U that are contained in \overline{KΔ_{2}(G)} =: A, such that g_{i} ∈ N_{A^{∗}}(V) for all 1 ≤ i ≤ s, one has V = U.

Call such a pair (U, S) a maximal pair and U a nearly maximal finite subgroup of A^{∗}. Note that if (U, S) is a maximal pair then, since N_{A^{∗}}(U) ⊆ N_{A^{∗}}(B^{∘}(U)), one has U = B^{∘}(U).

Table 10.2. Assume that A is a quaternion algebra with center ℚ and s = 1. Then the maximal pairs (U, {g}) may be derived from the following table:

\begin{matrix}U & norm(g) & A \\ ^{±}C_{3} & 2 & Q_{2,3} \\ ^{±}C_{3} & 1 & ℚ^{2×2} \\ C_{4} & 3 & Q_{2,3} \\ C_{4} & 1 & ℚ^{2×2} \\ S̃_{3} & 3 & Q_{∞,3} \\ SL_{2}(3) & 2 & Q_{∞,2} \\ D_{8} & 2 & ℚ^{2×2} \\ ^{±}S_{3} & 3 & ℚ^{2×2} \end{matrix}

Here the first column displays the matrix group U, the second column gives a norm of the element g = g_{1} in the normalizer of U in A^{∗} as defined in Definition 7.10, and the last column the central simple ℚ-algebra A = \overline{Δ_{2}(G)} generated by U and g.

For central simple algebras \overline{KN} the most important situation is that the g_{i} lie in the enveloping algebra (cf. Theorem 7.11). For the determination of the maximal possibilities for N, the following is helpful:

Remark 10.3. In the situation of Theorem 7.11, let a ∈ N\overline{KN}^{∗}(N) be normalized such that aF a^{t} = qF where q ∈ R is a norm of α (cf. Definition 7.10). Let L be an a^{2}q^{−1}-invariant RN-lattice and assume that F ∈ F_{>0}(N) is integral on L. Then qL ⊆ La ⊆ L. If the ideal generated by q and det(F, L) is the ring of integers of K, then F defines a bilinear form F̄ : L/qL × L/qL → R/qR. Since the dual lattice of La with respect to F is (La)^{#} = q^{−1}L^{#}a, the lattice La corresponds to a maximal isotropic subspace of L/qL.

Let N(≤ GL_{8}(ℚ)) be a finite matrix group such that the enveloping ℚ-algebra \overline{N} is a central simple ℚ-algebra of dimension 16. Assume that all abelian characteristic subgroups of N are cyclic. If the pair (N, \{g\}) with g ∈ N\overline{ℚN}^{∗}(N) is a maximal pair, then N is one of the groups in the following table:

Table 10.4

\begin{matrix}N & norm(g) & \overline{ℚN} \\ A_{4} & 1 & ℚ^{4×4} \\ ^{±}C_{5} : C_{4} & 5 & ℚ^{4×4} \\ F_{4} & 1, 2 & ℚ^{4×4} \\ C_{3} □^{2} SL_{2}(3) & 3, 6 & ℚ^{4×4} \\ S_{3} ⊗ D_{8} & 2, 6 & ℚ^{4×4} \\ SL_{2}(5) : 2 & 1, 5 & Q_{∞,5}^{2×2} \\ SL_{2}(5).24 & 1, 5 & Q_{∞,5}^{2×2} \\ 2_{−}^{1+4}.Alt_{5} & 1, 2 & Q_{∞,2}^{2×2} \\ C_{3} ⊃ × ⊏^{2(2)}D_{8} & 3, 6 & Q_{∞,2}^{2×2} \\ S_{3} ⊗ SL_{2}(3) & 1, 2, 3, 6 & Q_{∞,2}^{2×2} \\ C_{3} □^{2} SL_{2}(3) & 1, 2, 3, 6 & Q_{∞,3}^{2×2} \\ SL_{2}(9) & 1, 3 & Q_{∞,3}^{2×2} \\ S̃_{3} ⊗ D_{8} & 2, 6 & Q_{∞,3}^{2×2} \\ C_{3} ⊃ × ⊏^{2(2)}D_{8} & 1, 2, 3, 6 & Q_{2,3}^{2×2} \\ S̃_{3}⊗ \sqrt{-3} SL_{2}(3) & 1, 2, 3, 6 & Q_{2,3}^{2×2} \end{matrix}

In the first column of this table the finite matrix group N is given using the notation of Chapter 5. The last column displays the enveloping ℚ-algebra A := N̄ of N. The second column allows one to read off the elements g ∈ N_{A^{∗}}(N) such that (N, {g}) is a maximal pair, since these are modulo N uniquely determined by their norms (cf. Corollary 7.12) . In particular N is a maximal finite subgroup of A^{∗}, if and only if a “1” appears in this column.

Proof. The proof is divided into 5 cases according to the possible enveloping algebras \overline{ℚN} = Q^{2×2} where Q is either a definite or an indefinite quaternion algebra with center ℚ. If Q is definite, Theorem 12.1 below implies that Q is one of Q_{∞,2}, Q_{∞,3}, or Q_{∞,5}. In the indefinite case it follows from [Sou 94] that Q is either ℚ^{2×2} or Q_{2,3}. In the last case, N is already maximal finite, since the 2 maximal finite subgroups of GL_{2}(Q_{2,3}) are the generalized Bravais groups of their minimal absolutely irreducible subgroups.

If \overline{ℚN} = ℚ^{4×4}, then N is contained in one of the three r.i.m.f. subgroups G = A_{4}, F_{4}, or A_{2}^{2} of GL_{4}(ℚ). Let L be the natural G-lattice.

If N ≤ A_{4}, then the order of N is divisible by 5. Hence N = A_{4} ≅ ^{±}S_{5} or N = ^{±}C_{5} : C_{4} is the generalized Bravais group of one of the two minimal absolutely irreducible subgroups Alt_{5} or C_{5} : C_{4} of A_{4}. In the first case, N_{GL_{4}(ℚ)}(N) = ℚ^{∗}N, by Corollary 7.6. In the other case, N fixes additionally the lattices A_{4}(1 − x) and A_{4}(1 − x)^{2} where x generates O_{5}(N). Hence N_{GL_{4}(ℚ)}(N) additionally contains an element (x + x^{−1}) of norm 5 inducing a similarity A_{4} ∼ A_{4}(1 − x)^{2}.

Since the lattice F_{4} is 2-modular, the normalizer N_{GL_{4}(ℚ)}(F_{4}) contains an element of norm 2 ([Neb 97, Proposition 3]). Apart from 2L^{#}, there is no other sublattice M with 2L ⊂ M ⊂ L which is similar to L. Moreover 2 and 3 are the only primes dividing the group order. Hence by Remark 10.3, the absolutely irreducible nearly maximal finite subgroups N contained in F_{4} are the absolutely irreducible stabilizers of the maximal isotropic subspaces of L/3L. There are eight such subspaces lying in one orbit under the action of the group F_{4}. The stabilizer

of such a subspace is SL_{2}(3) □^{2} C_{3}.

Similarly, the absolutely irreducible nearly maximal finite proper subgroups of A_{2}^{2} stabilize one of the 6 maximal isotropic subspaces of (A_{2}^{2})/2(A_{2}^{2}). All these stabilizers are conjugate to S_{3} ⊗ D_{8}.

Now assume that Q is a definite quaternion algebra. Then N embeds into one of the six primitive a.i.m.f. groups of Theorem 12.1 or into SL_{2}(3) ≀ C_{2} or S̃_{3} ≀ C_{2}. Let 𝔐 be a maximal order of Q.

If Q = Q_{∞,5}, then N embeds into one of SL_{2}(5).2 or SL_{2}(5) : 2. As in the case N ≤ A_{4} the only other possibility for a nearly maximal finite group is ^{±}C_{5}.C_{4}. But now the outer automorphism of ^{±}C_{5}.C_{4} mapping an element x of order 8 in ±C_{5}.C_{4} onto −x extends to an automorphism of SL_{2}(5).2 and SL_{2}(5) : 2 stabilizing the character. Hence the nonsplit extension ^{±}C_{5}.C_{4} is not a nearly maximal finite group.

If Q = Q_{∞,2}, then N is a subgroup of one of the three a.i.m.f. groups 2_{−}^{1+4}.Alt_{5}, SL_{2}(3)⊗S_{3}, or SL_{2}(3)≀C_{2}. Since 2_{−}^{1+4}.Alt_{5} has an element of norm 2 in its normalizer, and the stabilizers of the other ten 𝔐/2𝔐-subspaces of L/2L corresponding to lattices which are similar to L are not absolutely irreducible, one finds with Remark 10.3 that N is a stabilizer of one of the forty 𝔐-sublattices corresponding to the maximal isotropic subspaces of the 𝔐/3𝔐-module L/3L, where L is a 𝔐2_{−}^{1+4}Alt_{5}-lattice. These lattices lie in one orbit under 2_{−}^{1+4}.Alt_{5}. One calculates

N = C_{3} ⊃ × ⊏^{2(2)}D_{8} in this case.

All prime divisors of the order of G := S_{3} ⊗ SL_{2}(3) arise as norms of elements of the normalizer of N in \overline{N}^{∗}. Since the absolutely irreducible subgroups of G are characteristic in G Corollary 7.12 implies that the group N has no proper nearly maximal finite subgroups.

If N is a subgroup of SL_{2}(3) ≀ C_{2}, the forty maximal isotropic 𝔐/3𝔐-subspaces of L/3L fall into two orbits of length 16 and 24. Their stabilizers are not absolutely irreducible. The stabilizers of the thirteen 𝔐/2𝔐-subspaces of L/2L corresponding to lattices which are similar to L are either SL_{2}(3)≀C_{2} which has a noncyclic abelian normal subgroup or the subgroup D_{8} ⊗ SL_{2}(3) of index 12. Since B^{∘}(D_{8} ⊗ Q_{8}) = 2_{−}^{1+4}.Alt_{5}, one finds no groups N here.

In the last case, Q = Q_{∞,3}. Now N is a subgroup of one of the a.i.m.f. groups G

conjugate to SL_{2}(3) □^{2} C_{3}, SL_{2}(9), or S̃_{3} ≀ C_{2}. All three groups admit an element of norm 3 in their normalizer.

In the first case, G itself admits an element of norm 2 in its normalizer. The

minimal absolutely irreducible subgroups of G are S̃_{4} and Q_{8} □^{2} C_{3}. For p = 2 and 3, there is only one proper 𝔐S̃_{4}-sublattice of L containing pL which is similar to

L. This lattice is also fixed by G, hence N ≠ S̃_{4}. Clearly N ≠ Q_{8} □^{2} C_{3}, because N ≠ B^{∘}(N) = G.

In the other two cases, the proper subspaces of the 𝔐/3𝔐 module L/3L give rise to thirty-one resp. thirty-seven lattices similar to L. In the first case, their stabilizers are either G or subgroups of index 20 resp. 10 in G normalizing a Sylow 3-subgroup (≅ C_{3} × C_{3}) of G. In the last case, the thirty-seven lattices fall into two orbits. The lattices which are not fixed by G have a reducible stabilizer ≅ C_{8}.

Hence by Remark 10.3 N is an absolutely irreducible stabilizer of one of the fifteen 𝔐 sublattices corresponding to the maximal isotropic subspaces of the 𝔐/2𝔐module L/2L, where L is a 𝔐G-lattice. In the first case, these subspaces form one orbit. The stabilizer of such a subspace is S̃_{4} and not absolutely irreducible. In the last case, the fifteen maximal isotropic subspaces fall into two orbits of length 9 respectively 6 under the action of G. Only a representative of the second orbit has an absolutely irreducible stabilizer N = D_{8} ⊗ S̃_{3}.

11. Special dimensions

There are some cases where it is easy to describe an infinite family of simplicial complexes M_{n}^{irr}(D). Two of them are dealt with in the next theorem.

Theorem 11.1. (i) Let p ≡ 1 (mod 4) be a prime. M_{\frac{p−1}{4}} (Q_{\sqrt{p},∞}) consists of a single vertex: _{\sqrt{p},∞}[SL_{2}(p)]_{\frac{p−1}{4}} . The group SL_{2}(p) fixes an even unimodular ℤ-lattice (of rank 2(p − 1)).

(ii) Let p be a prime. If p ≡ 1 (mod 4), then M_{\frac{p−1}{2}} (Q_{∞,p}) consists of one 1dimensional simplex:

[figure: _{∞,p}[SL_{2}(p).2]_{\frac{p−1}{2}}; _{∞,p}[SL_{2}(p) : 2]_{\frac{p−1}{2}}]

where the common absolutely irreducible subgroup of the two a.i.m.f. groups is ^{±}C_{p}.C_{p−1}. The corresponding ℤ-lattices are unimodular (for the nonsplit extension _{∞,p}[SL_{2}(p).2]_{\frac{p−1}{2}} ) resp. p-modular (for the split extension ∞,p[SL_{2}(p) : 2]_{\frac{p−1}{2}} ).

If p ≡ −1 (mod 4), then M_{\frac{p−1}{2}} (Q_{∞,p}) consists of one single vertex: ∞,p[^{±}L_{2}(p).2]_{\frac{p−1}{2}} . The group _{∞,p}[^{±}L_{2}(p).2]_{\frac{p−1}{2}} fixes an even p-modular ℤlattice (of rank 2(p − 1)).

To prove the theorem, we need a lemma which is also of independent interest in later chapters.

Lemma 11.2. Let D be a definite quaternion algebra with center K and d := [K : ℚ] = 1 or 2. Let p be an odd prime such that n := \frac{p−1}{2d} ∈ ℕ. If G ≤ GL_{n}(D) is an a.i.m.f. subgroup, then O_{p}(G) = 1.

Proof. Assume the O_{p}(G) > 1. Then by the formula in [Schu 05] P := O_{p}(G) ≅ C_{p} and in the case d = 2, K = ℚ[\sqrt{p}] and p ≡ 1 (mod 4). Since the commuting algebra C_{D^{n×n}}(P) is isomorphic to ℚ[ζ_{p}], the centralizer C_{G}(P) = ^{±}P. Now G is absolutely irreducible, so G/C_{G}(P) ≅ C_{\frac{p−1}{d}} is isomorphic to the subgroup of index d in the automorphism group of P. The split extension ^{±}P : C_{\frac{p−1}{d}} has real Schur index 1, and the nonsplit extension G = ^{±}P.C_{\frac{p−1}{d}} is a subgroup of _{\sqrt{p},∞}[SL_{2}(p)]_{\frac{p−1}{4}} (if d = 2), _{∞,p}[SL_{2}(p).2]_{\frac{p−1}{2}} (if d = 1 and p ≡ 1 (mod 4)), resp. _{∞,p}[^{±}L_{2}(p).2]_{\frac{p−1}{2}} (if d = 1 and p ≡ 3 (mod 4)) which is a contradiction.

Proof of Theorem 11.1. From the classification of a.i.m.f. subgroups of GL_{1}(Q_{\sqrt{5},∞}) and GL_{\frac{p−1}{2}} (Q_{∞,p}) for p ≤ 11 in this paper, the theorem is true for p ≤ 11. So we may assume p ≥ 13.

(i) Let Q := Q_{\sqrt{p},∞}, n := \frac{p−1}{4}, and G be an a.i.m.f. subgroup of GL_{n}(Q).

Then by Lemma 2.13 p divides the order of G. By Proposition 2.16 the Sylow p-subgroup P of G is ≅ C_{p}. Since the degree of the natural character of G is \frac{p−1}{2} [Fei 82, Theorem VIII.7.2] implies that either G/Z(G) ≅ P SL_{2}(p) or P ⊴ G is normal in G. The second case is excluded by Lemma 11.2. Since ^{±}1 are the only roots of unity contained in the center ℚ[\sqrt{p}], the results on the Schur indices of the characters of SL_{2}(p) in [Fei 83] yield that G = SL_{2}(p) in the first case. The SL_{2}(p)-invariant ℤ-lattices are described in [Neb 98].

(ii) Let −1 ∈ G ≤ GL_{\frac{p−1}{2}} (Q_{∞,p}) be absolutely irreducible.

Then by Lemma 2.13 p divides the order of G. By Proposition 2.16 the Sylow p-subgroup of G has order p. Let 𝔐 be a maximal order in Q_{∞,p}, P the maximal two-sided ideal of 𝔐 containing p, and L ∈ Z_{𝔐}(G) a 𝔐G-lattice in the natural

G-module Q_{∞,p}^{1× \frac{p−1}{2}} . Then L̄ := L/PL is a 𝔽_{p^{2}}G-module of dimension \frac{p−1}{2}. Since the kernel of the action of G on L̄ coincides with the one on PL/pL, this kernel is contained in O_{p}(G). By Lemma 11.2 O_{p}(G) = 1, so L̄ is a faithful 𝔽_{p^{2}}G-module and [Fei 82, Theorem (VIII.3.3)] implies that G is of type L_{2}(p), i.e. the unique composition factor O^{p′}(G)/(O^{p′}(G) ∩ O_{p′}(G)) of G of order divisible by p is either isomorphic to L_{2}(p) or to C_{p}. Here O^{p′}(G) is the smallest normal subgroup of G of index prime to p and O_{p′}(G) the largest normal subgroup of G of order prime to p. Let g be an element of order p in G. Then C_{G}(g) embeds into GL_{1}(ℚ[ζ_{p}]), hence is ⟨^{±}g⟩. Therefore g acts fixed point freely on O_{p′}(G)/⟨^{±}1⟩. By Thompson’s theorem (cf. [Hup 67], p. 505) O_{p′}(G)/⟨^{±}1⟩ is nilpotent. Let r ≠ p be a prime and A an abelian normal r-subgroup of G. Then A is cyclic by Corollary 2.4 and the enveloping algebra of A is contained in ℚ^{(p−1)×(p−1)}. Hence by Corollary 2.4 r < p and g centralizes A. Therefore r = 2 and A ≤ ⟨^{±}1⟩. Hence O_{p′}(G) = O_{2}(G) and the maximal abelian normal subgroup of G is ⟨^{±}1⟩.

If O_{2}(G) > ^{±}1, then Proposition 8.9 gives a contradiction to the fact that the p-adic Schur index of the natural representation of G is 2.

Hence O_{2}(G) = ^{±}1 and O^{p′}(G) is one of ^{±}L_{2}(p), SL_{2}(p), or ^{±}C_{p}. By Lemma 11.2 the latter possibility does not occur. [Fei 83, Theorem 6.1] yields that the padic Schur indices of the representations of SL_{2}(p) are 1. Hence the ℂ-constituents of the restriction of the natural representation of G to O^{p′}(G) are of degree \frac{p−1}{2}. Using the character tables in [Schu 07] one concludes that if p ≡ −1 (mod 4), then G is the unique extension of ^{±}L_{2}(p) ≤ GL_{\frac{p−1}{2}} (ℚ[\sqrt{-p}]) by C_{2} ≅ Out(L_{2}(p)) with real Schur index 2, and if p ≡ 1 (mod 4), then G is one of the two extensions of the matrix group SL_{2}(p) of (i).

For p ≡ 1 (mod 4), the SL_{2}(p).2-invariant and SL_{2}(p) : 2-invariant ℤ-lattices are described in [Neb 98, Remark 2.5]. For their determinants see [Tie 97, Section 5]. If p ≡ 3 (mod 4), the natural representation of the group L_{2}(p) ≤ _{∞,p}[^{±}L_{2}(p).2]_{\frac{p−1}{2}} is

a globally irreducible representation ([Gro 90, Chapter 11]). The ℤ[\frac{1+\sqrt{−p}}{2}]L_{2}(p)- lattices are unimodular Hermitian lattices over ℤ[\frac{1+\sqrt{−p}}{2}]. The lattices of which the endomorphism ring is a maximal order 𝔐 of Q_{∞,p} containing ℤ[\frac{1+\sqrt{−p}}{2}] and which are preserved by the a.i.m.f. group _{∞,p}[^{±}L_{2}(p).2]_{\frac{p−1}{2}} are scalar extensions of these unimodular lattices and therefore also unimodular Hermitian. Since the discriminant of 𝔐 is generated by \sqrt{−p}, they become p-modular ℤ-lattices.

12. The a.i.m.f. subgroups of GL_{2}(Q)

Z(Q) = ℚ

Theorem 12.1. Let Q be a definite quaternion algebra with center ℚ and G be a maximal finite primitive absolutely irreducible subgroup of GL_{2}(Q). Then Q is one of Q_{∞,2}, Q_{∞,3}, or Q_{∞,5} and G is conjugate to one of the groups in the following table.

List of the primitive a.i.m.f. subgroups of GL_{2}(Q).

lattice L  |Aut(L)| r.i.m.f. supergroups
∞,2[2_{−}^{1+4}.Alt_{5}]_{2}  2^{7} · 3 · 5 E_{8}
∞,2[SL_{2}(3)]_{1} ⊗ A_{2} 2^{4} · 3^{2} A_{2} ⊗ F_{4}
∞,3[SL_{2}(9)]_{2}  2^{4} · 3^{2} · 5 E_{8}
∞,3[SL_{2}(3) □^{2} C_{3}]_{2} 2^{4} · 3^{2} F_{4}^{2}
∞,5[SL_{2}(5).2]_{2}  2^{4} · 3 · 5 E_{8}
∞,5[SL_{2}(5) : 2]_{2}  2^{4} · 3 · 5 [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8}

Proof. If G contains a quasi-semi-simple normal subgroup, [HaS 85] and [CCNPW 85] show that either G^{(∞)} ≅ SL_{2}(5) and G = _{∞,5}[SL_{2}(5).2]_{2} or G = ∞,5[SL_{2}(5) : 2]_{2} or G = G^{(∞)} = _{∞,3}[SL_{2}(9)]_{2}. Now assume that G contains no quasi-semi-simple normal subgroup. By Lemma 11.2 O_{5}(G) = 1. If O_{3}(G) = 1, then O_{2}(G) is a self-centralizing normal subgroup of G. With Proposition 8.9 one finds that G = B^{∘}(O_{2}(G)) = _{∞,2}[2_{−}^{1+4}.Alt_{5}]_{2}.

If O_{3}(G) > 1, then O_{3}(G) ≅ C_{3}, G contains C := C_{G}(O_{3}(G)) of index two, and C is an absolutely irreducible subgroup of (ℚ[ζ_{3}]⊗Q)^{∗}. Using [Bli 17] one finds that C is one of C_{3}∘SL_{2}(3) or C_{3}⊗D_{8}. In both cases, one has two possible automorphisms of C yielding each a unique extension G = C.2 in GL_{2}(Q) (cf. Lemma 2.17). In the first case, G is one of _{∞,2}[SL_{2}(3)]_{1} ⊗ A_{2} or _{∞,3}[SL_{2}(3) □^{2} C_{3}]_{2}. In the second

case, one finds no groups G, since S̃_{3} ⊗ D_{8} is imprimitive and C_{3} ⊃ × ⊏^{2(2)}D_{8} is a proper subgroup of _{∞,2}[2_{−}^{1+4}.Alt_{5}]_{2}.

Theorem 12.2. M_{2}(Q_{∞,2})^{irr} is as follows.

[figure: _{∞,2}[2_{−}^{1+4}.Alt_{5}]_{2}; _{∞,2}[SL_{2}(3)]_{1}^{2}; _{∞,2}[SL_{2}(3)]_{1} ⊗ A_{2}]
List of the maximal simplices in M_{2}^{irr}(Q_{∞,2})
simplex  a common subgroup
(_{∞,2}[2_{−}^{1+4}.Alt_{5}]_{2},_{∞,2}[SL_{2}(3)]_{1}^{2}) D_{8} ⊗ Q_{8}

Proof. The a.i.m.f. subgroups of GL_{2}(Q_{∞,2}) can be deduced from Theorem 12.1 and Theorem 6.1. The completeness of the list of maximal simplices in M_{2}^{irr}(Q_{∞,2}) follows from the fact, that the unique minimal absolutely irreducible subgroup of ∞,2[SL_{2}(3)]_{1}⊗A_{2} is S_{3}⊗Q_{8} and does not fix any 3-unimodular lattice with maximal order as an endomorphism ring.

Theorem 12.3. M_{2}(Q_{∞,3})^{irr} is as follows.

[figure: _{∞,3}[SL_{2}(9)]_{2}; _{∞,3}[S̃_{3}]_{1}^{2}; _{∞,3}[SL_{2}(3) □^{2} C_{3}]_{2}]
List of the maximal simplices in M_{2}^{irr}(Q_{∞,3})
simplex  a common subgroup
(_{∞,3}[SL_{2}(9)]_{2},_{∞,3}[S̃_{3}]_{1}^{2})  (^{±}C_{3} × C_{3}).C_{4}
(_{∞,3}[SL_{2}(9)]_{2},_{∞,3}[SL_{2}(3) □^{2} C_{3}]_{2}) S̃_{4}

Proof. The a.i.m.f. subgroups of GL_{2}(Q_{∞,3}) can be deduced from Theorem 12.1 and Theorem 6.1. The completeness of the list of maximal simplices in M_{2}^{irr}(Q_{∞,3}) follows from the fact, that the minimal absolutely irreducible subgroups of

∞,3[SL_{2}(3) □^{2} C_{3}]_{2} are S̃_{4} and Q_{8} □^{2} C_{3}. Whereas the first group also embeds into _{∞,3}[SL_{2}(9)]_{2}, the second one has a unique a.i.m.f. supergroup.

Theorem 12.4. M_{2}(Q_{∞,5})^{irr} is as follows.

[figure: _{∞,5}[SL_{2}(5) : 2]_{2}; _{∞,5}[SL_{2}(5).2]_{2}]

List of the maximal simplices in M_{2}^{irr}(Q_{∞,5})

simplex  a common subgroup
(_{∞,5}[SL_{2}(5).2]_{2},_{∞,5}[SL_{2}(5) : 2]_{2}) (^{±}C_{5}).C_{4}

Proof. The theorem follows immediately from Theorem 12.1 and Theorem 6.1.

Z(Q) real quadratic.

Theorem 12.5. Let G be an absolutely irreducible maximal finite subgroup of GL_{2}(Q), where Q is a totally definite quaternion algebra with center K and [K : ℚ] = 2. Assume that G has a quasi-semi-simple normal subgroup. Then Q is isomorphic to Q_{\sqrt{5},∞} or Q_{\sqrt{3},∞} and G is conjugate to one of

\sqrt{5},∞[SL_{2}(5)⊗ \sqrt{5} D_{10}]_{2}, _{\sqrt{5},∞}[SL_{2}(5)]_{1} ⊗ A_{2}, _{\sqrt{3},∞}[2.S_{6}]_{2},

or the imprimitive group _{\sqrt{5},∞}[SL_{2}(5)]_{1}^{2}.

Proof. Let G be such a maximal finite group and N ⊴ G a quasi-semi-simple normal subgroup. If G is imprimitive Theorem 6.1 implies that G = _{\sqrt{5},∞}[SL_{2}(5)]_{1}^{2}. Assume now that G is primitive. By Table 9.1 N is either SL_{2}(5) or SL_{2}(9) (cf. also [HaS 85]). Assume first that N is SL_{2}(5). Then the enveloping algebra of N is \overline{ℚN} = Q_{\sqrt{5},∞}. If K ≠ ℚ[\sqrt{5}], then N is an irreducible subgroup of GL_{2}(Q). The centralizer C_{G}(N) embeds into C_{GL_{2}(Q)}(N) ≅ K[\sqrt{5}]^{∗}. Since K[\sqrt{5}] is a totally real field, one gets that C_{G}(N) = ^{±}1. Therefore G contains N of index 2 and by Lemma 2.14 the enveloping ℚ-algebra of G is of dimension 8 or 16, contradicting the assumption that G is an absolutely irreducible subgroup of GL_{2}(Q).

Hence K = ℚ[\sqrt{5}] is the center of the enveloping algebra \overline{ℚN}. Then N is primitively saturated over K and hence G = N⊗_{K} C, for some centrally irreducible maximal finite subgroup of (C_{Q^{2×2}}(N))^{∗}. The commuting algebra of N is an indefinite quaternion algebra with center ℚ[\sqrt{5}] ramified at those primes on which Q ramifies. The classification of finite subgroups of GL_{2}(ℂ) ([Bli 17]) now shows that C_{Q^{2×2}}(ℚN) is isomorphic to ℚ[\sqrt{5}]^{2×2}. Moreover C_{G}(N) is one of^{±}D_{10},^{±}S_{3}, or D_{8}. One computes that the two groups _{\sqrt{5},∞}[SL_{2}(5)⊗ \sqrt{5} D_{10}]_{2} and _{\sqrt{5},∞}[SL_{2}(5)]_{1} ⊗ A_{2} are maximal finite whereas the third group is a proper subgroup of the imprimitive maximal finite group _{\sqrt{5},∞}[SL_{2}(5)]_{1}^{2}.

If N = SL_{2}(9), the enveloping algebra \overline{ℚN} of N is Q_{∞,3}^{2×2}. Therefore the centralizer C := C_{G}(N) has to be contained in the center of GL_{2}(Q) ≅ K^{∗}. Since K is totally real, one has C = ^{±}1. The factor group G/N is a subgroup of Glide(N) = C_{2} and [CCNPW 85] implies that Q = Q_{\sqrt{3},∞} and G = N.2 = _{\sqrt{3},∞}[2.S_{6}]_{2}.

Theorem 12.6. Let G be an absolutely irreducible maximal finite subgroup of GL_{2}(Q), where Q is a totally definite quaternion algebra with center K and [K : ℚ] = 2. Assume that G has no quasi-semi-simple normal subgroup. Then Q is isomorphic to Q_{\sqrt{2},∞}, Q_{\sqrt{2},∞,2,3}, Q_{\sqrt{3},∞}, Q_{\sqrt{5},∞,2,5}, Q_{\sqrt{5},∞,5,3}, or Q_{\sqrt{6},∞} and G is conjugate to one of the primitive groups

\sqrt{2},∞[2_{−}^{1+4}.S_{5}]_{2}, _{\sqrt{2},∞}[S̃_{4}]_{1} ⊗ A_{2}, _{\sqrt{2},∞,2,3}[C_{3} ⊃ × ⊏^{2(2+\sqrt{2})}D_{16}]_{2},

\sqrt{2},∞,2,3[C_{3} ⊃ ⊏^{2(2+\sqrt{2})}Q_{16}]_{2}, _{\sqrt{3},∞}[D_{24} ⊗^{2(2)} SL_{2}(3)]_{2}, _{\sqrt{5},∞,2,5}[C_{5} ⊃ × ⊏^{2(2)}D_{8}],

\sqrt{5},∞,2,5[C_{5} ⊃ × ⊏^{2(2)}SL_{2}(3)], _{\sqrt{5},∞,5,3}[^{±}C_{5} ⊃ × ⊏^{2(3)}S_{3}],

\sqrt{5},∞,5,3[C_{5} ⊃ × ⊏^{2(3)} S̃_{3}], or _{\sqrt{6},∞}[GL_{2}(3) ⊠^{2} C_{3}]_{2},

or to one of the imprimitive groups _{\sqrt{2},∞}[S̃_{4}]_{1}^{2} or _{\sqrt{3},∞}[C_{12}.C_{2}]_{1}^{2}.

Proof. The imprimitive a.i.m.f. groups may be determined with Theorem 6.1 so assume that G is primitive. If p is a prime with O_{p}(G) ≠ 1, then by Corollary 2.4 one has that p ∈ {2, 3, 5}.

Assume first that O_{5}(G) ≠ 1. Then N := O_{5}(G) ≅ C_{5}. The centralizer C := C_{G}(O_{5}(G)) embeds into the commuting algebra C_{Q^{2×2}}(O_{5}(G)) which is either isomorphic to an indefinite quaternion algebra Q′ with center ℚ[ζ_{5}] if K = ℚ[\sqrt{5}] or to K[ζ_{5}] if the center K of Q is not the maximal real subfield of ℚ[ζ_{5}]. In the latter case one finds C = ^{±}C_{5} which contradicts the assumption that G is absolutely irreducible. Therefore K = ℚ[\sqrt{5}]. Since the prime divisors of |G| lie in \{2, 3, 5\} the only finite places, on which Q is ramified contain one of these 3 primes. Therefore, ℚ[ζ_{5}] splits Q and one has Q′ = ℚ[ζ_{5}]^{2×2}. Moreover G contains C of index 2. Hence by Lemma 2.14, C is an absolutely irreducible subgroup of GL_{2}(ℚ[ζ_{5}]). Using the classification of finite subgroups of P GL_{2}(ℂ) in [Bli 17] together with the assumption that G contains no quasi-semi-simple normal subgroup, O_{5}(G) = C_{5} and C = B^{∘}(C), one finds that C is one of C_{5} ⊗ D_{8}, C_{5}⊗ \sqrt{5}_{′} SL_{2}(3), ^{±}C_{5} ⊗ S_{3}, or

C_{5}⊗ \sqrt{5}_{′} S̃_{3}.

If G centralizes C/O_{5}(G), then G is a proper subgroup of one of the three groups involving SL_{2}(5) of Theorem 12.5.

If G induces the nontrivial outer automorphism of C/O_{5}(G), one has 2 = |H^{2}(C_{2}, C_{2})| possible extensions G = C.2, only one of which has real Schur index 1, by Lemma 2.17. One computes that G is

\sqrt{5},∞,2,5[C_{5} ⊃ × ⊏^{2(2)}D_{8}],_{\sqrt{5},∞,2,5}[C_{5} ⊃ × ⊏^{2(2)}SL_{2}(3)],_{\sqrt{5},∞,5,3}[^{±}C_{5} ⊃ × ⊏^{2(3)}S_{3}],_{\sqrt{5},∞,5,3}[C_{5} ⊃ × ⊏^{2(3)} S̃_{3}],

respectively.

Assume now that O_{5}(G) = 1 and O_{3}(G) ≠ 1. Then O_{3}(G) ≅ C_{3}. The centralizer C := C_{G}(O_{3}(G)) embeds into the commuting algebra C_{Q^{2×2}}(O_{3}(G)) which is isomorphic to an indefinite quaternion algebra Q′ over K[ζ_{3}]. Since G is absolutely irreducible and contains C of index 2, Lemma 2.14 implies that C is an absolutely irreducible subgroup of GL_{1}(Q′). Hence Q′ = K[ζ_{3}]^{2×2}. The classification of finite subgroups of P GL_{2}(ℂ) ([Bli 17]) together with the assumption that G is primitive, has no quasi-semi-simple normal subgroup, and satisfies O_{5}(G) = 1, one finds that C is one of C_{3} ⊗D_{16}, C_{3} ⊗QD_{16}, C_{3} ⊗GL_{2}(3), C_{3} ∘Q_{16}, C_{3} ∘S̃_{4}, or O_{2}(C) = C_{4} ∘Q_{8} and C = C_{3}B^{∘}(O_{2}(G)) = C_{3} ⊗ (C_{4} ∘ SL_{2}(3)).2.

If C = C_{3} ⊗ D_{16} or C_{3} ∘ Q_{16}, the normalizer N_{(\overline{ℚC})^{∗}}(C) of C in the unit group of its enveloping algebra contains CK[ζ_{3}]^{∗} of index 2. In the other cases one has a unique outer automorphism of C inducing the Galois automorphism of K[ζ_{3}] over the maximal totally real subfield K of K[ζ_{3}]. Therefore one finds in these cases only two possible extensions G = C.2, only one of which has real Schur index 1.

One concludes that G is one of S̃_{3} ⊗ D_{16}, C_{3} ⊃ × ⊏^{2(2+\sqrt{2})}D_{16}, C_{3} ⊠^{2} QD_{16}, C_{3} ⊠^{2} GL_{2}(3),

C_{3} ⊃ ⊏^{2(2+\sqrt{2})}Q_{16}, S_{3} ⊗Q_{16}, S_{3} ⊗S̃_{4}, respectively C_{3} ⊠^{2}(C_{4} ∘SL_{2}(3)).2. The first group

is a proper subgroup of _{\sqrt{2},∞}[2_{−}^{1+4}.S_{5}]_{2}, the third is contained in the fourth group

\sqrt{6},∞[GL_{2}(3) ⊠^{2} C_{3}]_{2}, the sixth one in the seventh group _{\sqrt{2},∞}[S̃_{4}]_{1} ⊗ A_{2}, and the

last group is _{\sqrt{3},∞}[C_{3} ⊠^{2} (C_{4} ∘ SL_{2}(3)).2]_{2} = _{\sqrt{3},∞}[D_{24} ⊗^{2(2)} SL_{2}(3)]_{2}.

If O_{p}(G) = 1 for all odd primes p, then Proposition 8.9 gives that O_{2}(G) is one of D_{8} ⊗ Q_{8} or D_{8} ⊗ Q_{16}. In the first case G = _{\sqrt{2},∞}[2_{−}^{1+4}.S_{5}]_{2} is maximal finite. In the last case, O_{2}(G) is an absolutely irreducible subgroup of GL_{2}(Q_{\sqrt{2},∞}). Let 𝔐 be the up to conjugacy unique maximal order in Q_{\sqrt{2},∞}. One computes the Bravais group on a normal critical 𝔐O_{2}(G)-lattice (cf. Definition 2.7) to be_{\sqrt{2},∞}[2_{−}^{1+4}.S_{5}]_{2}, contradicting the assumptions on O_{2}(G).

Table 12.7. List of the primitive a.i.m.f. subgroups of GL_{2}(Q) where Q is a totally definite quaternion algebra over a real quadratic number field Z(Q).

lattice L  |Aut(L)| r.i.m.f. supergroups
\sqrt{2},∞[2_{−}^{1+4}.S_{5}]_{2}  2^{8} · 3 · 5 F_{4} ˜⊗F_{4}, E_{8}^{2}
\sqrt{2},∞,2,3[C_{3} ⊃ × ⊏^{2(2+\sqrt{2})}D_{16}]_{2}  2^{5} · 3 (A_{2} ⊗ F_{4})^{2}, A_{2}^{8}
\sqrt{2},∞,2,3[C_{3} ⊃ ⊏^{2(2+\sqrt{2})}Q_{16}]_{2}  2^{5} · 3 F_{4}^{4}, E_{8}^{2}
\sqrt{3},∞[D_{24} ⊗^{2(2)} SL_{2}(3)]_{2}  2^{6} · 3^{2} (A_{2} ⊗ F_{4})^{2}, E_{8}^{2}
\sqrt{3},∞[2.S_{6}]_{2}  2^{5} · 3^{2} · 5 E_{8}^{2}, [(Sp_{4}(3) ∘ C_{3}) ⊃ × ⊏_{\sqrt{-3}}^{2} SL_{2}(3)]_{16}

[SL_{2}(9) ⊗_{∞,3}^{2(3)} SL_{2}(9) : 2]_{16}, F_{4} ˜⊗F_{4}

\sqrt{5},∞[SL_{2}(5)⊗ \sqrt{5} D_{10}]_{2}  2^{4} · 3 · 5^{2} [(SL_{2}(5) ∘ SL_{2}(5)) : 2 ⊃ × ⊏_{\sqrt{5}}^{2} D_{10}]_{16}
\sqrt{5},∞[SL_{2}(5)]_{1} ⊗ A_{2}  2^{4} · 3^{2} · 5 A_{2} ⊗ [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8}, A_{2} ⊗ E_{8}
\sqrt{5},∞,2,5[C_{5} ⊃ × ⊏^{2(2)}D_{8}]  2^{4} · 5 A_{4}^{4}
\sqrt{5},∞,2,5[C_{5} ⊃ × ⊏^{2(2)}SL_{2}(3)]  2^{4} · 3 · 5 E_{8}^{2}, [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8}^{2}
\sqrt{5},∞,5,3[^{±}C_{5} ⊃ × ⊏^{2(3)}S_{3}]  2^{3} · 3 · 5 (A_{2} ⊗ A_{4})^{2}
\sqrt{5},∞,5,3[C_{5} ⊃ × ⊏^{2(3)} S̃_{3}]  2^{3} · 3 · 5 E_{8}^{2}, [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8}^{2}

E_{8}^{2}, [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8}^{2}

(A_{2} ⊗ F_{4})^{2}, F_{4}^{4}, [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8}^{2}

A_{2} ⊗ E_{8}, E_{8}^{2}, [SL_{2}(5) ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+4}.Alt_{5}]_{16}

\sqrt{2},∞[S̃_{4}]_{1} ⊗ A_{2}  2^{5} · 3^{2} (A_{2} ⊗ F_{4})^{2}, A_{2} ⊗ E_{8}

A_{2} ⊗ E_{8}, F_{4} ˜⊗F_{4} (A_{2} ⊗ A_{4})^{2} \sqrt{6},∞[(S_{3} ⊗ SL_{2}(3)).2]_{2}  2^{5} · 3^{2} (A_{2} ⊗ F_{4})^{2}, F_{4} ˜⊗F_{4}

The first column contains representatives G of the conjugacy classes of a.i.m.f. subgroups of GL_{2}(Q), the second the order of the corresponding groups. In the third column the r.i.m.f. supergroups of G that act on a lattice L ∈ Z_{𝔐}(G) for some maximal order of Q are given. There is one line for each conjugacy class of maximal orders in Q which come in the same order as in Table 4.1.

Theorem 12.8. M_{2}^{irr}(Q_{\sqrt{2},∞}) is as follows.

[figure: _{\sqrt{2},∞}[2_{−}^{1+4}.S_{5}]_{2}; _{\sqrt{2},∞}[S̃_{4}]_{1}^{2}; _{\sqrt{2},∞}[S̃_{4}]_{1} ⊗ A_{2}]

List of the maximal simplices in M_{2}^{irr}(Q_{\sqrt{2},∞})
simplex  a common subgroup
(_{\sqrt{2},∞}[2_{−}^{1+4}.S_{5}]_{2},_{\sqrt{2},∞}[S̃_{4}]_{1}^{2}) Q_{16} ⊗ D_{8}

Proof. The completeness of the list of a.i.m.f. subgroups in GL_{2}(Q_{\sqrt{2},∞}) follows from Theorems 6.1, 12.5 and 12.6. To see that the list of maximal simplices in M_{2}^{irr}(Q_{\sqrt{2},∞}) is complete one has to note that the unique minimal absolutely irreducible subgroup of _{\sqrt{2},∞}[S̃_{4}]_{1} ⊗ A_{2} is Q_{16} ⊗ S_{3} and not contained in one of the other a.i.m.f. groups.

Theorem 12.9. M_{2}^{irr}(Q_{\sqrt{2},∞,2,3}) consists of two 0-simplices

[figure: _{\sqrt{2},∞,2,3}[C_{3} ⊃ × ⊏^{2(2+\sqrt{2})}D_{16}]_{2}; _{\sqrt{2},∞,2,3}[C_{3} ⊃ ⊏^{2(2+\sqrt{2})}Q_{16}]_{2}]

Proof. The completeness of the list of a.i.m.f. subgroups in GL_{2}(Q_{\sqrt{2},∞,2,3}) follows from Theorems 6.1, 12.5, and 12.6. Both a.i.m.f. groups are minimal absolutely irreducible whence the theorem follows.

Theorem 12.10. M_{2}^{irr}(Q_{\sqrt{3},∞}) is as follows.

[figure: _{\sqrt{3},∞}[D_{24} ⊗^{2(2)} SL_{2}(3)]_{2}; _{\sqrt{3},∞}[Q_{24}]_{1}^{2}; _{\sqrt{3},∞}[2.S_{6}]_{2}]

List of the maximal simplices in M_{2}^{irr}(Q_{\sqrt{3},∞})
simplex  a common subgroup
(_{\sqrt{3},∞}[D_{24} ⊗^{2(2)} SL_{2}(3)]_{2},_{\sqrt{3},∞}[Q_{24}]_{1}^{2}) D_{24} ⊗ Q_{8}
(_{\sqrt{3},∞}[D_{24} ⊗^{2(2)} SL_{2}(3)]_{2},_{\sqrt{3},∞}[2.S_{6}]_{2}) S̃_{4}.2
(_{\sqrt{3},∞}[Q_{24}]_{1}^{2},_{\sqrt{3},∞}[2.S_{6}]_{2})  (^{±}C_{3} × C_{3}).D_{8}

Proof. The completeness of the list of a.i.m.f. subgroups in GL_{2}(Q_{\sqrt{3},∞}) follows from Theorems 6.1, 12.5, and 12.6. The list of maximal simplices in M_{2}^{irr}(Q_{\sqrt{3},∞}) is complete as one can see by computing the lattices of (^{±}C_{3} × C_{3}).D_{8} and S̃_{4}.2, the minimal absolutely irreducible subgroups of _{\sqrt{3},∞}[2.S_{6}]_{2} of order not divisible by 5.

Remark 12.11. The simplicial complex M_{2}(Q_{\sqrt{2},∞}) contains M_{2}(Q_{∞,2}), in the sense, that for every vertex v ∈ M_{2}(Q_{∞,2}) there is a vertex of v′ ∈ M_{2}(Q_{\sqrt{2},∞}) with representatives G_{v} respectively G_{v′}, such that G_{v} ≤ G_{v′} and for every simplex (v_{1}, …, v_{s}) ∈ M_{2}(Q_{∞,2}), the corresponding simplex (v′_{1}, …, v′_{s}) is a simplex in M_{2}(Q_{\sqrt{2},∞}).

In this sense the simplicial complex M_{2}(Q_{\sqrt{3},∞}) contains M_{2}(Q_{∞,3}).

Theorem 12.12. M_{2}^{irr}(Q_{\sqrt{5},∞}) is as follows.

[figure: _{\sqrt{5},∞}[SL_{2}(5)⊗ \sqrt{5} D_{10}]_{2}; _{\sqrt{5},∞}[SL_{2}(5)]_{1}^{2}; _{\sqrt{5},∞}[SL_{2}(5)]_{1} ⊗ A_{2}]

List of the maximal simplices in M_{2}^{irr}(Q_{\sqrt{5},∞})
simplex  a common subgroup
(_{\sqrt{5},∞}[SL_{2}(5)⊗ \sqrt{5} D_{10}]_{2},_{\sqrt{5},∞}[SL_{2}(5)]_{1}^{2}) Q_{20}⊗ \sqrt{5} D_{10}

Proof. The completeness of the list of a.i.m.f. subgroups in GL_{2}(Q_{\sqrt{5},∞}) follows from Theorems 6.1, 12.5, and 12.6. To see that the list of maximal simplices in M_{2}^{irr}(Q_{\sqrt{5},∞}) is complete one has to note that the unique minimal absolutely irreducible subgroup of _{\sqrt{5},∞}[SL_{2}(5)]_{1} ⊗ A_{2} is Q_{20} ⊗ S_{3} and not contained in one of the other a.i.m.f. groups.

Theorem 12.13. M_{2}^{irr}(Q_{\sqrt{5},∞,2,5}) consists of two 0-simplices

[figure: _{\sqrt{5},∞,2,5}[C_{5} ⊃ × ⊏^{2(2)}D_{8}]_{2}; _{\sqrt{5},∞,2,5}[C_{5} ⊃ × ⊏^{2(2)}SL_{2}(3)]_{2}]

Proof. The completeness of the list of a.i.m.f. subgroups in GL_{2}(Q_{\sqrt{5},∞,2,5}) follows from Theorems 6.1, 12.5, and 12.6. The completeness of the list of maximal sim-

plices in M_{2}^{irr}(Q_{\sqrt{5},∞,2,5}) follows from the fact that the group _{\sqrt{5},∞,2,5}[C_{5} ⊃ × ⊏^{2(2)}D_{8}]_{2} is minimal absolutely irreducible.

Similarly one gets:

Theorem 12.14. M_{2}^{irr}(Q_{\sqrt{5},∞,5,3}) consists of two 0-simplices

[figure: _{\sqrt{5},∞,5,3}[^{±}C_{5} ⊃ × ⊏^{2(2)}S_{3}]_{2}; _{\sqrt{5},∞,5,3}[C_{5} ⊃ × ⊏^{2(2)} S̃_{3}]_{2}]

Z(Q) real cubic.

Theorem 12.15. Let Q be a definite quaternion algebra with center K of degree 3 over ℚ and G a primitive a.i.m.f. subgroup of GL_{2}(Q). Then G is one of the groups in the following table, which is built up as in Table 12.7:

List of the primitive a.i.m.f. subgroups of GL_{2}(Q).

lattice L  |Aut(L)| r.i.m.f. supergroups
θ_{9},∞,3[C_{9} ⊃ × ⊏_{\sqrt{-3}}^{2(2)}SL_{2}(3)]_{2} 2^{4} · 3^{3} F_{4}^{6}, [3_{+}^{1+2} : SL_{2}(3) ⊠_{\sqrt{-3}}^{2} SL_{2}(3)]_{12}^{2}
θ_{9},∞,2[D_{18} ⊗ SL_{2}(3)]_{2} 2^{4} · 3^{3} (A_{2} ⊗ F_{4})^{3} , F_{4} ⊗ E_{6}

F_{4} ⊗ E_{6}

θ_{9},∞,2[C_{9} ⊃ × ⊏^{2(2)}D_{8}]_{2}  2^{4} · 3^{2} E_{8}^{2}, [Sp_{4}(3) ⊠_{\sqrt{-3}}^{2} 3_{+}^{1+2} : SL_{2}(3)]_{24}

(A_{2}^{12}), (E_{6}^{4})

θ_{7},∞,7[Q_{28}]_{1} ⊗ A_{2}  2^{3} · 3 · 7 (A_{6} ⊗ A_{2})^{2}, (A_{6}^{(2)} ⊗ A_{2})^{2}
θ_{7},∞,7[C_{7} ⊃ × ⊏_{\sqrt{-7}}^{2(3)} S̃_{3}]_{2}  2^{3} · 3 · 7 [6.U_{4}(3).2^{2}]_{12}^{2}
θ_{7},∞,7[C_{7} ⊃ × ⊏^{2(2)}D_{8}]_{2}  2^{4} · 7 [L_{2}(7) ⊗^{2(2)} D_{8}]_{12}^{2}, [L_{2}(7) ⊃ × ⊏^{2(2)}D_{8}]_{12}^{2}
θ_{7},∞,2[D_{14} ⊗ SL_{2}(3)]_{2} 2^{4} · 3 · 7 A_{6} ⊗ F_{4}, A_{6}^{(2)} ⊗ F_{4}
θ_{7},∞,2[C_{7} ⊃ × ⊏^{2(2)}SL_{2}(3)]_{2} 2^{4} · 3 · 7 [L_{2}(7) ⊗^{2(2)} F_{4}]_{24}, [L_{2}(7) ⊃ × ⊏^{2(2)}F_{4}]_{24}
θ_{7},∞,3[^{±}C_{7} ⊃ × ⊏^{2(3)}S_{3}]_{2}  2^{3} · 3 · 7 [6.U_{4}(3).2^{2}]_{12}^{2}

((A_{2} ⊗ A_{6})^{2}), ((A_{2} ⊗ A_{6}^{(2)})^{2})

θ_{7},∞,3[D_{14} ⊗ S̃_{3}]_{2}  2^{3} · 3 · 7 (A_{2} ⊗ A_{6})^{2}, (A_{2} ⊗ A_{6}^{(2)})^{2}

([6.U_{4}(3).2^{2}]_{12}^{2})

ω_{13},∞,13[^{±}C_{13}.C_{4}]_{2}  2^{3} · 13 (A_{12}^{2})

[2.Co_{1}]_{24} , [SL_{2}(13) ⊃ ⊏^{2(2)}SL_{2}(3)]_{24}

Proof. Let Q be a definite quaternion algebra with center K of degree 3 over ℚ and G a primitive a.i.m.f. subgroup of GL_{2}(Q). Then K is contained in a cyclotomic field of degree ≤ 12, hence K ≅ ℚ[θ_{7}], ℚ[θ_{9}], or ℚ[ω_{13}], where ω_{13} is a generator of the subfield of degree 3 over ℚ of the cyclotomic field ℚ[ζ_{13}] and the θ_{i} = ζ_{i} + ζ_{i}^{−1} generate the maximal real subfield of ℚ[ζ_{i}] (cf. Notation 4.2). By [CCNPW 85], G has no quasi-semi-simple normal subgroup. If K = ℚ[ω_{13}], then 13 divides |G|. One concludes that O_{13}(G) ≅ C_{13} and G = _{ω_{13},∞,13}[^{±}C_{13}.C_{4}]_{2}. Now assume that K = ℚ[θ_{7}]. Then 7 divides the order of G. Since the possible normal 2- and 3subgroups have no automorphism of order 7, one has O_{7}(G) ≅ C_{7}. The centralizer C := C_{G}(C_{7}) is a centrally irreducible subgroup of GL_{1}(D) for a quaternion algebra D with center ℚ[ζ_{7}]. One only has the possibilities D = ℚ[ζ_{7}]^{2×2} and C = ^{±}C_{7} ×U, where U is one of D_{8}, S_{3}, or S̃_{3}, or D = Q_{ζ_{7},2}, where C = C_{7} ⊗ SL_{2}(3). Since |Out(^{±}U)| = 2, one has in each case 2 possibilities for G = C.2, where there is always a unique extension yielding a representation with real Schur index 2 (cf. Lemma 2.17). Since Q_{28} ⊗ D_{8} is imprimitive, one finds the groups of the proposition. The case K = ℚ[θ_{9}] is dealt with analogously.

Corollary 12.16. Let Q be a definite quaternion algebra with center K of degree 3 over ℚ and G an a.i.m.f. subgroup of GL_{2}(Q). Then Q is one of Q_{θ_{9},∞,3}, Q_{θ_{9},∞,2}, Q_{θ_{7},∞,7}, Q_{θ_{7},∞,2}, Q_{θ_{7},∞,3}, or Q_{ω_{13},∞,13}. The simplicial complexes M_{2}^{irr}(Q) consist of 0-simplices each.

Proof. Theorems 6.1 and 12.15 prove the completeness of the list of quaternion algebras Q and of a.i.m.f. subgroups of GL_{2}(Q). That there are no common

absolutely irreducible subgroups, may be easily seen, since the groups _{θ_{9},∞,2}[C_{9} ⊃ × ⊏^{2(2)}

D_{8}]_{2}, _{θ_{7},∞,7}[Q_{28}]_{1} ⊗ A_{2}, _{θ_{7},∞,7}[C_{7} ⊃ × ⊏_{\sqrt{-7}}^{2(3)} S̃_{3}]_{2}, _{θ_{7},∞,7}[C_{7} ⊃ × ⊏^{2(2)} D_{8}]_{2}, _{θ_{7},∞,3}[^{±}C_{7} ⊃ × ⊏^{2(3)} S_{3}]_{2},

and_{θ_{7},∞,3}[D_{14}⊗S̃_{3}]_{2} are minimal absolutely irreducible and the minimal absolutely

subgroups of _{θ_{7},∞,2}[D_{14} ⊗ SL_{2}(3)]_{2} resp. _{θ_{7},∞,2}[C_{7} ⊃ × ⊏^{2(2)}SL_{2}(3)]_{2} are D_{14} ⊗ Q_{8} resp.

C_{7} ⊃ × ⊏^{2(2)}Q_{8} and not isomorphic.

Let 𝔐 be a maximal order of Q_{θ_{9},∞,3} and U a minimal absolutely irreducible

subgroup of _{θ_{9},∞,3}[C_{9} ⊃ × ⊏_{\sqrt{-3}}^{2(2)} SL_{2}(3)]_{2}. Then U/O_{2}(U) ≅ D_{18}. Hence the 2-modular

constituents of the natural representation of U ⊗_{θ_{9}} 𝔐 ≤ GL_{24}(ℚ) are of degree 12, so U cannot fix a 2-modular and a 2-unimodular lattice. So there is no common

absolutely irreducible subgroup of _{θ_{9},∞,3}[Q_{36}]_{1}^{2} and _{θ_{9},∞,3}[C_{9} ⊃ × ⊏_{\sqrt{-3}}^{2(2)}SL_{2}(3)]_{2}.

Z(Q) real quartic.

Theorem 12.17. Let Q be a definite quaternion algebra with center K of degree 4 over ℚ and G a primitive a.i.m.f. subgroup of GL_{2}(Q). Then G is one of the groups in the following table:

148 GABRIELE NEBE

List of the primitive a.i.m.f. subgroups of GL_{2}(Q), where Q is a definite

quaternion algebra with center K and [K : ℚ] = 4. θ_{16},∞[Q_{32}]_{1} ⊗ A_{2} ( 2^{6} · 3)

O_{1} (A_{2}^{16})
O_{2} (F_{4} ⊗ A_{2})^{4}, (E_{8} ⊗ A_{2})^{2}

θ_{16},∞[S̃_{4}⊗ \sqrt{2} D_{32}]_{2} (2^{8} · 3)

O_{1} (F_{4}^{8})
O_{2} (F_{4} ˜⊗F_{4})^{2}, E_{8}^{4}

θ_{24},∞[S̃_{4}⊗ \sqrt{2} D_{48}]_{2} (2^{7} · 3^{2})

O_{1} (A_{2} ⊗ F_{4})^{4}, (A_{2} ⊗ E_{8})^{2} (F_{4} ˜⊗F_{4})^{2}, E_{8}^{4}
O_{2} (E_{8}^{4}), ((A_{2} ⊗ F_{4})^{4})
O_{3} (F_{4} ˜⊗F_{4})^{2}, (A_{2} ⊗ E_{8})^{2}, A_{2} ⊗ F_{4} ˜⊗F_{4}, [2_{+}^{1+10}.O_{10}^{+}(2)]_{32}

θ_{20},∞[Q_{40}]_{1} ⊗ A_{2} (2^{4} · 3 · 5)

O_{1} (A_{2} ⊗ A_{4})^{4}, (A_{2} ⊗ [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8})^{2}, (A_{2} ⊗ E_{8})^{2}
O_{2} A_{2} ⊗ A_{4} ⊗ F_{4}, A_{2} ⊗ F_{4} ˜⊗F_{4},
O_{3} (A_{2} ⊗ [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8})^{2}, (A_{2} ⊗ E_{8})^{2},

A_{2} ⊗ [SL_{2}(5) ∘ SL_{2}(5) : 2 ⊠_{\sqrt{5}}^{2} D_{10}]_{16}

θ_{20},∞[SL_{2}(5)⊗ \sqrt{5} D_{40}]_{2} (2^{5} · 3 · 5^{2})

O_{1} E_{8}^{4}, [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8}^{4}, [SL_{2}(5) ∘ SL_{2}(5) : 2 ⊠_{\sqrt{5}}^{2} D_{10}]_{16}^{2}
O_{2} E_{8} ⊗ F_{4}, [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8} ⊗ F_{4}, [2_{−}^{1+4}.Alt_{5}⊗_{∞,2} SL_{2}(5) ⊠_{\sqrt{5}}^{2(2)} D_{10}]_{32}
O_{3} [((SL_{2}(5) ∘ SL_{2}(5)) ⊠_{\sqrt{5}}^{2} (SL_{2}(5) ∘ SL_{2}(5))) : S_{4}]_{32,i} (i = 1, 2)

[SL_{2}(5) ∘ SL_{2}(5) : 2 ⊠_{\sqrt{5}}^{2} D_{10}]_{16}^{2}

θ_{20},∞[C_{5} ⊠^{2} (C_{4} ∘ SL_{2}(3).2)]_{2} (2^{6} · 3 · 5)

O_{1} (F_{4} ˜⊗F_{4})^{2}, (A_{4} ⊗ F_{4})^{2}, [SL_{2}(5) ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+4}.Alt_{5}]_{16}^{2}
O_{2} A_{4} ⊗ E_{8}, [2_{+}^{1+10}.O_{10}^{+}(2)]_{32}, [SL_{2}(5) ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+6}.O_{6}^{−}(2)]_{32}
O_{3} E_{8} ⊗ F_{4}, [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8} ⊗ F_{4}, [2_{−}^{1+4}.Alt_{5}⊗_{∞,2} SL_{2}(5) ⊠_{\sqrt{5}}^{2(2)} D_{10}]_{32}

θ_{15},∞[SL_{2}(5)⊗ \sqrt{5} D_{30}]_{2} (2^{4} · 3^{2} · 5^{2})

O_{1} [SL_{2}(5) ∘ SL_{2}(5) : 2 ⊠_{\sqrt{5}}^{2} D_{10}]_{16}^{2}, (A_{2} ⊗ E_{8})^{2},

(A_{2} ⊗ [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8})^{2}

O_{2} A_{2} ⊗ [SL_{2}(5) ∘ SL_{2}(5) : 2 ⊠_{\sqrt{5}}^{2} D_{10}]_{16},

[((SL_{2}(5) ∘ SL_{2}(5)) ⊠_{\sqrt{5}}^{2} (SL_{2}(5) ∘ SL_{2}(5))) : S_{4}]_{32,i} (i = 1, 2)

θ_{15},∞[C_{15} ⊃ × ⊏_{\sqrt{-3}}^{2(2)}SL_{2}(3)]_{2} (2^{4} · 3^{2} · 5)

O_{1} (A_{4} ⊗ F_{4})^{2}, [SL_{2}(5) ⊃ × ⊏_{∞,3}^{2(3)}(SL_{2}(3) □^{2} C_{3})]_{16}^{2}, [Sp_{4}(3) ∘ C_{3} ⊠_{\sqrt{-3}}^{2} SL_{2}(3)]_{16}^{2}
O_{2} E_{8} ⊗ F_{4}, F_{4} ⊗ [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8},

[(SL_{2}(5)⊗ \sqrt{5} D_{10}) ⊃ × ⊏_{∞,3}^{2(3)}(SL_{2}(3) □^{2} C_{3})]_{32}

θ_{15},∞[C_{15} ⊃ × ⊏^{2(2)}D_{8}]_{2} (2^{4} · 3 · 5)

O_{1} (F_{4} ˜⊗F_{4})^{2}, [D_{120}.(C_{4} × C_{2})]_{16}^{2}, [SL_{2}(5) ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+4}.Alt_{5}]_{16}^{2}
O_{2} [(2_{−}^{1+4}.Alt_{5}⊗_{∞,2} SL_{2}(5)) ⊠_{\sqrt{5}}^{2(2)} D_{10}]_{32}

[((SL_{2}(5) ∘ SL_{2}(5)) : 2 ⊃ × ⊏_{\sqrt{5}}^{2(6)}(C_{3} ⊃ × ⊏^{2(2)}D_{8})]_{32,i} (i = 1, 2)

η_{17},∞[^{±}C_{17}.C_{4}]_{2} (2^{3} · 17)

O_{1} [SL_{2}(17) ⊃ ⊏^{2(3)} S̃_{3}]_{32,i} (i = 1, 2), [SL_{2}(17)^{2(3)} ∘ S̃_{3}]_{32}
O_{2} [2_{+}^{1+10}.O_{10}^{+}(2)]_{32}

\sqrt{2},∞[S̃_{4}]_{1} ⊗ \sqrt{5}[^{±}D_{10}]_{2} (2^{5} · 3 · 5)

O_{1} (A_{4} ⊗ F_{4})^{2}, A_{4} ⊗ E_{8}, [C_{15} : C_{4} ⊃ × ⊏^{2(2)}F_{4}]_{32}
O_{2} [SL_{2}(5) ∘ SL_{2}(5) : 2 ⊠_{\sqrt{5}}^{2} D_{10}]_{16}^{2}, [(2_{−}^{1+4}.Alt_{5}⊗_{∞,2} SL_{2}(5)) ⊠_{\sqrt{5}}^{2(2)} D_{10}]_{32},

[SL_{2}(9) ⊗ D_{10} □^{2} SL_{2}(5)]_{32}, [(SL_{2}(5)⊗ \sqrt{5} D_{10}) ⊃ × ⊏_{∞,3}^{2(3)}(SL_{2}(3) □^{2} C_{3})]_{32}

\sqrt{5},∞[SL_{2}(5)]_{1} ⊗ \sqrt{2}[D_{16}]_{2} (2^{6} · 3 · 5)

O_{1} [2_{+}^{1+10}.O_{10}^{+}(2)]_{32}, [(Sp_{4}(3)⊗ \sqrt{-3} Sp_{4}(3)) : 2 □^{2} C_{3}]_{32},

[SL_{2}(9).2 ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+4}.Alt_{5}]_{32}, [SL_{2}(5) ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+6}.O_{6}^{−}(2)]_{32},

(F_{4} ˜⊗F_{4})^{2}, [SL_{2}(5) ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+4}.Alt_{5}]_{16}^{2}

O_{2} [((SL_{2}(5) ∘ SL_{2}(5)) : 2 ⊃ × ⊏_{\sqrt{5}}^{2(6)}(C_{3} ⊃ × ⊏^{2(2)}D_{8})]_{32,i} (i = 1, 2), E_{8} ⊗ F_{4},

[(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8}^{4}, [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8} ⊗ F_{4}, E_{8}^{4}

\sqrt{2}+\sqrt{5},∞,2,5[C_{5} ⊃ × ⊏^{2(2)}D_{16}]_{2} (2^{5} · 5)

O_{1} A_{4}^{8}, (A_{4} ⊗ F_{4})^{2}, [D_{120}.(C_{4} × C_{2})]_{16}^{2}
O_{2} ([D_{120}.(C_{4} × C_{2})]_{16}^{2})

\sqrt{2}+\sqrt{5},∞,2,5[C_{5} ⊃ × ⊏^{2(2)}Q_{16}]_{2} (2^{5} · 5)

O_{1} [SL_{2}(5) ⊃ × ⊏_{∞,3}^{2(3)}SL_{2}(9)]_{16}^{2}, [SL_{2}(9) ⊗_{∞,3}^{2(3)} SL_{2}(9)]_{16}^{2}, [SL_{2}(5) ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+4}.Alt_{5}]_{16}^{2},

[Sp_{4}(3) ∘ C_{3} ⊠_{\sqrt{-3}}^{2} SL_{2}(3)]_{16}^{2}, [SL_{2}(5) ⊃ × ⊏_{∞,3}^{2(3)}(SL_{2}(3) □^{2} C_{3})]_{16}^{2},

E_{8}^{4}, (F_{4} ˜⊗F_{4})^{2}, [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8}^{4}

O_{2} ([Sp_{4}(3) ∘ C_{3} ⊠_{\sqrt{-3}}^{2} SL_{2}(3)]_{16}^{2})

η_{40},∞[C_{5} ⊠^{2} QD_{16}]_{2} (2^{5} · 5)

O_{1} A_{4}^{8}, (A_{4} ⊗ F_{4})^{2}, (F_{4} ˜⊗F_{4})^{2}, E_{8}^{4}, [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8}^{4},

[SL_{2}(5) ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+4}.Alt_{5}]_{16}^{2}

O_{2} [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8}^{4}, E_{8}^{4}, F_{4} ⊗ [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8},

F_{4} ⊗ E_{8}, [SL_{2}(5) ∘ SL_{2}(5) : 2 ⊠_{\sqrt{5}}^{2} D_{10}]_{16}^{2},

[(2_{−}^{1+4}.Alt_{5}⊗_{∞,2} SL_{2}(5)) ⊠_{\sqrt{5}}^{2(2)} D_{10}]_{32}

O_{3} (A_{4}^{8}), ([(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8}^{4}), (E_{8}^{4})
O_{4} F_{4} ⊗ [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8}, F_{4} ⊗ E_{8}, [2_{+}^{1+10}.O_{10}^{+}(2)]_{32}

[(2_{−}^{1+4}.Alt_{5}⊗_{∞,2} SL_{2}(5)) ⊠_{\sqrt{5}}^{2(2)} D_{10}]_{32}, [SL_{2}(5) ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+6}.O_{6}^{−}(2)]_{32}

O_{5} [2_{+}^{1+10}.O_{10}^{+}(2)]_{32}, (A_{4} ⊗ F_{4})^{2}, [SL_{2}(5) ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+6}.O_{6}^{−}(2)]_{32}

A_{4} ⊗ E_{8}, (F_{4} ˜⊗F_{4})^{2}, [SL_{2}(5) ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+4}.Alt_{5}]_{16}^{2}

O_{6} A_{4} ⊗ E_{8}, (F_{4} ˜⊗F_{4})^{2}, [SL_{2}(5) ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+4}.Alt_{5}]_{16}^{2}

[2_{+}^{1+10}.O_{10}^{+}(2)]_{32}, (A_{4} ⊗ F_{4})^{2}, [SL_{2}(5) ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+6}.O_{6}^{−}(2)]_{32}

\sqrt{5},∞[SL_{2}(5)]_{1} ⊗ \sqrt{3}[D_{24}]_{2} (2^{5} · 3^{2} · 5)

O_{1} (A_{2} ⊗ E_{8})^{2}, (A_{2} ⊗ [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8})^{2},

F_{4} ⊗ [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8},

[((SL_{2}(5) ∘ SL_{2}(5)) ⊠_{\sqrt{5}}^{2} (SL_{2}(5) ∘ SL_{2}(5))) : S_{4}]_{32,i} (i = 1, 2),

F_{4} ⊗ E_{8}, [((SL_{2}(5) ∘ SL_{2}(5)) : 2 ⊃ × ⊏_{\sqrt{5}}^{2(6)}(C_{3} ⊃ × ⊏^{2(2)}D_{8})]_{32,i} (i = 1, 2)

O_{2} F_{4} ⊗ E_{8}, F_{4} ⊗ [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8}, A_{2} ⊗ F_{4} ˜⊗F_{4}, [2_{+}^{1+10}.O_{10}^{+}(2)]_{32},

A_{2} ⊗ [SL_{2}(5) ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+4}.Alt_{5}]_{16}, [SL_{2}(5) ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+6}.O_{6}^{−}(2)]_{32},

[(Sp_{4}(3)⊗ \sqrt{-3} Sp_{4}(3)) : 2 □^{2} C_{3}]_{32}, [SL_{2}(5) ⊃ × ⊏_{∞,3}^{2(3)}(Sp_{4}(3) □^{2} C_{3})]_{32}

O_{3} (A_{2} ⊗ E_{8})^{2}, (A_{2} ⊗ [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8})^{2}, F_{4} ˜⊗F_{4}^{2},

[SL_{2}(5) ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+4}.Alt_{5}]_{16}^{2}, E_{8}^{4},

[(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8}^{4},

[SL_{2}(5) ⊃ × ⊏_{∞,3}^{2(3)}(SL_{2}(3) □^{2} C_{3})]_{16}^{2}, [Sp_{4}(3) ∘ C_{3} ⊠_{\sqrt{-3}}^{2} SL_{2}(3)]_{16}^{2}

O_{4} [SL_{2}(5) ∘ C_{5} ⊃ × ⊏^{2(3)}D_{24}]_{32}

\sqrt{3},∞[Q_{24}]_{1} ⊗ \sqrt{5}[^{±}D_{10}]_{2} (2^{4} · 3 · 5)

O_{1} [(2_{−}^{1+4}.Alt_{5}⊗_{∞,2} SL_{2}(5)) ⊠_{\sqrt{5}}^{2(2)} D_{10}]_{32}, [(SL_{2}(5)⊗ \sqrt{5} D_{10}) ⊃ × ⊏_{∞,3}^{2(3)}(SL_{2}(3) □^{2} C_{3})]_{32},

[SL_{2}(5) ∘ SL_{2}(5) : 2 ⊠_{\sqrt{5}}^{2} D_{10}]_{16}^{2}, A_{2} ⊗ [SL_{2}(5) ∘ SL_{2}(5) : 2 ⊠_{\sqrt{5}}^{2} D_{10}]_{16}

O_{2} A_{4} ⊗ E_{8}, A_{2} ⊗ A_{4} ⊗ F_{4}, [C_{15} : C_{4} ⊃ × ⊏^{2(2)}F_{4}]_{32},

[(2_{−}^{1+4}.Alt_{5}⊗_{∞,2} SL_{2}(5)) ⊠_{\sqrt{5}}^{2(2)} D_{10}]_{32}

O_{3} (A_{2} ⊗ A_{4})^{4}, (F_{4} ⊗ A_{4})^{2}, [D_{120}.(C_{4} × C_{2})]_{16}^{2},

[SL_{2}(5) ∘ SL_{2}(5) : 2 ⊠_{\sqrt{5}}^{2} D_{10}]_{16}^{2}

O_{4} ([D_{120}.2]_{16,1}^{2})

\sqrt{3}+\sqrt{5},∞[C_{5} ⊃ × ⊏^{2}D_{24}]_{2} (2^{4} · 3 · 5)

O_{1} [SL_{2}(5) ∘ C_{5} ⊃ × ⊏^{2(3)}D_{24}]_{32}
O_{2} [SL_{2}(3)^{2(2)} ∘ C_{5} ⊃ × ⊏^{2(3)}D_{24}]_{32}
O_{3} ([D_{120}.2]_{16,1}^{2})
O_{4} [((SL_{2}(5) ∘ SL_{2}(5)) ⊠_{\sqrt{5}}^{2} (SL_{2}(5) ∘ SL_{2}(5))) : S_{4}]_{32,1},

[((SL_{2}(5) ∘ SL_{2}(5)) : 2 ⊃ × ⊏_{\sqrt{5}}^{2(6)}(C_{3} ⊃ × ⊏^{2(2)}D_{8})]_{32,1}

\sqrt{3}+\sqrt{5},∞[C_{5} ⊃ × ⊏_{\sqrt{5}′}^{2} Q_{24}]_{2} (2^{4} · 3 · 5)

O_{1} [SL_{2}(5) ∘ C_{5} ⊃ × ⊏_{\sqrt{5}′}^{2(3)}Q_{24}]_{32}
O_{2} [SL_{2}(3)^{2(2)} ∘ (C_{5} ⊃ × ⊏_{\sqrt{5}′}^{2(3)}Q_{24})]_{32}
O_{3} ([D_{120}.2]_{16,2}^{2})
O_{4} ([(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8}^{4}), ((F_{4} ˜⊗F_{4})^{2})

η_{48},∞[C_{3} ⊠^{2} QD_{32}]_{2} (2^{6} · 3)

O_{1} A_{2}^{16}, (A_{2} ⊗ F_{4})^{4}, F_{4}^{8}, E_{8}^{4}
O_{2} (F_{4}^{8}), (A_{2}^{16})
O_{3} (A_{2}^{16}), (F_{4}^{8}), (E_{8}^{4})
O_{4} E_{8}^{4}, (F_{4} ˜⊗F_{4})^{2}, (A_{2} ⊗ F_{4})^{4}, (A_{2} ⊗ E_{8})^{2}
O_{5} (A_{2}^{16}), (F_{4}^{8})
O_{6} (A_{2}^{16}), [2_{+}^{1+10}.O_{10}^{+}(2)]_{32}
O_{7} (A_{2} ⊗ E_{8})^{2}, (A_{2} ⊗ F_{4})^{4}, (F_{4} ˜⊗F_{4})^{2}, E_{8}^{4}
O_{8} [2_{+}^{1+10}.O_{10}^{+}(2)]_{32}
O_{9} (A_{2}^{16}), (F_{4}^{8})
O_{10} (A_{2}^{16}), (F_{4}^{8})

In the first line of each box the a.i.m.f. group G and its order is given. In the next lines some r.i.m.f. supergroups fixing a G-lattice with maximal order as endomorphism ring are given. If I did not find such groups, at least one r.i.m.f. supergroup of G is specified in brackets. If there is more than one isomorphism class of maximal orders in Q they are listed in the following lines, headed by the symbols O_{1}, O_{2}, … to distinguish the different ℤ-isomorphism classes of maximal orders, in the same order as they are displayed in Table 4.1.

Proof. Let Q be a definite quaternion algebra with center K of degree 4 over ℚ and G a primitive a.i.m.f. subgroup of GL_{2}(Q). Then K is contained in a cyclotomic field of degree 8 or 16 over ℚ. If K is the maximal real subfield of a cyclotomic field of degree 8 over ℚ, then K is one of ℚ[θ_{15}], ℚ[θ_{20}], ℚ[θ_{24}], or ℚ[θ_{16}].

Assume first that K = ℚ[θ_{15}]. If G contains a quasi-semi-simple normal subgroup, then SL_{2}(5) ⊴ G. The centralizer C := C_{G}(SL_{2}(5)) embeds into the commuting algebra D := C_{Q^{2×2}}(SL_{2}(5)), which is an indefinite quaternion algebra with center K. Since SL_{2}(5) is primitively saturated over K the group G is of the form G = SL_{2}(5)C. Hence C is a maximal finite subgroup of D^{∗} and the enveloping ℚ[\sqrt{5}]-algebra \overline{ℚ[\sqrt{5}]C} of C is D. By the classification of finite subgroups of P GL_{2}(ℂ) in [Bli 17], this implies that D = K^{2×2} and C = ^{±}D_{30}. Hence G = _{θ_{15},∞}[SL_{2}(5)⊗ \sqrt{5} D_{30}]_{2} in this case. Now assume that G does not contain a quasi-semi-simple normal subgroup. Since K is the character field of the natural character of G, an inspection of the relevant groups in Table 8.7 yields that O_{5}(G) > 1. Hence O_{5}(G) ≅ C_{5} and G contains the normal subgroup N := B_{K}^{∘}(O_{5}(G)) = ^{±}C_{15}. The centralizer C_{G}(N) = B^{∘}(C_{G}(N)) is a centrally irreducible subgroup of (Q ⊗_{K} ℚ[ζ_{15}])^{∗} and G/C_{G}(N) ≅ C_{2}. By Theorem 8.1 C_{G}(N) is either C_{15}⊗ \sqrt{-3} SL_{2}(3) or C_{15} ⊗ D_{8}. In each case there are two possible automorphisms. Since the group Q_{60} ⊗ D_{8} is imprimitive and D_{30} ⊗ SL_{2}(3) em-

beds into _{θ_{15},∞}[SL_{2}(5)⊗ \sqrt{5} D_{30}]_{2} one finds that G is one of _{θ_{15},∞}[C_{15} ⊃ × ⊏^{2(2)}SL_{2}(3)]_{2} or

θ_{15},∞[C_{15} ⊃ × ⊏^{2(2)}D_{8}]_{2}.

The case where K = ℚ[θ_{20}] is similar. If G contains a quasi-semi-simple normal subgroup, one easily concludes that G = _{θ_{20},∞}[SL_{2}(5)⊗ \sqrt{5} D_{40}]_{2}. If G has no quasisemi-simple normal subgroup, then as above, O_{5}(G) ≅ C_{5} and N := B_{K}^{∘}(O_{5}(G)) = C_{20} ⊴ G. Let C := C_{G}(N).

If O_{2}(G) > C_{4}, then O_{2}(C) = C_{4} ⊗ D_{8} ≅ C_{4} ∘ Q_{8} and C = B_{K}^{∘}(C) = C_{5} ⊗ (C_{4} ∘ SL_{2}(3).2). There is only one possible automorphism and therefore G =

θ_{20},∞[C_{5} ⊠^{2} (C_{4} ∘ SL_{2}(3).2)]_{2} in this case.

If O_{3}(C) > 1, one has to remark that Q_{40} ⊗ S_{3} ≅ C_{20} ⊃ × ⊏_{\sqrt{-1}}^{2(3)} S̃_{3} and D_{40} ⊗ S̃_{3} ≅

C_{20} ⊃ × ⊏^{2(3)} S_{3}. Since the last group is a subgroup of _{θ_{20},∞}[SL_{2}(5)⊗ \sqrt{5} D_{40}]_{2}, one finds that G is _{θ_{20},∞}[Q_{40}]_{1} ⊗ A_{2} in this case.

In the last two cases, K does not contain a subfield ℚ[\sqrt{5}]. Since K is the character field of the natural character of G, one finds that O_{5}(G) = 1 and that G does not contain a quasi-semi-simple normal subgroup.

If K = ℚ[θ_{24}] clearly O_{3}(G) ≠ 1. Hence O_{3}(G) = C_{3} and G contains a normal subgroup N := B_{K}^{∘}(O_{3}(G)) = C_{24}. One concludes that O_{2}(G) = C_{8} ∘ Q_{8} and C_{G}(N) = B_{K}^{∘}(C_{24}⊗ \sqrt{-1} Q_{8}) = C_{48}⊗ \sqrt{2} S̃_{4}. Now S̃_{4} is primitively saturated over K

and therefore G = _{θ_{24},∞}[S̃_{4}⊗ \sqrt{2} D_{48}]_{2} In the last case K = ℚ[θ_{16}]. If O_{3}(G) > 1, then O_{3}(G) ≅ C_{3} and C_{G}(O_{3}(G))

is an absolutely irreducible subgroup of ℚ[\sqrt{-3}] ⊗ Q. Since S̃_{3} ⊗ D_{32} = C_{16} ⊃ × ⊏^{2(3)} S_{3}

embeds into _{θ_{16},∞}[S̃_{4}⊗ \sqrt{2} D_{32}]_{2}, G is _{θ_{16},∞}[Q_{32}]_{1} ⊗ A_{2} = _{θ_{16},∞}[C_{16} ⊃ × ⊏^{2(3)} S̃_{3}]_{2}.

If O_{3}(G) = 1, then O_{2}(G) is a self-centralizing normal subgroup of G and G = B^{∘}(O_{2}(G)) = _{θ_{16},∞}[S̃_{4}⊗ \sqrt{2} D_{32}]_{2} with Proposition 8.9.

Now we consider the case, where K does not embed into a cyclotomic field of degree 8 over ℚ. Since K is real and of degree 4, this implies that K is contained in one of the cyclotomic fields ℚ[ζ_{i}] for i = 17, 40, 60, 48 of degree 16 over ℚ and that K is one of ℚ[η_{17}], ℚ[\sqrt{2}, \sqrt{5}], ℚ[η_{40}], ℚ[\sqrt{3}, \sqrt{5}], or ℚ[η_{48}]. The fields ℚ[η_{i}] denote subfields of ℚ[ζ_{i}] with Gal(ℚ[η_{i}]/ℚ) ≅ C_{4}.

In all cases i divides the exponent of G. If K = ℚ[η_{i}] is a cyclic extension of ℚ, then K is generated by a single character value. So in these cases G contains an element x of order i. Since K is the character field of the natural character of G, the whole Galois group Γ := Gal(ℚ[ζ_{i}]/ℚ[η_{i}]) is induced by conjugation with elements in the normalizer N_{G}(⟨x⟩). Hence G contains the irreducible subgroup ±C_{i}.Γ. Computing the automorphism group of the invariant lattices one gets G =

±C_{i}.Γ is one of _{η_{17},∞}[^{±}C_{17}.C_{4}]_{2}, _{η_{40},∞}[C_{5} ⊠^{2}QD_{16}]_{2}, respectively _{η_{48},∞}[C_{3} ⊠^{2}QD_{32}]_{2}.

Now let K = ℚ[\sqrt{2}, \sqrt{5}]. If G contains a quasi-semi-simple normal subgroup one easily concludes that G = _{\sqrt{5},∞}[SL_{2}(5)]_{1} ⊗ \sqrt{2}[D_{16}]_{2}. Otherwise G contains a normal subgroup N ≅ C_{5}. The centralizer C_{G}(N) is an absolutely irreducible subgroup of GL_{1}(Q ⊗ ℚ[ζ_{5}]) and G contains C_{G}(N) of index 2.

Hence, clearly O_{3}(G) = 1 and by Table 8.7 O_{2}(G) is one of Q_{8}, D_{16}, or Q_{16}. In the first case, G contains the normal subgroup B_{K}^{∘}(O_{2}(G)) = S̃_{4}. This group is primitively saturated over K and therefore G = _{\sqrt{2},∞}[S̃_{4}]_{1} ⊗ \sqrt{5}[^{±}D_{10}]_{2}.

In the other two cases the elements in G − C_{G}(N) may induce two different automorphisms. Since the groups Q_{20} ⊗ D_{16} resp. D_{10} ⊗ Q_{16} are contained in \sqrt{5},∞[SL_{2}(5)]_{1} ⊗ \sqrt{2}[D_{16}]_{2} resp. _{\sqrt{2},∞}[S̃_{4}]_{1} ⊗ \sqrt{5}[^{±}D_{10}]_{2}, one finds that G is one of

\sqrt{2}+\sqrt{5},∞,2,5[C_{5} ⊃ × ⊏^{2(2)}D_{16}]_{2} resp. _{\sqrt{2}+\sqrt{5},∞,2,5}[C_{5} ⊃ × ⊏^{2(2)}Q_{16}]_{2}.

In the case K = ℚ[\sqrt{5}, \sqrt{3}], one analogously gets that G is one of

\sqrt{5},∞[SL_{2}(5)]_{1} ⊗ \sqrt{3}[D_{24}]_{2}, _{\sqrt{3},∞}[Q_{24}]_{1} ⊗ \sqrt{5}[^{±}D_{10}]_{2},

\sqrt{3}+\sqrt{5},∞[C_{5} ⊃ × ⊏^{2}D_{24} ]_{2}, or _{\sqrt{3}+\sqrt{5},∞}[C_{5} ⊃ × ⊏_{\sqrt{5}′}^{2} Q_{24}]_{2}.

Theorem 12.18. Let Q be a definite quaternion algebra with center K and [K : ℚ] = 4. If G is an a.i.m.f. subgroup of GL_{2}(Q), then Q is one of Q_{θ_{16},∞}, Q_{θ_{24},∞}, Q_{θ_{20},∞}, Q_{θ_{15},∞}, Q_{η_{17},∞}, Q_{\sqrt{2}+\sqrt{5},∞}, Q_{\sqrt{2}+\sqrt{5},∞,2,5}, Q_{η_{40},∞}, Q_{\sqrt{3}+\sqrt{5},∞}, or Q_{η_{40},∞}. If K is not the maximal real subfield of a cyclotomic field, the simplicial complexes M_{2}^{irr}(Q) consist of zero simplices. For K = ℚ[θ_{i}] (i = 16, 24, 20, 15) the simplicial complexes M_{2}^{irr}(Q) are as follows.

[figure: _{θ_{16},∞}[S̃_{4}⊗ \sqrt{2} D_{32}]_{2}; _{θ_{16},∞}[Q_{32}]_{1}^{2}; _{θ_{16},∞}[Q_{32}]_{1} ⊗ A_{2}]

[figure: _{θ_{24},∞}[S̃_{4}⊗ \sqrt{2} D_{48}]_{2}; _{θ_{24},∞}[Q_{48}]_{1}^{2}]

[figure: _{θ_{20},∞}[SL_{2}(5)⊗ \sqrt{5} D_{40}]_{2}; _{θ_{20},∞}[C_{5} ⊠^{2} (C_{4} ∘ SL_{2}(3)).2]_{2}; _{θ_{20},∞}[Q_{40}]_{1}^{2}; _{θ_{20},∞}[Q_{40}]_{1} ⊗ A_{2}]

[figure: _{θ_{15},∞}[SL_{2}(5)⊗ \sqrt{5} D_{30}]_{2}; _{θ_{15},∞}[Q_{60}]_{1}^{2}; _{θ_{15},∞}[C_{15} ⊃ × ⊏_{\sqrt{-3}}^{2(2)}SL_{2}(3)]_{2}; _{θ_{15},∞}[C_{15} ⊃ × ⊏^{2} D_{8}]_{2}]

List of the maximal simplices in M_{2}^{irr}(Q)

simplex

a common subgroup

(_{θ_{16},∞}[S̃_{4}⊗ \sqrt{2} D_{32}]_{2},_{θ_{16},∞}[Q_{32}]_{1}^{2})  Q_{32} ⊗ D_{8}
(_{θ_{24},∞}[S̃_{4}⊗ \sqrt{2} D_{48}]_{2},_{θ_{24},∞}[Q_{48}]_{1}^{2})  Q_{48} ⊗ D_{8}
(_{θ_{20},∞}[SL_{2}(5)⊗ \sqrt{5} D_{40}]_{2},_{θ_{20},∞}[Q_{40}]_{1}^{2},_{θ_{20},∞}[C_{5} ⊠^{2} (C_{4} ∘ SL_{2}(3)).2]_{2}) Q_{40} ⊗ D_{8}
(_{θ_{15},∞}[SL_{2}(5)⊗ \sqrt{5} D_{30}]_{2},_{θ_{15},∞}[Q_{60}]_{1}^{2})  Q_{20}⊗ \sqrt{5} D_{30}

Z(Q) real quintic.

Analogously one finds:

Theorem 12.19. Let Q be a definite quaternion algebra with center K of degree 5 over ℚ and G a primitive a.i.m.f. subgroup of GL_{2}(Q). Then G is one of the groups in the following table, which is built up as Table 12.7. The simplicial complexes consist of zero simplices each.

FINITE QUATERNIONIC MATRIX GROUPS 155

List of the primitive a.i.m.f. subgroups of GL_{2}(Q), where Q is a definite

quaternion algebra with center K and [K : ℚ] = 5.

lattice L  |Aut(L)| some r.i.m.f. supergroups
θ_{11},∞,11[Q_{44}]_{1} ⊗ A_{2}  2^{3} · 3 · 11 (A_{2} ⊗ A_{10})^{2}, (A_{2} ⊗ A_{10}^{(2)})^{2}, (A_{2} ⊗ A_{10}^{(3)})^{2}

([L_{2}(11) ⊗^{2(3)} D_{12}]_{20}^{2}), ([L_{2}(11) ⊃ × ⊏^{2(3)}D_{12}]_{20}^{2})

θ_{11},∞,11[C_{11} ⊃ × ⊏^{2(2)}SL_{2}(3)]_{2} 2^{4} · 3 · 11 [SL_{2}(11)^{2(2)} ∘ SL_{2}(3)]_{20}^{2}, [U_{5}(2)^{2(2)} ∘ SL_{2}(3)]_{20}^{2}

[L_{2}(11)⊗ \sqrt{-11} SL_{2}(3) ⊗ S_{3}).2]_{40}

θ_{11},∞,11[^{±}C_{11} ⊃ × ⊏^{2(3)}S_{3}]_{2}  2^{3} · 3 · 11 [L_{2}(11) ⊃ × ⊏^{2(3)}D_{12}]_{20}^{2}, [L_{2}(11) ⊗^{2(3)} D_{12}]_{20}^{2}

((A_{2} ⊗ A_{10})^{2})

θ_{11},∞,2[^{±}C_{11} ⊃ × ⊏^{2(2)}D_{8}]_{2}  2^{4} · 11 [SL_{2}(11) ⊗_{∞,2}^{2(2)} 2_{−}^{1+4}.Alt_{5}]_{40},

[U_{5}(2) ⊗_{∞,2}^{2(2)} 2_{−}^{1+4}.Alt_{5}]_{40}

([2.M_{12}.2⊗ \sqrt{-2} GL_{2}(3)]_{40}) ((A_{10}^{4})), (((A_{10}^{(2)})^{4})), (((A_{10}^{(3)})^{4}))

θ_{11}[^{±}D_{22}]_{1} ⊗_{∞,2}[SL_{2}(3)]_{1} 2^{4} · 3 · 11 A_{10} ⊗ F_{4}, A_{10}^{(2)} ⊗ F_{4}, A_{10}^{(3)} ⊗ F_{4}

(A_{10} ⊗ F_{4}) [SL_{2}(11)^{2(S)} ∘ L_{2}(3)]_{20}^{2}

θ_{11}[^{±}D_{22}]_{1} ⊗_{∞,3}[S̃_{3}]_{1}  2^{3} · 3 · 11 (A_{10} ⊗ A_{2})^{2}, (A_{10}^{(2)} ⊗ A_{2})^{2}, (A_{10}^{(3)} ⊗ A_{2})^{2}

((A_{10} ⊗ A_{2})^{2}) ((A_{10} ⊗ A_{2})^{2}) ((A_{10} ⊗ A_{2})^{2})

θ_{11},∞,3[^{±}C_{11} ⊃ × ⊏^{2(3)} S̃_{3}]_{2}  2^{3} · 3 · 11 [L_{2}(11) ⊃ × ⊏^{2(3)}D_{12}]_{20}^{2}, [L_{2}(11) ⊗^{2(3)} D_{12}]_{20}^{2}

([L_{2}(11) ⊃ × ⊏^{2(3)}D_{12}]_{20}^{2}), ([L_{2}(11) ⊗^{2(3)} D_{12}]_{20}^{2})

([L_{2}(11) ⊃ × ⊏^{2(3)}D_{12}]_{20}^{2}), ([L_{2}(11) ⊗^{2(3)} D_{12}]_{20}^{2})

([L_{2}(11) ⊃ × ⊏^{2(3)}D_{12}]_{20}^{2}), ([L_{2}(11) ⊗^{2(3)} D_{12}]_{20}^{2})

σ_{25},∞,5[^{±}C_{25}.C_{4}]_{2}  2^{3} · 5^{2} E_{8}^{5}, [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8}^{5}

(E_{8}^{5}), ([(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8}^{5})

(E_{8}^{5}), ([(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8}^{5})

(E_{8}^{5}), ([(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8}^{5})

(E_{8}^{5}), ([(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8}^{5})

13. The a.i.m.f. subgroups of GL_{3}(Q)

Z(Q) = ℚ

Theorem 13.1. Let Q be a definite quaternion algebra with center ℚ and G be a primitive a.i.m.f. subgroup of GL_{3}(Q). Then G is one of the groups in the following table.

List of the primitive a.i.m.f. subgroups of GL_{3}(Q).

lattice L  |Aut(L)| r.i.m.f. supergroups
∞,2[SL_{2}(5)]_{3}  2^{3} · 3 · 5 [SL_{2}(5)^{2(2)} ∘ SL_{2}(3)]_{12}
∞,3[^{±}U_{3}(3)]_{3}  2^{6} · 3^{3} · 7 [6.U_{4}(3).2^{2}]_{12}
∞,3[^{±}3_{+}^{1+2}.GL_{2}(3)]_{3} 2^{5} · 3^{4} E_{6}^{2}
∞,7[^{±}L_{2}(7).2]_{3}  2^{5} · 3 · 7 (A_{6}^{(2)})^{2}

Proof. Let Q be a definite quaternion algebra with center ℚ and G be a primitive a.i.m.f. subgroup of GL_{3}(Q). Assume that 1 ≠ N ⊴G is a quasi-semi-simple normal subgroup of G. With [CCNPW 85] one finds that N is one of Alt_{5}, SL_{2}(5), L_{2}(7), or U_{3}(3). The centralizer C := C_{G}(N) in G of N embeds into the commuting algebra C_{Q^{3×3}}(N), which is isomorphic to ℚ[\sqrt{5}], ℚ, ℚ[\sqrt{-7}], resp. ℚ in the respective cases. Therefore C = ^{±}1 in all cases and G/^{±}N embeds into C_{2}, the outer automorphism group of N. This gives a contradiction in the first case, since both groups (^{±}Alt_{5}).2 are subgroups of GL_{6}(ℚ). In the second and fourth cases, one finds that N = G = ∞,2[SL_{2}(5)]_{3} resp. N = G = _{∞,3}[^{±}U_{3}(3)]_{3}, because the extensions of the natural character of N to N.2 are not rational. In the third case G = _{∞,7}[^{±}L_{2}(7).2]_{3} has to be isomorphic to a nonsplit extension of ^{±}N by C_{2}.

Now assume that G does not contain a quasi-semi-simple normal subgroup and let p be a prime with O_{p}(G) ≠ 1. Then by Corollary 2.4 one has p ∈ {2, 3, 7}. By Lemma 11.2 O_{7}(G) = 1.

Therefore Proposition 8.11 gives that O_{3}(G) ≠ 1. From Table 8.7 one gets that O_{3}(G) is one of C_{3}, C_{9}, or 3_{+}^{1+2}. In the first two cases, C_{G}(O_{3}(G)) = ^{±}O_{3}(G) and G contains C_{G}(O_{3}(G)) of index 2, contradicting the irreducibility of G. In the last case, G contains the generalized Bravais group B^{∘}(O_{3}(G)) = ^{±}3_{+}^{1+2} : SL_{2}(3) of index 2. The split extension is a subgroup of GL_{6}(ℚ), so G has to be isomorphic to the nonsplit extension G = _{∞,3}[^{±}3_{+}^{1+2}.GL_{2}(3)]_{3}.

Theorem 13.2. Let Q be a definite quaternion algebra with center ℚ and let G be an a.i.m.f. subgroup of GL_{3}(Q). Then Q is one of Q_{∞,2}, Q_{∞,3}, or Q_{∞,7}. The simplicial complexes M_{3}^{irr}(Q) are as follows:

[figure: _{∞,2}[SL_{2}(5)]_{3}; _{∞,2}[SL_{2}(3)]_{1}^{3}]

[figure: _{∞,3}[^{±}U_{3}(3)]_{3}; _{∞,3}[S̃_{3}]_{1}^{3}; _{∞,3}[^{±}3_{+}^{1+2}.GL_{2}(3)]_{3}]

{ \atop _{∞,7}[^{±}L_{2}(7).2]_{3}}

List of maximal simplices in M_{3}^{irr}(Q_{∞,3}):

simplex  a common subgroup
(_{∞,3}[^{±}U_{3}(3)],_{∞,3}[^{±}3_{+}^{1+2}.GL_{2}(3)]_{3}) 3_{+}^{1+2} : C_{8}
(_{∞,3}[S̃_{3}]_{1}^{3},_{∞,3}[^{±}3_{+}^{1+2}.GL_{2}(3)]_{3})  (^{±}3_{+}^{1+2}).C_{2}

Proof. Theorems 13.1 and 6.1 prove the completeness of the list of quaternion algebras Q and of a.i.m.f. subgroups of GL_{3}(Q). The completeness of the list of maximal simplices in M_{3}^{irr}(Q) for the respective quaternion algebras Q can be seen as follows: M_{3}^{irr}(Q_{∞,2}) consists of two 0-simplices, because the group ∞,2[SL_{2}(5)]_{3} is minimal absolutely irreducible. The unique minimal absolutely irreducible subgroup of_{∞,3}[^{±}U_{3}(3)]_{3} is 3_{+}^{1+2} : C_{8} as one sees from the list of maximal subgroups of U_{3}(3) in [CCNPW 85]. This group does not embed into _{∞,3}[S̃_{3}]_{1}^{3} so the list of maximal simplices in M_{3}^{irr}(Q_{∞,3}) is complete.

Z(Q) real quadratic.

Theorem 13.3. Let Q be a definite quaternion algebra with center K, such that [K : ℚ] = 2 and let G be a primitive a.i.m.f. subgroup of GL_{3}(Q). Then G is conjugate to one of the groups in the following table, which is built up as Table 12.7.

List of the primitive a.i.m.f. subgroups of GL_{3}(Q).

lattice L  |Aut(L)| r.i.m.f. supergroups
\sqrt{2},∞[SL_{2}(7)]_{3}  2^{4} · 3 · 7 [SL_{2}(7) ∘ S̃_{4}]_{24}
\sqrt{2},∞[SL_{2}(5).2]_{3}  2^{4} · 3 · 5 [SL_{2}(5)^{2(2)} ∘ SL_{2}(3)]_{12}^{2}, [SL_{2}(5) ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+4′}.Alt_{5}]_{24}
\sqrt{3},∞[(^{±}U_{3}(3)).2]_{3}  2^{7} · 3^{3} · 7 [6.U_{4}(3).2^{2}]_{12}^{2}, [6.U_{4}(3).2 ⊠_{\sqrt{-3}}^{2} SL_{2}(3)]_{24}

[2.Co_{1}]_{24}, [(C_{4} ∘ SL_{2}(3)).2 ⊃ × ⊏_{\sqrt{-1}}^{2(3)}U_{3}(3)]_{24}

\sqrt{3},∞[C_{4} ⊠^{2} 3_{+}^{1+2} : SL_{2}(3)]_{3} 2^{6} · 3^{4} E_{6}^{4}, [3_{+}^{1+2} : SL_{2}(3) ⊃ × ⊏_{\sqrt{-3}}^{2(2)}SL_{2}(3)]_{12}^{2}

F_{4} ⊗ E_{6}, [Sp_{4}(3) ⊠_{\sqrt{-3}}^{2} 3_{+}^{1+2} : SL_{2}(3)]_{24}

\sqrt{5},∞[2.J_{2}]_{3}  2^{8} · 3^{3} · 5^{2} · 7[2.J_{2} □^{2} SL_{2}(5)]_{24}, [2.Co_{1}]_{24}
\sqrt{5},∞[Alt_{5}⊗ \sqrt{5} SL_{2}(5)]_{3}  2^{5} · 3^{2} · 5^{2} [(SL_{2}(5) ∘ SL_{2}(5)) : 2 ⊠_{\sqrt{5}}^{2} Alt_{5}]_{24,i} (i = 1, 2)
\sqrt{7},∞[C_{4} ⊠^{2} L_{2}(7)]_{3}  2^{6} · 3 · 7 (A_{6}^{(2)})^{4}, [L_{2}(7) ⊃ × ⊏^{2(2)}D_{8}]_{12}^{2}, [6.U_{4}(3).2^{2}]_{12}^{2}

[2.Co_{1}]_{24}, [6.U_{4}(3).2 ⊠_{\sqrt{-3}}^{2} SL_{2}(3)]_{24}

\begin{matrix}[L_{2}(7) ⊃ × ⊏^{2(2)}F_{4}]_{24}, A_{6}^{(2)} ⊗ F_{4}, \\ [(C_{4} ∘ SL_{2}(3)).2 ⊃ × ⊏_{\sqrt{-1}}^{2(3)}U_{3}(3)]_{24} \end{matrix}

\sqrt{13},∞[SL_{2}(13)]_{3}  2^{3} · 3 · 7 · 13 [SL_{2}(13) ⊃ ⊏^{2(2)}SL_{2}(3)]_{24}, [2.Co_{1}]_{24}
\sqrt{21},∞[^{±}C_{3} ⊠^{2} L_{2}(7)]_{3}  2^{5} · 3^{2} · 7 (A_{2} ⊗ A_{6}^{(2)})^{2}, [6.U_{4}(3).2^{2}]_{12}^{2}

[2.Co_{1}]_{24}, [L_{2}(7) ⊃ × ⊏^{2(2)}F_{4}]_{24}

Proof. Let G be a primitive a.i.m.f. subgroup of GL_{3}(Q). Assume that 1 ≠ N ⊴ G is a quasi-semi-simple normal subgroup of G. With [CCNPW 85] one finds that N is one of Alt_{5}, SL_{2}(5) (2 groups), L_{2}(7), SL_{2}(7), 3.Alt_{6}, SL_{2}(13), U_{3}(3), or 2.J_{2}. If N is isomorphic to SL_{2}(7), SL_{2}(13), or 2.J_{2}, one computes that G = N is an a.i.m.f. group.

If N is U_{3}(3) or SL_{2}(5) (where the restriction of the natural character of G to N is 2χ_{6}), the centralizer C_{G}(N) embeds into C_{Q^{3×3}}(N) = Z(Q) = K. Since K is a (totally) real field, one concludes that C_{G}(N) = ^{±}1 and G/^{±}N ≅ C_{2} is isomorphic to the outer automorphism group of N. Using [CCNPW 85] one finds that G = _{\sqrt{2},∞}[SL_{2}(5).2]_{3} resp. G = _{\sqrt{3},∞}[(^{±}U_{3}(3)).2]_{3}.

If N = SL_{2}(5), where the restriction of the natural character χ of G to N contains χ_{2a}, one has Q = Q_{\sqrt{5},∞} and χ_{|N} = 3χ_{2a}. The centralizer C_{G}(N), embedding into C_{Q^{3×3}}(N) = ℚ[\sqrt{5}]^{3×3}, is either^{±}1 or Alt_{5}. Since 3 does not divide the order of the outer automorphism group of N, the first possibility contradicts the irreducibility of G. In the second case one computes G = NC_{G}(N) = _{\sqrt{5},∞}[Alt_{5}⊗ \sqrt{5} SL_{2}(5)]_{3}.

Now assume that G contains a simple normal subgroup N isomorphic to Alt_{5}. Since the maximal real subfield of Q is the center K = Z(Q), one finds that K ≅ ℚ[\sqrt{5}] and the restriction of the natural character of G to N is (w.l.g.) 2χ_{3a}. The centralizer C := C_{G}(N) embeds into GL_{1}(Q). Therefore it is isomorphic to a subgroup of SL_{2}(5). Since the outer automorphism of N induces the Galois automorphism on the center of the enveloping algebra ℚN one concludes that G = CN = _{\sqrt{5},∞}[Alt_{5}⊗ \sqrt{5} SL_{2}(5)]_{3}.

Now let N = L_{2}(7) be a normal subgroup of G. The centralizer C_{G}(N) embeds into K[\sqrt{-7}] and therefore is one of ^{±}1, C_{4}, or ^{±}C_{3}. Since G contains NC_{G}(N) of index 2, one concludes, that in the first case G cannot be absolutely irreducible, because that character field is only ℚ. In the remaining two cases one constructs

G to be _{\sqrt{7},∞}[C_{4} ⊠^{2} L_{2}(7)]_{3} resp. _{\sqrt{21},∞}[^{±}C_{3} ⊠^{2} L_{2}(7)]_{3}.

In the last case N = 3.Alt_{6} and C_{G}(N) = ^{±}C_{3}. Using [CCNPW 85] one finds that G = ^{±}3.P GL_{2}(9) is not maximal finite but contained in _{\sqrt{5},∞}[2.J_{2}]_{3}.

Assume for the rest of the proof, that G does not contain a quasi-semi-simple normal subgroup. By Corollary 2.4 O_{p}(G) = 1 for p ∉ {2, 3, 5, 7, 13} and by Lemma 11.2 O_{13}(G) = 1.

If O_{7}(G) ≠ 1, then O_{7}(G) = C_{7}. Because K = Z(Q) is a real quadratic number field, one has that C := C_{G}(O_{7}(G)) embeds into K[ζ_{7}], hence is one of ^{±}C_{7}, C_{28} or ^{±}C_{21} and G contains C of index 6. In the first case, the character field of the natural character of G is ℚ contradicting the absolute irreducibility of G. In the other two cases one has a unique possibility for G ≤ GL_{3}(Q). Both groups are not

maximal finite but contained in _{\sqrt{7},∞}[C_{4} ⊠^{2} L_{2}(7)]_{3} resp. _{\sqrt{21},∞}[^{±}C_{3} ⊠^{2} L_{2}(7)]_{3}.

Next assume that O_{5}(G) ≠ 1. Then O_{5}(G) = C_{5} and K = Z(Q) is isomorphic to ℚ[\sqrt{5}]. The centralizer C_{G}(O_{5}(G)) embeds into C_{Q^{3×3}}(O_{5}(G)) = ℚ[ζ_{5}]^{3×3}. Since G does not contain a quasi-semi-simple normal subgroup and 3 does not divide the order of the automorphism group of O_{5}(G), this contradicts the irreducibility of G.

Assume now, that O_{3}(G) > 1. Then O_{3}(G) is one of C_{3}, C_{9}, or 3_{+}^{1+2}. In the first case C_{G}(O_{3}(G)) embeds into K[ζ_{3}]^{3×3}. Since G does not contain a quasisemi-simple normal subgroup and 3 does not divide the order of the automorphism group of O_{5}(G), this contradicts the irreducibility of G. In the second case, C := C_{G}(O_{3}(G)) embeds into K[ζ_{9}], hence is one of ^{±}C_{9} or C_{36}. The assumption that O_{3}(G) = C_{9} implies in both cases that the index of C in G is not divisible by 3, which contradicts the irreducibility of G. In the last case, the group G contains the normal subgroup B := B^{∘}(O_{3}(G)) = ^{±}3_{+}^{1+2} : SL_{2}(3). The centralizer C_{G}(B) embeds into K[ζ_{3}], hence is one of ^{±}C_{3} or C_{12}. The first possibility contradicts

the absolutely irreducibility of G, and in the second case, G = _{\sqrt{3},∞}[C_{4} ⊠^{2} 3_{+}^{1+2} : SL_{2}(3)]_{3}.

If O_{p}(G) = 1 for all odd primes p, O_{2}(G) is self-centralizing in G contradicting Proposition 8.9.

Theorem 13.4. Let Q be a definite quaternion algebra with real quadratic center K and let G be an a.i.m.f. subgroup of GL_{3}(Q). Then Q is one of Q_{\sqrt{2},∞}, Q_{\sqrt{3},∞}, Q_{\sqrt{5},∞}, Q_{\sqrt{7},∞}, Q_{\sqrt{13},∞}, or Q_{\sqrt{21},∞}. The simplicial complexes M_{3}^{irr}(Q) are as follows:

[figure: _{\sqrt{2},∞}[SL_{2}(5).2]_{3}; _{\sqrt{2},∞}[SL_{2}(7)]_{3}; _{\sqrt{2},∞}[S̃_{4}]_{1}^{3}]

[figure: _{\sqrt{3},∞}[(^{±}U_{3}(3)).2]_{3}; _{\sqrt{3},∞}[C_{4} ⊠^{2} 3_{+}^{1+2} : SL_{2}(3)]_{3}; _{\sqrt{3},∞}[C_{12}.C_{2}]_{1}^{3}]

[figure: _{\sqrt{5},∞}[2.J_{2}]_{3}; _{\sqrt{5},∞}[Alt_{5}⊗ \sqrt{5} SL_{2}(5)]_{3}; _{\sqrt{5},∞}[SL_{2}(5)]_{1}^{3}]

\sqrt{7},∞[C_{4} ⊠^{2} L_{2}(7)]_{3}

\sqrt{13},∞[SL_{2}(13)]_{3}

\sqrt{21},∞[^{±}C_{3} ⊠^{2} L_{2}(7)]_{3}
List of maximal simplices in M_{3}^{irr}(Q_{\sqrt{3},∞}):
simplex  a common subgroup
(_{\sqrt{3},∞}[(^{±}U_{3}(3)).2]_{3},_{\sqrt{3},∞}[C_{4} ⊠^{2} ^{±}3_{+}^{1+2} : SL_{2}(3)]_{3}) (^{±}3_{+}^{1+2} : C_{8}).C_{2}
(_{\sqrt{3},∞}[C_{12}.C_{2}]_{1}^{3},_{\sqrt{3},∞}[C_{4} ⊠^{2} ^{±}3_{+}^{1+2} : SL_{2}(3)]_{3})  C_{4} ⊠^{2} 3_{+}^{1+2}

List of maximal simplices in M_{3}^{irr}(Q_{\sqrt{5},∞}):
simplex  a common subgroup
(_{\sqrt{5},∞}[2.J_{2}]_{3},_{\sqrt{5},∞}[Alt_{5}⊗ \sqrt{5} SL_{2}(5)]_{3})  Alt_{5} ⊗ Q_{8}
(_{\sqrt{5},∞}[SL_{2}(5)]_{1}^{3},_{\sqrt{5},∞}[Alt_{5}⊗ \sqrt{5} SL_{2}(5)]_{3}) Q_{20} ⊗ Alt_{4}
(_{\sqrt{5},∞}[2.J_{2}]_{3},_{\sqrt{5},∞}[SL_{2}(5)]_{1}^{3})  (^{±}C_{5} × C_{5}).D_{12}

Proof. Theorems 13.3 and 6.1 prove the completeness of the list of quaternion algebras Q and of a.i.m.f. subgroups of GL_{3}(Q). The completeness of the list of maximal simplices in M_{3}^{irr}(Q) for the respective quaternion algebras Q can be seen as follows:

M_{3}^{irr}(Q_{\sqrt{2},∞}) consists of three 0-simplices, because the groups _{\sqrt{2},∞}[SL_{2}(7)]_{3} and _{\sqrt{2},∞}[SL_{2}(5).2]_{3} are minimal absolutely irreducible groups.

The unique minimal absolutely irreducible subgroup of _{\sqrt{3},∞}[(^{±}U_{3}(3)).2]_{3} is (^{±}3_{+}^{1+2} : C_{8}).2 as one sees from the list of maximal subgroups of U_{3}(3) in [CCNPW 85]. Therefore, there is no common absolutely irreducible subgroup of

\sqrt{3},∞[(^{±}U_{3}(3)).2]_{3} and _{\sqrt{3},∞}[C_{12}.C_{2}]_{1}^{3}

and one sees that the list of maximal simplices in M_{3}^{irr}(Q_{\sqrt{3},∞}) is complete.

From the list of maximal subgroups in [CCNPW 85] one finds that the absolutely irreducible maximal subgroups of _{\sqrt{5},∞}[2.J_{2}]_{3} are ^{±}3.P GL_{2}(9), SL_{2}(3) ⊗ Alt_{5}, and (^{±}C_{5}×C_{5}).D_{12}. The first group has no absolutely irreducible subgroup of which the only nonabelian composition factors are isomorphic to Alt_{5}. The unique minimal absolutely irreducible subgroup of the second group is Q_{8} ⊗ Alt_{5}, and the two minimal absolutely irreducible subgroups ((^{±}C_{5} ×C_{5}).C_{6} and (^{±}C_{5} ×C_{5}).S_{3}) of the third group do not embed into _{\sqrt{5},∞}[Alt_{5}⊗ \sqrt{5} SL_{2}(5)]_{3}.

Z(Q) real cubic.

Theorem 13.5. Let Q be a definite quaternion algebra with center K of degree 3 over ℚ and G a primitive a.i.m.f. subgroup of GL_{3}(Q). Then G is conjugate to

one of _{θ_{9},∞,3}[^{±}C_{9} ⊠_{\sqrt{-3}}^{2} 3_{+}^{1+2} : SL_{2}(3)]_{3}, _{θ_{7},∞,7}[^{±}C_{7} ⊠_{\sqrt{-7}}^{2} L_{2}(7)]_{3}, or _{ω_{19},∞,19}[^{±}C_{19}.C_{6}]_{3}.

List of the primitive a.i.m.f. subgroups of GL_{3}(Q).

lattice L  |Aut(L)| some r.i.m.f. supergroups
θ_{9},∞,3[^{±}C_{9} ⊠_{\sqrt{-3}}^{2} 3_{+}^{1+2} : SL_{2}(3)]_{3} 2^{5} · 3^{5} [^{±}3^{1+4} : Sp_{4}(3).2]_{18}^{2}, E_{6}^{6}
θ_{7},∞,7[^{±}C_{7} ⊠_{\sqrt{-7}}^{2} L_{2}(7)]_{3}  2^{5} · 3 · 7^{2} [^{±}L_{2}(7) ⊠_{\sqrt{-7}}^{2} L_{2}(7)]_{18}^{2}
ω_{19},∞,19[^{±}C_{19}.C_{6}]_{3}  2^{2} · 3 · 19 A_{18}^{2}, (A_{18}^{(5)})^{2}

Proof. Let Q be a definite quaternion algebra with center K of degree 3 over ℚ and let G be a primitive a.i.m.f. subgroup of GL_{3}(Q). Then K is contained in a cyclotomic field of degree ≤ 18, hence K ≅ ℚ[θ_{7}], ℚ[θ_{9}], ℚ[ω_{13}], or ℚ[ω_{19}], where θ_{i} are generators of the maximal totally real subfield of the corresponding cyclotomic field ℚ[ζ_{i}] and the ω_{i} generators of the subfield of degree 3 over ℚ of the corresponding cyclotomic field ℚ[ζ_{i}] (cf. notation 4.2). By Table 9.1 the only possibility for a quasi-semi-simple normal subgroup N of G is N = L_{2}(7). If L_{2}(7) ⊴ G, then clearly

K = ℚ[θ_{7}], C_{G}(L_{2}(7)) = ^{±}C_{7}, and G = _{θ_{7},∞,7}[^{±}C_{7} ⊠_{\sqrt{-7}}^{2} L_{2}(7)]_{3}.

Assume that K ≠ ℚ[ω_{13}]. Then 13 divides |G| and one concludes that O_{13}(G) ≅ C_{13}. To get the character field K, 4 has to divide the degree of the natural character of G, which is a contradiction. If K = ℚ[ω_{19}], one similarly gets that G = _{ω_{19},∞,19}[^{±}C_{19}.C_{6}]_{3}. Now assume that K = ℚ[θ_{7}]. Then 7 divides the order of G. Since the possible normal 2- and 3-subgroups have no automorphism of order 7, one has either L_{2}(7) ⊴ G (which is dealt with above) or O_{7}(G) ≅ C_{7}. In the last case ℚ[ζ_{7}] is a maximal subfield of Q and the centralizer C_{G}(C_{7}) is a centrally irreducible subgroup of GL_{3}(ℚ[ζ_{7}]). Since O_{7}(G) ≠ C_{7} × C_{7}, this implies

L_{2}(7) ⊴ G. One finds G = _{θ_{9},∞,3}[^{±}C_{9} ⊠_{\sqrt{-3}}^{2} 3_{+}^{1+2} : SL_{2}(3)]_{3} completely analogous, if

K = ℚ[θ_{9}].

Corollary 13.6. Let Q be a definite quaternion algebra with center K of degree 3 over ℚ and G an a.i.m.f. subgroup of GL_{3}(Q). Then Q is one of Q_{θ_{9},∞,3}, Q_{θ_{7},∞,7}, or Q_{ω_{19},∞,19}. The simplicial complexes M_{3}^{irr}(Q) are as follows:

[figure: _{θ_{9},∞,3}[^{±}C_{9} ⊠_{\sqrt{-3}}^{2} 3_{+}^{1+2} : SL_{2}(3)]_{3}; _{θ_{9},∞,3}[^{±}C_{9}.C_{2}]_{1}^{3}]

[figure: _{θ_{7},∞,7}[^{±}C_{7} ⊠_{\sqrt{-7}}^{2} L_{2}(7)]_{3}; _{θ_{7},∞,7}[^{±}C_{7}.C_{2}]_{1}^{3}]

{ \atop _{ω_{19},∞,19}[^{±}C_{19}.C_{6}]_{3}}

simplex  a common subgroup
(_{θ_{9},∞,3}[^{±}C_{9} ⊠_{\sqrt{-3}}^{2} 3_{+}^{1+2} : SL_{2}(3)]_{3}, _{θ_{9},∞,3}[^{±}C_{9}.C_{2}]_{1}^{3}) ^{±}C_{9} ⊠_{\sqrt{-3}}^{2} 3_{+}^{1+2}
(_{θ_{7},∞,7}[^{±}C_{7} ⊠_{\sqrt{-7}}^{2} L_{2}(7)]_{3}, _{θ_{7},∞,7}[^{±}C_{7}.C_{2}]_{1}^{3})  ^{±}C_{7} ⊠_{\sqrt{-7}}^{2} C_{7} : C_{3}

Proof. Theorems 6.1 and 13.5 give the list of quaternion algebras Q and the a.i.m.f. subgroups of GL_{3}(Q). Since all simplicial complexes M_{3}^{irr}(Q) for the respective quaternion algebras Q consist of one simplex, it is clear that the list of maximal simplices in M_{3}^{irr}(Q) is complete.

14. The a.i.m.f. subgroups of GL_{4}(Q)

Z(Q) = ℚ

Theorem 14.1. Let Q be a totally definite quaternion algebra with center ℚ and G be a maximal finite primitive absolutely irreducible subgroup of GL_{4}(Q). Then Q is one of Q_{∞,2}, Q_{∞,3}, Q_{∞,5}, or Q_{∞,7}. The primitive a.i.m.f. subgroups G of GL_{4}(Q) are given in the following table:

List of the primitive a.i.m.f. subgroups of GL_{4}(Q).

lattice L  |Aut(L)| r.i.m.f. supergroups
∞,2[2_{−}^{1+4}Alt_{5}]_{2} ⊗ A_{2}  2^{8} · 3^{2} · 5 A_{2} ⊗ E_{8}
∞,2[SL_{2}(3)]_{1} ⊗ A_{4}  2^{6} · 3^{2} · 5 A_{4} ⊗ F_{4}
∞,2[2_{−}^{1+6}.O_{6}^{−}(2)]_{4}  2^{13} · 3^{4} · 5 F_{4} ˜⊗F_{4}
∞,2[SL_{2}(5) ⊃ × ⊏^{2(2)}D_{8}]_{4}  2^{6} · 3 · 5 [SL_{2}(5) ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+4′}.Alt_{5}]_{16}
∞,3[S̃_{3}]_{1} ⊗ A_{4}  2^{5} · 3^{2} · 5 (A_{2} ⊗ A_{4})^{2}
∞,3[S̃_{3}]_{1} ⊗ F_{4}  2^{8} · 3^{3}  (A_{2} ⊗ F_{4})^{2}
∞,3[Sp_{4}(3) □^{2} C_{3}]_{4}  2^{8} · 3^{5} · 5 E_{8}^{2}
∞,3[SL_{2}(5) ⊃ × ⊏^{2(3)}S_{3}]_{4}  2^{5} · 3^{2} · 5 [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8}^{2}
∞,3[SL_{2}(7)]_{4}  2^{4} · 3 · 7 [SL_{2}(7)^{2(3)} ∘ S̃_{3}]_{16}
∞,5[SL_{2}(5).2]_{2} ⊗ A_{2}  2^{5} · 3^{2} · 5 A_{2} ⊗ E_{8}
∞,5[SL_{2}(5) : 2]_{2} ⊗ A_{2} 2^{5} · 3^{2} · 5 A_{2} ⊗ [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8}
∞,5[SL_{2}(5) ⊠_{\sqrt{5}}^{2} D_{10}]_{4}  2^{5} · 3 · 5^{2} [(SL_{2}(5) ∘ SL_{2}(5)) : 2 ⊠_{\sqrt{5}}^{2} D_{10}]_{16}
∞,7[SL_{2}(7).2]_{4}  2^{5} · 3 · 7 (B_{16})
∞,7[2.S_{7}]_{4}  2^{5} · 3^{2} · 5 · 7 E_{8}^{2}

The proof of this theorem is split up into eight lemmata. For the rest of this paragraph let Q be a definite quaternion algebra with center ℚ and let G be a primitive a.i.m.f. subgroup of GL_{4}(Q). Then G has a complex representation of degree 8 of which the character values lie in ℚ. By [Schu 05] this implies that the prime divisors of the order of G and hence the finite primes, at which Q ramifies, lie in {2, 3, 5, 7}.

Lemma 14.2. If the order of G is divisible by 7, then G is one of _{∞,3}[SL_{2}(7)]_{4}, ∞,7[SL_{2}(7).2]_{4}, or _{∞,7}[2.S_{7}]_{4}.

Proof. Assume that 7 divides |G|. Since O_{7}(G) = 1 and the possible normal psubgroups of G have no automorphism of order 7 (cf. Chapter 8), G contains a quasisemi-simple normal subgroup N of order divisible by 7. According to [CCNPW 85] N is one of SL_{2}(7) (2 representations) or 2.Alt_{7} (cf. Table 9.1).

If N is conjugate to SL_{2}(7), where the enveloping ℚ-algebra of N is Q_{∞,3}^{4×4}, the group N is already an absolutely irreducible subgroup of GL_{4}(Q_{∞,3}). One computes that G = N = _{∞,3}[SL_{2}(7)]_{4}.

Next assume that N is conjugate to SL_{2}(7), where the enveloping ℚ-algebra of N is ℚ[\sqrt{-7}]^{4×4}. Then the centralizer C_{G}(N) embeds into C_{Q^{4×4}}(N) ≅ ℚ[\sqrt{-7}], hence is ^{±}1. Therefore G is isomorphic to SL_{2}(7).2. Since there is an element x of order 7 in G, such that χ(x) ∈ ℚ for all irreducible characters χ of G and χ_{o}(x) = −1 for the natural character χ_{o} (of degree 8) of G, Theorem A of [Fei 83] implies that Q can only be ramified at ∞ and 7. Let 𝔐 denote the maximal order in Q_{∞,7}, which is unique up to conjugacy. Then N fixes up to isomorphism five 𝔐-lattices, three of which form a set of normal critical lattices (in the sense of Definition 2.7), and a one-dimensional space of Hermitian forms. The automorphism group on the three normal critical lattices is _{∞,7}[SL_{2}(7).2]_{4} whereas the automorphism groups of the two 𝔐N-lattices, which are not invariant under the outer automorphism of N are conjugate to _{∞,7}[2.S_{7}]_{4}. One concludes that G is conjugate to _{∞,7}[SL_{2}(7).2]_{4} in this case.

If N is isomorphic to 2.Alt_{7}, one concludes as above that G = N.2 and Q is isomorphic to Q_{∞,7}. Since N contains the subgroup SL_{2}(7) of the last case, one gets that G is conjugate to _{∞,7}[2.S_{7}]_{4} in this case.

We now may assume that 7 does not divide the order of G. Hence the only finite primes on which Q ramifies lie in {2, 3, 5}.

Lemma 14.3. If G contains a quasisimple normal subgroup N isomorphic to

Sp_{4}(3), then G is _{∞,3}[Sp_{4}(3) □^{2} C_{3}]_{4}.

Proof. If Sp_{4}(3) ≅ N ⊴ G, then G contains the normal subgroup B^{∘}(N) = Sp_{4}(3)∘ C_{3} = NC_{G}(N) of index 2. Since there is an element x of order 3 in G, such that χ(x) ∈ ℚ for all irreducible characters χ of G and χ_{o}(x) = −1 for the natural character χ_{o} (of degree 8) of G, Theorem A of [Fei 83] implies that Q = Q_{∞,3}. Let 𝔐 denote the maximal order in Q_{∞,3}, which is unique up to conjugacy. N fixes only one isomorphism class of 𝔐-lattices and a one-dimensional space of Hermitian forms and therefore has at most one a.i.m.f. supergroup. One computes that G is

∞,3[Sp_{4}(3) □^{2} C_{3}]_{4}.

Imediately from Corollary 7.6 one gets:

Lemma 14.4. If G contains a normal subgroup N isomorphic to Alt_{5}, then G is ∞,2[SL_{2}(3)]_{1} ⊗ A_{4} or _{∞,3}[S̃_{3}]_{1} ⊗ A_{4}.

Lemma 14.5. G does not contain a normal subgroup SL_{2}(9).

Proof. Assume that SL_{2}(9) ≅ N ⊴G. Then the restriction of the natural character of G to N is 4χ_{4a} (or 4χ_{4b}) and the centralizer C := C_{G}(N) embeds into C_{Q^{4×4}}(N) which is an indefinite quaternion algebra C with center ℚ. Since the two outer automorphisms of N, not contained in S_{6} ≤ Aut(N) interchange the two characters χ_{4a} and χ_{4b}, the group G contains CN of index ≤ 2. Since B^{∘}(C_{3} ∘ SL_{2}(9)) = C_{3} ∘ Sp_{4}(3) one has O_{3}(C) = 1. If G = CN, then C is an a.i.m.f. subgroup of GL_{1}(C). Hence by Corollary 6.2 C ≅ ℚ^{2×2}, Q ≅ Q_{∞,3}, C = D_{8} and G = _{∞,3}[SL_{2}(9)]_{2} ⊗ D_{8} is imprimitive and contained in _{∞,3}[SL_{2}(9)]_{2}^{2}. Hence G contains CN of index

2. Since O_{3}(C) = 1, one finds O_{2}(C) ≅ C_{4} and G = C_{4} ⊃ ⊏^{2(3)} N is contained in ∞,2[2_{−}^{1+6}.O_{6}^{−}(2)]_{4}.

Lemma 14.6. If G contains a normal subgroup N isomorphic to SL_{2}(5), then G is one of

∞,2[SL_{2}(5) ⊃ × ⊏^{2(2)}D_{8}]_{4}, _{∞,3}[SL_{2}(5) ⊃ × ⊏^{2(3)}S_{3}]_{4}, _{∞,5}[SL_{2}(5).2]_{2} ⊗ A_{2},

∞,5[SL_{2}(5) : 2]_{2} ⊗ A_{2}, or _{∞,5}[SL_{2}(5) ⊠_{\sqrt{5}}^{2} D_{10}]_{4}.

Proof. By Table 9.1 the restriction of the natural character of G to N is 4(χ_{2a}+χ_{2b}). The centralizer C := C_{G}(N) embeds into C := C_{Q^{4×4}}(N) which is an indefinite quaternion algebra with center ℚ[\sqrt{5}]. Moreover, the center of the enveloping algebra \overline{ℚCN} is ℚ[\sqrt{5}] and G contains CN of index 2. With Lemma 2.14 this implies that dim_{ℚ}(\overline{ℚCN}) = 32. Therefore dim_{ℚ[\sqrt{5}]}(\overline{ℚ[\sqrt{5}]C}) = 4. One concludes that C is isomorphic to ℚ[\sqrt{5}]^{2×2} and C is one of D_{10}, S_{3}, or D_{8}. Let α be an element of G − CN. In the first case, α does not centralize C. Computing the two possible extensions CN.2 = G one finds that they are isomorphic and G is conjugate

to _{∞,5}[SL_{2}(5) ⊠_{\sqrt{5}}^{2} D_{10}]_{4} in this case. In the other two cases one has two possibilities:

Either α centralizes C or it induces the unique nontrivial outer automorphism of C. If α centralizes C one concludes that G is one of

∞,5[SL_{2}(5).2]_{2} ⊗ A_{2} or _{∞,5}[SL_{2}(5) : 2]_{2} ⊗ A_{2},

since the two groups_{∞,5}[SL_{2}(5).2]_{2}⊗D_{8} resp. _{∞,5}[SL_{2}(5) : 2]_{2}⊗D_{8} are imprimitive and contained in _{∞,5}[SL_{2}(5).2]_{2}^{2} resp. _{∞,5}[SL_{2}(5) : 2]_{2}^{2}.

If α does not centralize C, one finds in each case two nonisomorphic exten-

sions: _{∞,3}[SL_{2}(5) ⊃ × ⊏^{2(3)} S_{3}]_{4} and a proper subgroup of _{∞,3}[Sp_{4}(3) □^{2} S_{3}]_{4} resp.

∞,2[SL_{2}(5) ⊃ × ⊏^{2(2)} D_{8}]_{4} and a proper subgroup of _{∞,2}[2_{−}^{1+6}.O_{6}^{−}(2)]_{4}. Hence G is con-

jugate to _{∞,3}[SL_{2}(5) ⊃ × ⊏^{2(3)}S_{3}]_{4} resp. _{∞,2}[SL_{2}(5) ⊃ × ⊏^{2(2)}D_{8}]_{4} in these cases.

Lemma 14.7. If G does not contain a quasi-semi-simple normal subgroup, then O_{5}(G) = 1.

Proof. Assume that O_{5}(G) > 1. Then O_{5}(G) ≅ C_{5}. Since ℚ[ζ_{5}] splits all possible quaternion algebras Q (which are Q_{∞,2}, Q_{∞,3}, Q_{∞,5}, and Q_{∞,2,3,5} since by Lemma 14.2 ramification at 7 is excluded) the centralizer C := C_{G}(O_{5}(G)) embeds into C_{Q^{4×4}}(O_{5}(G)) ≅ ℚ[ζ_{5}]^{2×2}. Moreover G contains C of index 4. Applying Lemma 2.14 two times, one sees that the enveloping algebra \overline{ℚC} is isomorphic to ℚ[ζ_{5}]^{2×2}.

If O_{3}(C) ≠ 1, then C is one of^{±}C_{5}⊗S_{3} or C_{5}⊗S̃_{3}. Since the outer automorphism groups of S̃_{3} resp. ^{±}S_{3} are ≅ C_{2} and Q is totally definite one concludes that G contains one of the groups Q_{20} ⊗S_{3} or D_{10} ⊗S̃_{3} of index 2. Computing the possible extensions one finds that G is not maximal finite but contained in _{∞,5}[SL_{2}(5).2]_{2} ⊗

A_{2} and _{∞,5}[SL_{2}(5) : 2]_{2} ⊗ A_{2} resp. _{∞,3}[Sp_{4}(3) □^{2} C_{3}]_{4} and _{∞,3}[SL_{2}(5) ⊃ × ⊏^{2(3)} S_{3}]_{4} or

∞,3[S̃_{3}]_{1} ⊗ A_{4} resp. _{∞,5}[SL_{2}(5) ⊠_{\sqrt{5}}^{2} D_{10}]_{4}.

If O_{3}(G) = 1, then C is either D_{8} or SL_{2}(3) and as above one finds that G is a proper subgroup of _{∞,5}[SL_{2}(5).2]_{2}^{2} and _{∞,5}[SL_{2}(5) : 2]_{2}^{2} resp. _{∞,2}[2_{−}^{1+6}.O_{6}^{−}(2)]_{4}

and _{∞,2}[SL_{2}(5) ⊃ × ⊏^{2(2)}D_{8}]_{4} or _{∞,2}[SL_{2}(3)]_{1} ⊗ A_{4} resp. _{∞,5}[SL_{2}(5) ⊠_{\sqrt{5}}^{2} D_{10}]_{4}.

Lemma 14.8. If G does not contain a quasi-semi-simple normal subgroup, O_{5}(G) = 1, and O_{3}(G) ≠ 1, then G is conjugate to _{∞,2}[2_{−}^{1+4}.Alt_{5}]_{2} ⊗ A_{2} or _{∞,3}[S̃_{3}]_{1} ⊗ F_{4}.

Proof. Assume that O_{3}(G) > 1. Then O_{3}(G) ≅ C_{3}. Since ℚ[ζ_{3}] splits all possible quaternion algebras Q (which are Q_{∞,2}, Q_{∞,3}, Q_{∞,5}, and Q_{∞,2,3,5} since by Lemma 14.2 ramification at 7 is excluded) the centralizer C := C_{G}(O_{3}(G)) embeds into C_{Q^{4×4}}(O_{3}(G)) ≅ ℚ[ζ_{3}]^{2×2}. Moreover G contains C of index 2. Lemma 2.14 implies that the enveloping algebra \overline{ℚC} is isomorphic to ℚ[ζ_{3}]^{4×4}.

With Theorem 8.1 one finds that O_{2}(G) is one of Q_{8}, Q_{8} ∘ Q_{8}, or Q_{8} ⊗ D_{8}. and C is one of C_{3} ⊗ GL_{2}(3), C_{3} ∘ S̃_{4}, C_{3} ⊗ F_{4} or C_{3} ∘ 2_{−}^{1+4}.Alt_{5}. Constructing the possible extensions one gets that G is either one of the two groups in the lemma or

a proper subgroup of _{∞,2}[2_{−}^{1+6}.O_{6}^{−}(2)]_{4} or _{∞,3}[Sp_{4}(3) □^{2} C_{3}]_{4}.

Proposition 8.9 yields the following:

Lemma 14.9. If G does not contain a quasi-semi-simple normal subgroup and O_{p}(G) = 1 for all odd primes p, then G is conjugate to _{∞,2}[2_{−}^{1+6}.O_{6}^{−}(2)]_{4}.

Proof of Theorem 14.1. Assume first that G contains a quasi-semi-simple normal subgroup N. According to Table 9.1 N is one of Alt_{5}, SL_{2}(5), SL_{2}(9), SL_{2}(7) (2 representations), 2.Alt_{7}, or Sp_{4}(3). These cases are dealt with in Lemma 14.4, 14.6, 14.5, 14.2, respectively 14.3. The remaining three lemmata treat the case, that G does not contain a quasi-semi-simple normal subgroup.

Theorem 14.10. M_{4}^{irr}(Q_{∞,2}) is as follows:

[figure: _{∞,2}[SL_{2}(5) ⊃ × ⊏^{2(2)}D_{8}]_{4}; _{∞,2}[SL_{2}(3)]_{1} ⊗ A_{4}; (_{∞,2}[SL_{2}(3)]_{1} ⊗ A_{2})^{2}; _{∞,2}[2_{−}^{1+4}.Alt_{5}]_{2} ⊗ A_{2}; _{∞,2}[2_{−}^{1+4}.Alt_{5}]_{2}^{2}; _{∞,2}[2_{−}^{1+6}.O_{6}^{−}(2)]_{4}; _{∞,2}[SL_{2}(3)]_{1}^{4}]

List of the maximal simplices in M_{4}^{irr}(Q_{∞,2})
simplex  a common subgroup
(_{∞,2}[2_{−}^{1+4}.Alt_{5}]_{2}^{2}, _{∞,2}[SL_{2}(3)]_{1}^{4},_{∞,2}[2_{−}^{1+6}.O_{6}^{−}(2)]_{4}) D_{8} ⊗ D_{8} ⊗ Q_{8}
(_{∞,2}[2_{−}^{1+6}.O_{6}^{−}(2)]_{4},_{∞,2}[SL_{2}(5) ⊃ × ⊏^{2(2)}D_{8}]_{4})  Q_{20} ⊃ × ⊏^{2(2)}D_{8}
((_{∞,2}[SL_{2}(3)]_{1} ⊗ A_{2})^{2},_{∞,2}[2_{−}^{1+6}.O_{6}^{−}(2)]_{4})  ((S_{3} × S_{3}) ⊗ SL_{2}(3)).2
((_{∞,2}[SL_{2}(3)]_{1} ⊗ A_{2})^{2},_{∞,2}[2_{−}^{1+4}.Alt_{5}]_{2} ⊗ A_{2})  A_{2} ⊗ D_{8} ⊗ Q_{8}
(_{∞,2}[2_{−}^{1+4}.Alt_{5}]_{2}^{2},_{∞,2}[2_{−}^{1+4}.Alt_{5}]_{2} ⊗ A_{2})  (C_{3} ≀ C_{2}) ⊃ × ⊏^{2(2)}D_{8}

Proof. The list of a.i.m.f. subgroups of GL_{4}(Q_{∞,2}) is obtained from Theorems 14.1, 12.1, and 6.1. The vertex _{∞,2}[SL_{2}(3)]_{1} ⊗ A_{4} forms a component on its own, as it can be seen from the proof of Theorem (VI.13) in [NeP 95]. There it is shown that for every absolutely irreducible subgroup U ≤ GL_{16}(ℚ) of Aut(F_{4} ⊗A_{4}) the degrees of the 5-modular constituents of the natural representation of U are divisible by 4. Assume that there is a common absolutely irreducible subgroup V ≤ GL_{4}(Q_{∞,2}) of _{∞,2}[SL_{2}(3)]_{1} ⊗ A_{4} and one of the other a.i.m.f. subgroups of GL_{4}(Q_{∞,2}). Let H ≅ SL_{2}(3) be the unit group of the endomorphism ring 𝔐 of the V -lattice ∞,2[SL_{2}(3)]_{1} ⊗ A_{4}. Then the group H ∘ V ≤ GL_{16}(ℚ) is an absolutely irreducible subgroup of Aut(F_{4} ⊗ A_{4}) acting on the ℤ-lattices of the r.i.m.f. supergroups that one obtains from the 𝔐-lattices of the a.i.m.f. supergroups of V. Hence H ∘V fixes a 5-unimodular ℤ-lattice or a ℤ-lattice with elementary divisors 2^{8} ·5^{8} contradicting Theorem (VI.13) of [NeP 95].

Now consider the vertex G := _{∞,2}[SL_{2}(5) ⊃ × ⊏^{2(2)} D_{8}]_{4}. The minimal absolutely

irreducible subgroup of this group is easily seen to be Q_{20} ⊃ × ⊏^{2(2)} D_{8}. Since the only other a.i.m.f. supergroup of this group is _{∞,2}[2_{−}^{1+6}.O_{6}^{−}(2)]_{4}, the list of maximal simplices in M_{4}^{irr}(Q_{∞,2}) with vertex G is complete.

To finish the proof it remains to show that there are no other simplices with one vertex G := (_{∞,2}[SL_{2}(3)]_{1} ⊗ A_{2})^{2} or H := _{∞,2}[2_{−}^{1+4}.Alt_{5}]_{2} ⊗ A_{2} and one vertex in {_{∞,2}[2_{−}^{1+4}.Alt_{5}]_{2}^{2},_{∞,2}[SL_{2}(3)]_{1}^{4},_{∞,2}[2_{−}^{1+6}.O_{6}^{−}(2)]_{4}}. Assume that there is such an additional simplex and let U be an absolutely irreducible common subgroup of the groups belonging to the vertices of the simplex. First assume that one of the vertices of the simplex is H. Let 𝔐 be the maximal order in Q_{∞,2} and L ∈ Z_{𝔐}(H) be a natural 𝔐-lattice of U. For p = 2 and 3 let L_{p} be the full preimage of the Sylow p-subgroup of the finite abelian group L^{#}/L. Then for both primes p = 2 and 3, the 𝔐/p𝔐U-module L_{p}/L is not simple, hence U fixes a 𝔐-lattice M_{p} with L ⊂ M_{p} ⊂ L_{p}. Computing the stabilizers in H of all the possible lattices one finds no such absolutely irreducible group U. In an analogous way, one checks the completeness of the list of maximal simplices in M_{4}^{irr}(Q_{∞,2}) with vertex G. Since the unique G-orbit of lattices M_{2} having an absolutely irreducible stabilizer Stab_{G}(M_{2}) satisfies M_{2} ∼ _{∞,2}[2_{−}^{1+4}.Alt_{5}]_{2} ⊗ A_{2} one concludes that every simplex with vertex G not listed in the theorem also contains a vertex H. Therefore the list of maximal simplices in M_{4}^{irr}(Q_{∞,2}) with vertex G is complete.

Theorem 14.11. M_{4}^{irr}(Q_{∞,3}) is as follows.

[figure: _{∞,3}[SL_{2}(9)]_{2}^{2}; _{∞,3}[Sp_{4}(3) □^{2} C_{3}]_{4}; _{∞,3}[S̃_{3}]_{1}^{4}; _{∞,3}[S̃_{3}]_{1} ⊗ F_{4}; _{∞,3}[SL_{2}(5) ⊃ × ⊏^{2(3)}S_{3}]_{4}; _{∞,3}[S̃_{3}]_{1} ⊗ A_{4}; _{∞,3}[SL_{2}(7)]_{4}; _{∞,3}[SL_{2}(3) ⊃ ⊏^{2(2)}C_{3}]_{2}^{2}]

List of the maximal simplices in M_{4}^{irr}(Q_{∞,3})

simplex  a common subgroup
\begin{matrix}(_{∞,3}[SL_{2}(3) ⊃ ⊏^{2(2)}C_{3}]_{2}^{2}_{∞,3}[S̃_{3}]_{1}^{4}, \\ _{∞,3}[S̃_{3}]_{1} ⊗ F_{4},_{∞,3}[Sp_{4}(3) □^{2} C_{3}]_{4}) \end{matrix}  GL_{2}(3) ⊗ S̃_{3}

(_{∞,3}[SL_{2}(9)]_{2}^{2},_{∞,3}[SL_{2}(3) ⊃ ⊏^{2(2)}C_{3}]_{2}^{2}_{∞,3}[Sp_{4}(3) □^{2} C_{3}]_{4}) D_{8} ⊗ S̃_{4}
(_{∞,3}[S̃_{3}]_{1}^{4},_{∞,3}[SL_{2}(9)]_{2}^{2})  ((^{±}C_{3} × C_{3}).C_{4}) ≀ C_{2}
(_{∞,3}[Sp_{4}(3) □^{2} C_{3}]_{4},_{∞,3}[SL_{2}(5) ⊃ × ⊏^{2(3)}S_{3}]_{4})  Q_{20} ⊃ × ⊏^{2(3)}S_{3}

Proof. The list of a.i.m.f. subgroups of GL_{4}(Q_{∞,3}) is obtained from Theorems 14.1, 12.1, and 6.1. The group _{∞,3}[SL_{2}(7)]_{4} forms a simplex on its own, because it is minimal absolutely irreducible. The minimal absolutely irreducible subgroups U of _{∞,3}[S̃_{3}]_{1} ⊗ A_{4} either satisfy U^{(∞)} = Alt_{5} or U = S̃_{3} ⊗ C_{5} : C_{4}. In both cases U is not a subgroup of one of the other a.i.m.f. groups. The minimal absolutely

irreducible subgroup of _{∞,3}[SL_{2}(5) ⊃ × ⊏^{2(3)}S_{3}]_{4} is Q_{20} ⊃ × ⊏^{2(3)}S_{3} and its only other a.i.m.f.

supergroup is _{∞,3}[Sp_{4}(3) □^{2} C_{3}]_{4}. To prove the theorem it remains to show that the list of maximal simplices in M_{4}^{irr}(Q_{∞,3}) with vertex G := _{∞,3}[SL_{2}(9)]_{2}^{2} is complete. Assume that there is an absolutely irreducible subgroup U ≤ G such that the a.i.m.f. supergroups of U lie not in one of the two maximal simplices in M_{4}^{irr}(Q_{∞,3}) with vertex G listed in the theorem. Then U embeds into one of ∞,3[S̃_{3}]_{1}^{4} or _{∞,3}[S̃_{3}]_{1} ⊗ F_{4} and hence the order of U is not divisible by 5. Moreover U contains a normal subgroup N ⊴ U of index 2, such that the restriction of the natural representation Δ of U to N is Δ_{|N} = Δ_{1} + Δ_{2} with Δ_{i}(N) ≤ _{∞,3}[SL_{2}(9)]_{2} absolutely irreducible (i = 1, 2). Therefore Δ_{1}(N) is one of the two absolutely irreducible subgroups of _{∞,3}[SL_{2}(9)]_{2} of order not divisible by 5, which are S̃_{4} and (^{±}C_{3} × C_{3}).C_{4}. By Lemma 2.14 one also finds that the enveloping algebra \overline{ℚN} of N is Q_{∞,3}^{2×2} ⊕ Q_{∞,3}^{2×2}. Hence Δ_{1} and Δ_{2} are inequivalent. Let 𝔐 be the maximal order in Q_{∞,3} and L ∈ Z_{𝔐}(Δ_{1}(N)).

If Δ_{1}(N) = (^{±}C_{3}×C_{3}).C_{4}, then 2L is a maximal 𝔐Δ_{1}(N) sublattice of L. Hence

U cannot embed into one of _{∞,3}[SL_{2}(3) ⊃ ⊏^{2(2)}C_{3}]_{2}^{2} or _{∞,3}[S̃_{3}]_{1} ⊗ F_{4}. Therefore U is a

subgroup of _{∞,3}[Sp_{4}(3) □^{2} C_{3}]_{4} in this case. Since this primitive a.i.m.f. group has a normal subgroup ≅ C_{3}, there is a normal subgroup N_{1} of U of index 2 such that \overline{ℚN_{1}} ≅ ℚ[\sqrt{-3}]^{4×4} Therefore N_{1} ∩ N =: N_{2} is a normal subgroup of index 2 in N such that the enveloping algebra \overline{ℚΔ_{1}(N_{2})} is ℚ[\sqrt{-3}]^{2×2}. But Δ_{1}(N) has only one subgroup of index 2 and the enveloping algebra of this subgroup is isomorphic to Q_{∞,3} ⊕ Q_{∞,3} which is a contradiction.

Hence Δ_{1}(N) = S̃_{4}. If ℘_{3} denotes the maximal ideal of 𝔐 containing 3, then L/℘L is a simple 𝔽_{9}Δ_{1}(N)-module. Since Δ_{1} and Δ_{2} are inequivalent, one concludes that U cannot fix one of the 𝔐-lattices of _{∞,3}[S̃_{3}]_{1} ⊗ F_{4} or _{∞,3}[S̃_{3}]_{1}^{4}.

Theorem 14.12. M_{4}^{irr}(Q_{∞,5}) is as follows.

[figure: _{∞,5}[SL_{2}(5).2]_{2}^{2}; _{∞,5}[SL_{2}(5) : 2]_{2}^{2}; _{∞,5}[SL_{2}(5) ⊠_{\sqrt{5}}^{2} D_{10}]_{4}; _{∞,5}[SL_{2}(5) : 2]_{2} ⊗ A_{2}; _{∞,5}[SL_{2}(5).2]_{2} ⊗ A_{2}]
List of the maximal simplices in M_{4}^{irr}(Q_{∞,5})
simplex  a common subgroup
(_{∞,5}[SL_{2}(5).2]_{2}^{2},_{∞,5}[SL_{2}(5) : 2]_{2}^{2},_{∞,5}[SL_{2}(5) ⊠_{\sqrt{5}}^{2} D_{10}]_{4}) Q_{20} ⊠_{\sqrt{5}}^{2} D_{10}
(_{∞,5}[SL_{2}(5) : 2]_{2} ⊗ A_{2},_{∞,5}[SL_{2}(5).2]_{2} ⊗ A_{2})  (^{±}C_{5}.C_{4}) ⊗ A_{2}

Proof. The list of a.i.m.f. subgroups of GL_{4}(Q_{∞,5}) may be obtained from Theorems 14.1, 12.1, and 6.1. To see that the list of maximal simplices in M_{4}^{irr}(Q_{∞,5}) is complete, one has to note, that the minimal uniform subgroup of both groups ∞,5[SL_{2}(5) : 2]_{2} ⊗ A_{2} and _{∞,5}[SL_{2}(5).2]_{2} ⊗ A_{2} is (^{±}C_{5}.C_{4}) ⊗ A_{2}. Since this group does not embed into one of the other 3 a.i.m.f. groups one easily deduces the theorem.

Theorem 14.13. M_{4}^{irr}(Q_{∞,7}) is as follows.

[figure: _{∞,7}[SL_{2}(7).2]_{4}; _{∞,7}[2.S_{7}]_{4}]

Proof. By Theorems 14.1, 12.1, and 6.1. M_{4}^{irr}(Q_{∞,7}) has two vertices. These two a.i.m.f. groups have no common absolutely irreducible subgroup since both groups are minimal absolutely irreducible as one sees from the list of maximal subgroups of the two groups given in [CCNPW 85].

Z(Q) real quadratic.

Theorem 14.14. Let Q be a definite quaternion algebra with center K, such that [K : ℚ] = 2 and let G be a primitive a.i.m.f. subgroup of GL_{4}(Q). Then G is conjugate to one of the a.i.m.f. groups given in the following table.

The table is built up as the one in Theorem 12.17.

List of the primitive a.i.m.f. subgroups of GL_{4}(Q), where Q is a definite quaternion algebra over a real quadratic field.

\sqrt{2},∞[S̃_{4}]_{1} ⊗ A_{4} (2^{7} · 3^{2} · 5)

(F_{4} ⊗ A_{4})^{2}, E_{8} ⊗ A_{4}

\sqrt{2},∞[2_{−}^{1+4}.S_{5}]_{2} ⊗ A_{2} (2^{9} · 3^{2} · 5)

(E_{8} ⊗ A_{2})^{2}, (F_{4} ⊗ A_{2})^{4}

∞,3[SL_{2}(9)]_{2} ⊗ \sqrt{2}[D_{16}]_{2} (2^{7} · 3^{2} · 5)

[(Sp_{4}(3)⊗ \sqrt{-3} Sp_{4}(3)) : 2 □^{2} C_{3}]_{32}, [SL_{2}(9).2 ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+4}.Alt_{5}]_{32}

∞,5[SL_{2}(5).2]_{2} ⊗ \sqrt{2}[D_{16}]_{2} (2^{7} · 3 · 5)

[SL_{2}(5) ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+6}.O_{6}^{−}(2)]_{32}, [SL_{2}(5) ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+4}.Alt_{5}]_{16}^{2}

\sqrt{2},∞[2_{−}^{1+6}.O_{6}^{−}(2).2]_{4} (2^{14} · 3^{4} · 5)

[2_{+}^{1+10}.O_{10}^{+}(2)]_{32}, (F_{4} ˜⊗F_{4})^{2}

\sqrt{2},∞,2,3[C_{3} ⊃ ⊏^{2(2+\sqrt{2})} S̃_{4}⊗ \sqrt{2} D_{16}]_{4} (2^{8} · 3^{2})

E_{8}^{4}, (F_{4} ˜⊗F_{4})^{2}

\sqrt{2},∞,2,3[C_{3} ⊃ × ⊏^{2(2+\sqrt{2})} S̃_{4} ∘ Q_{16}]_{4} (2^{8} · 3^{2})

(A_{2} ⊗ E_{8})^{2}, (A_{2} ⊗ F_{4})^{4}

\sqrt{2},∞,2,5[SL_{2}(5) ⊃ × ⊏^{2(2+\sqrt{2})}D_{16}]_{4,1} (2^{7} · 3 · 5)

(E_{8}^{4})

\sqrt{2},∞,2,5[SL_{2}(5) ⊃ × ⊏^{2(2+\sqrt{2})}D_{16}]_{4,2} (2^{7} · 3 · 5)

([(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8}^{4})

\sqrt{2},∞,2,5[D_{10} ⊃ × ⊏^{2(2+\sqrt{2})}Q_{16}]_{4} (2^{6} · 5)

(A_{4}^{8})

\sqrt{3},∞[Q_{24}]_{1} ⊗ A_{4} (2^{6} · 3^{2} · 5)

O_{1} (A_{4} ⊗ A_{2})^{4}, (A_{4} ⊗ F_{4})^{2}
O_{2} F_{4} ⊗ A_{4} ⊗ A_{2}, E_{8} ⊗ A_{4}

\sqrt{3},∞[SL_{2}(7) : 2]_{4} (2^{5} · 3 · 7)

O_{1} [SL_{2}(7)^{2(3)} ∘ S̃_{3}]_{16}^{2}, [SL_{2}(7) ⊗_{∞,3}^{2(3)} (SL_{2}(3) □^{2} C_{3})]_{32}
O_{2} [(SL_{2}(3) ∘ C_{4}).2 ⊃ × ⊏_{\sqrt{-1}}^{2(3)}SL_{2}(7)]_{32}, [SL_{2}(7) ⊗_{∞,3}^{2(3)} SL_{2}(9)]_{32}

\sqrt{3},∞[Sp_{4}(3) ⊠_{\sqrt{-3}}^{2} C_{12}]_{4} (2^{9} · 3^{5} · 5)

O_{1} E_{8}^{4}, [(Sp_{4}(3) ∘ C_{3}) ⊠_{\sqrt{-3}}^{2} SL_{2}(3)]_{16}^{2}
O_{2} F_{4} ⊗ E_{8}, [(Sp_{4}(3)⊗ \sqrt{-3} Sp_{4}(3)) : 2 □^{2} C_{3}]_{32}

∞,5[SL_{2}(5).2]_{2} ⊗ \sqrt{3}[D_{24}]_{2} (2^{6} · 3^{2} · 5)

O_{1} [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8}^{4}, [SL_{2}(5) ⊃ × ⊏_{∞,3}^{2(3)}(SL_{2}(3) □^{2} C_{3})]_{16}^{2}
O_{2} [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8} ⊗ F_{4}, [SL_{2}(5) ⊃ × ⊏_{∞,3}^{2(3)}(Sp_{4}(3) □^{2} C_{3})]_{32}

\sqrt{3},∞[D_{8} ⊗ D_{8} ⊗ C_{4}.S_{6} ⊠^{2} C_{3}]_{4} (2^{11} · 3^{3} · 5)

O_{1} (F_{4} ˜⊗F_{4})^{2}, (A_{2} ⊗ E_{8})^{2}
O_{2} (F_{4} ˜⊗F_{4}) ⊗ A_{2}, [2_{+}^{1+10}.O_{10}^{+}(2)]_{32}

\sqrt{3},∞[SL_{2}(5) ⊃ × ⊏^{2(2+\sqrt{3})}D_{24}]_{4} (2^{6} · 3^{2} · 5)

O_{1} [SL_{2}(5) ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+6}.O_{6}^{−}(2)]_{32}, [SL_{2}(5) ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+4}.Alt_{5}]_{16} ⊗ A_{2}
O_{2} [SL_{2}(5) ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+4}.Alt_{5}]_{16}^{2}, ([(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8} ⊗ A_{2})^{2}

\sqrt{3},∞[D_{10} ⊃ × ⊏^{2} Q_{24}]_{4} (2^{5} · 3 · 5)

O_{1} [(SL_{2}(5) ∘ SL_{2}(5)) : 2 ⊠_{\sqrt{5}}^{2} D_{10}]_{16}^{2}, [D_{120}.(C_{4} × C_{2})]_{16}^{2}
O_{2} [C_{15} : C_{4} ⊃ × ⊏^{2(2)}F_{4}]_{32}, [2_{−}^{1+4}.Alt_{5}⊗_{∞,2} SL_{2}(5) ⊠_{\sqrt{5}}^{2} D_{10}]_{32}

\sqrt{5},∞[SL_{2}(5)]_{1} ⊗ F_{4} (2^{9} · 3^{3} · 5)

[(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8} ⊗ F_{4}, F_{4} ⊗ E_{8}

\sqrt{5},∞[SL_{2}(5)⊗ \sqrt{5} D_{10}]_{2} ⊗ A_{2} (2^{5} · 3^{2} · 5^{2})

[(SL_{2}(5) ∘ SL_{2}(5)) : 2 ⊠_{\sqrt{5}}^{2} D_{10}]_{16} ⊗ A_{2}

\sqrt{5},∞[SL_{2}(9)]_{4} (2^{4} · 3^{2} · 5)

[SL_{2}(9) □^{2} SL_{2}(5)]_{32}, ([4.L_{3}(4).2^{2}]_{32,1}, [4.L_{3}(4).2^{2}]_{32,2})

∞,3[SL_{2}(9)]_{2} ⊗ \sqrt{5}[^{±}D_{10}]_{2} (2^{5} · 3^{2} · 5^{2})

[SL_{2}(9) ⊗ D_{10} □^{2} SL_{2}(5)]_{32}

\sqrt{5},∞[(SL_{2}(5) ∘ SL_{2}(5)⊗ \sqrt{5} SL_{2}(5)) : S_{3}]_{4} (2^{8} · 3^{4} · 5^{3})

[((SL_{2}(5) ∘ SL_{2}(5)) ⊠_{\sqrt{5}}^{2} (SL_{2}(5) ∘ SL_{2}(5))) : S_{4}]_{32,i} (i = 1, 2)

∞,3[SL_{2}(3) □^{2} C_{3}]_{2} ⊗ \sqrt{5}[^{±}D_{10}]_{2} (2^{5} · 3^{2} · 5)

[(SL_{2}(5)⊗ \sqrt{5} D_{10}) ⊃ × ⊏_{∞,3}^{2(3)}(SL_{2}(3) □^{2} C_{3})]_{32}

∞,2[2_{−}^{1+4}.Alt_{5}]_{2} ⊗ \sqrt{5}[^{±}D_{10}]_{2} (2^{8} · 3 · 5^{2})

[(2_{−}^{1+4}.Alt_{5}⊗_{∞,2} SL_{2}(5)) ⊠_{\sqrt{5}}^{2(2)} D_{10}]_{32}

\sqrt{5},∞[SL_{2}(5)]_{1}⊗ \sqrt{5} _{\sqrt{5},2,5}[C_{5} ⊃ × ⊏_{\sqrt{5}′}^{2(2)}SL_{2}(3)]_{2} (2^{6} · 3^{2} · 5^{2})

[(SL_{2}(5) ∘ SL_{2}(5)) : 2 ⊠_{\sqrt{5}}^{2} D_{10}]_{16}^{2}

\sqrt{5},∞[SL_{2}(5)]_{1}⊗ \sqrt{5} _{\sqrt{5},2,5}[C_{5} ⊃ × ⊏^{2(2)}D_{8}]_{2} (2^{6} · 3 · 5^{2})

E_{8}^{4}, [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8}^{4}

\sqrt{5},∞,2,5[C_{5} ⊃ × ⊏^{2(2)}F_{4}]_{4} (2^{8} · 3^{2} · 5)

(A_{4} ⊗ F_{4})^{2}

\sqrt{5},∞,2,5[C_{5} ⊃ × ⊏_{\sqrt{5}′}^{2(2)}2_{−}^{1+4}.Alt_{5}]_{4} (2^{8} · 3^{2} · 5)

(F_{4} ˜⊗F_{4})^{2}, [SL_{2}(5) ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+4}.Alt_{5}]_{16}^{2}

\sqrt{5},∞,2,5[C_{5} ⊃ × ⊏^{2(2)}D_{8}]_{2} ⊗ A_{2} (2^{5} · 3 · 5)

(A_{2} ⊗ A_{4})^{4}

\sqrt{5},∞,2,5[C_{5} ⊃ × ⊏_{\sqrt{5}′}^{2(2)}SL_{2}(3)]_{2} ⊗ A_{2} (2^{5} · 3^{2} · 5)

(A_{2} ⊗ E_{8})^{2}, (A_{2} ⊗ [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8})^{2}

\sqrt{5},∞,2,5[C_{5} ⊃ × ⊏_{\sqrt{5}}^{2(2)}(C_{3} ⊃ × ⊏^{2(2)}D_{8})]_{32} (2^{5} · 3 · 5)

[D_{120}.(C_{4} × C_{2})]_{16}^{2}

\sqrt{5},∞,2,5[C_{5} ⊃ × ⊏_{\sqrt{5}′}^{2(2)}(SL_{2}(3) □^{2} C_{3})]_{4} (2^{5} · 3^{2} · 5)

[(Sp_{4}(3) ∘ C_{3}) ⊠_{\sqrt{-3}}^{2} SL_{2}(3)]_{16}^{2}, [SL_{2}(5) ⊃ × ⊏_{∞,3}^{2(3)}(SL_{2}(3) □^{2} C_{3})]_{16}^{2}

\sqrt{5},∞,5,3[C_{5} ⊃ × ⊏^{2(3)}SL_{2}(9)]_{4} (2^{5} · 3^{2} · 5^{2})

O_{1} [SL_{2}(9) ⊗^{2(3)} SL_{2}(9) : 2]_{16}^{2}, [SL_{2}(5) ⊗^{2(3)} SL_{2}(9)]_{16}^{2}
O_{2} [(Sp_{4}(3)⊗ \sqrt{-3} Sp_{4}(3)) : 2 □^{2} C_{3}]_{32}, [SL_{2}(5) ⊃ × ⊏^{2(3)}(Sp_{4}(3) □^{2} C_{3})]_{32}

\sqrt{5},∞[SL_{2}(5)]_{1}⊗ \sqrt{5} _{\sqrt{5},5,3}[C_{5} ⊃ × ⊏_{\sqrt{5}′}^{2(3)} S̃_{3}]_{2} (2^{5} · 3^{2} · 5^{2})

O_{1} [(SL_{2}(5) ∘ SL_{2}(5)) : 2 ⊠_{\sqrt{5}}^{2} D_{10}]_{16}^{2}
O_{2} [(SL_{2}(5) ∘ SL_{2}(5)) : 2 ⊠_{\sqrt{5}}^{2} D_{10}]_{16}^{2}

\sqrt{5},∞[SL_{2}(5)]_{1}⊗ \sqrt{5} _{\sqrt{5},5,3}[C_{5} ⊃ × ⊏^{2(3)}S_{3}]_{2} (2^{5} · 3^{2} · 5^{2})

O_{1} (A_{2} ⊗ E_{8})^{2}, (A_{2} ⊗ [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8})^{2}
O_{2} (A_{2} ⊗ E_{8})^{2}, (A_{2} ⊗ [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8})^{2}

\sqrt{5},∞,5,3[C_{5} ⊃ × ⊏_{\sqrt{5}}^{2(3)}(C_{3} ⊃ × ⊏^{2(2)}D_{8})]_{32} (2^{5} · 3 · 5)

O_{1} [D_{120}.(C_{4} × C_{2})]_{16}^{2}, ([C_{15} : C_{4} ⊃ × ⊏^{2(2)}F_{4}]_{32})
O_{2} [D_{120}.(C_{4} × C_{2})]_{16}^{2}

\sqrt{5},∞,5,3[C_{5} ⊃ × ⊏_{\sqrt{5}′}^{2(3)}(C_{3} ⊃ × ⊏^{2(2)}D_{8})]_{32} (2^{5} · 3 · 5)

O_{1} [SL_{2}(5) ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+4}.Alt_{5}]_{16}^{2}, (F_{4} ˜⊗F_{4})^{2}
O_{2} [SL_{2}(5) ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+4}.Alt_{5}]_{16}^{2}, (F_{4} ˜⊗F_{4})^{2}

\sqrt{5},∞,5,3[C_{5} ⊃ × ⊏_{\sqrt{5}′}^{2(3)}(SL_{2}(3) □^{2} C_{3})]_{4} (2^{5} · 3^{2} · 5)

O_{1} [(Sp_{4}(3) ∘ C_{3}) ⊠_{\sqrt{-3}}^{2} SL_{2}(3)]_{16}^{2}, [SL_{2}(5) ⊃ × ⊏_{∞,3}^{2(3)}(SL_{2}(3) □^{2} C_{3})]_{16}^{2}
O_{2} [(Sp_{4}(3) ∘ C_{3}) ⊠_{\sqrt{-3}}^{2} SL_{2}(3)]_{16}^{2}, [SL_{2}(5) ⊃ × ⊏_{∞,3}^{2(3)}(SL_{2}(3) □^{2} C_{3})]_{16}^{2}

\sqrt{5},∞,5,3[C_{5} ⊃ × ⊏^{2(3)}(SL_{2}(3) □^{2} C_{3})]_{4} (2^{5} · 3^{2} · 5)

O_{1} (A_{4} ⊗ F_{4})^{2}
O_{2} (A_{4} ⊗ F_{4})^{2}

\sqrt{5},∞,2,3[C_{5} ⊃ × ⊏_{\sqrt{5}′}^{2(6)}(S̃_{3} ⊗ D_{8})]_{32} (2^{5} · 3 · 5)

(E_{8}^{4}), ([(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8}^{4})

\sqrt{5},∞,2,3[C_{5} ⊃ × ⊏^{2(6)}(S_{3} ⊗ D_{8})]_{32} (2^{5} · 3 · 5)

((A_{2} ⊗ A_{4})^{4})

\sqrt{5},∞,2,3[C_{5} ⊃ × ⊏_{\sqrt{5}}^{2(6)}(C_{3} ⊃ × ⊏^{2(2)}D_{8})]_{32} (2^{5} · 3 · 5)

[D_{120}.(C_{4} × C_{2})]_{16}^{2}, [((SL_{2}(5) ∘ SL_{2}(5)) : 2 ⊃ × ⊏_{\sqrt{5}}^{2(6)}(C_{3} ⊃ × ⊏^{2(2)}D_{8})]_{32,1}

\sqrt{5},∞,2,3[C_{5} ⊃ × ⊏_{\sqrt{5}′}^{2(6)}(C_{3} ⊃ × ⊏^{2(2)}D_{8})]_{32} (2^{5} · 3 · 5)

([SL_{2}(5) ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+4}.Alt_{5}]_{16}^{2})

\sqrt{5},∞,2,3[C_{5} ⊃ × ⊏_{\sqrt{5}′}^{2(6)}S_{3} ⊗ SL_{2}(3))]_{4} (2^{5} · 3^{2} · 5)

((A_{2} ⊗ E_{8})^{2}), ((A_{2} ⊗ [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8})^{2})

\sqrt{5},∞,2,3[C_{5} ⊃ × ⊏_{\sqrt{5}}^{2(6)}(S̃_{3}⊗ \sqrt{-3} SL_{2}(3))]_{4} (2^{5} · 3^{2} · 5)

[((SL_{2}(5) ∘ SL_{2}(5)) ⊠_{\sqrt{5}}^{2} (SL_{2}(5) ∘ SL_{2}(5))) : S_{4}]_{32,1}

\sqrt{5},∞,2,3[C_{5} ⊃ × ⊏^{2(6)}(SL_{2}(3) □^{2} C_{3})]_{4} (2^{5} · 3^{2} · 5)

((A_{4} ⊗ F_{4})^{2})

\sqrt{5},∞,2,3[C_{5} ⊃ × ⊏_{\sqrt{5}′}^{2(6)}(SL_{2}(3) □^{2} C_{3})]_{4} (2^{5} · 3^{2} · 5)

([SL_{2}(5) ⊃ × ⊏_{∞,3}^{2(3)}(SL_{2}(3) □^{2} C_{3})]_{16}^{2})

\sqrt{6},∞[(SL_{2}(9) ⊗ D_{8}).2]_{4} (2^{7} · 3^{2} · 5)

O_{1} [2_{+}^{1+10}.O_{10}^{+}(2)]_{32}, [(Sp_{4}(3)⊗ \sqrt{-3} Sp_{4}(3)) : 2 □^{2} C_{3}]_{32}
O_{2} [(Sp_{4}(3)⊗ \sqrt{-3} Sp_{4}(3)) : 2 □^{2} C_{3}]_{32}, E_{8}^{4}
O_{3} (F_{4} ˜⊗F_{4})^{2}, [SL_{2}(9).2 ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+4}.Alt_{5}]_{32}

\sqrt{6},∞[(S_{3} ⊗ 2_{−}^{1+4}.Alt_{5}).2]_{4} (2^{9} · 3^{2} · 5)

O_{1} (A_{2} ⊗ E_{8})^{2}, [2_{+}^{1+10}.O_{10}^{+}(2)]_{32}
O_{2} (A_{2} ⊗ E_{8})^{2}, [SL_{2}(5) ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+4}.Alt_{5}]_{16}^{2}, E_{8}^{4}
O_{3} (F_{4} ˜⊗F_{4})^{2}, A_{2} ⊗ F_{4} ˜⊗F_{4}, [SL_{2}(5) ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+6}.O_{6}^{−}(2)]_{32}

\sqrt{6},∞[(S_{3} ⊗ F_{4}).2]_{4} (2^{9} · 3^{3})

O_{1} (F_{4} ˜⊗F_{4})^{2}, A_{2} ⊗ F_{4} ˜⊗F_{4}, [SL_{2}(5) ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+6}.O_{6}^{−}(2)]_{32}
O_{2} (A_{2} ⊗ F_{4})^{4}, (F_{4} ˜⊗F_{4})^{2}
O_{3} (A_{2} ⊗ E_{8})^{2}, [2_{+}^{1+10}.O_{10}^{+}(2)]_{32}

\sqrt{7},∞[2.Alt_{7} ⊠^{2} C_{4}]_{4} (2^{6} · 3^{2} · 5 · 7)

O_{1} E_{8}^{4}, (F_{4} ˜⊗F_{4})^{2}, [2.Alt_{7} ⊃ × ⊏_{\sqrt{-7}}^{2(3)} S̃_{3}]_{16}^{2}
O_{2} [(2.Alt_{7} ⊠_{\sqrt{-7}}^{2} 2.Alt_{7}) : 2]_{32}, [2.Alt_{7} ⊃ × ⊏_{\sqrt{-7}}^{2(3)}(SL_{2}(3) □^{2} C_{3})]_{32}
O_{3} F_{4} ⊗ E_{8}, [2_{+}^{1+10}.O_{10}^{+}(2)]_{32}

\sqrt{7},∞[SL_{2}(7) ⊠^{2} C_{4}]_{4} (2^{6} · 3 · 7)

O_{1} (F_{4} ˜⊗F_{4})^{2}, [SL_{2}(7) ⊃ × ⊏_{\sqrt{-7}}^{2(3)} S̃_{3}]_{16}^{2}
O_{2} [SL_{2}(7) ⊠_{\sqrt{-7}}^{2} 2.Alt_{7}]_{32}, [SL_{2}(7) ⊃ × ⊏_{\sqrt{-7}}^{2(3)}(SL_{2}(3) □^{2} C_{3})]_{32}
O_{3} [2_{+}^{1+10}.O_{10}^{+}(2)]_{32}

\sqrt{10},∞[SL_{2}(5) ⊠^{2} D_{16}]_{4} (2^{7} · 3 · 5)

O_{1} E_{8}^{4}, [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8}^{4}, F_{4} ⊗ [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8},

F_{4} ⊗ E_{8}, [(SL_{2}(5) ∘ SL_{2}(5)) : 2 ⊃ × ⊏_{\sqrt{5}}^{2(6)}(C_{3} ⊃ × ⊏^{2(2)}D_{8})]_{32,i}(i = 1, 2)

O_{2} [SL_{2}(5) ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+6}.O_{6}^{−}(2)]_{32}, [(Sp_{4}(3)⊗ \sqrt{-3} Sp_{4}(3)) : 2 □^{2} C_{3}]_{32},

(F_{4} ˜⊗F_{4})^{2}, [SL_{2}(5) ⊃ × ⊏_{∞,3}^{2(3)}(Sp_{4}(3) □^{2} C_{3})]_{32}, [2_{+}^{1+10}.O_{10}^{+}(2)]_{32},

[SL_{2}(9).2 ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+4}.Alt_{5}]_{32}, [SL_{2}(5) ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+4}.Alt_{5}]_{16}^{2}

O_{3} ∼
O_{4} ∼

\sqrt{10},∞[D_{10} ⊠^{2} S̃_{4}]_{4} (2^{6} · 3 · 5)

O_{1} [SL_{2}(9) ⊗ D_{10} □^{2} SL_{2}(5)]_{32}, [(SL_{2}(5) ∘ SL_{2}(5)) : 2 ⊠_{\sqrt{5}}^{2} D_{10}]_{16}^{2},

[(2_{−}^{1+4}.Alt_{5}⊗_{∞,2} SL_{2}(5)) ⊠_{\sqrt{5}}^{2} D_{10}]_{32}, [(SL_{2}(5)⊗ \sqrt{5} D_{10}) ⊃ × ⊏_{∞,3}^{2(3)}(SL_{2}(3) □^{2} C_{3})]_{32}

O_{2} (A_{4} ⊗ F_{4})^{2}, A_{4} ⊗ E_{8}, [C_{15} : C_{4} ⊃ × ⊏^{2(2)}F_{4}]_{32}
O_{3} ∼
O_{4} ∼

\sqrt{15},∞[SL_{2}(5) ⊠^{2} D_{24}]_{4,1} (2^{6} · 3^{2} · 5)

O_{1} (A_{2} ⊗ E_{8})^{2}, [(SL_{2}(5) ∘ SL_{2}(5)) : 2 ⊃ × ⊏_{\sqrt{5}}^{2(6)}(C_{3} ⊃ × ⊏^{2(2)}D_{8})]_{32,2}
O_{2} A_{2} ⊗ (F_{4} ˜⊗F_{4}), [SL_{2}(5) ⊃ × ⊏_{∞,3}^{2(3)}(Sp_{4}(3) □^{2} C_{3})]_{32}
O_{3} E_{8}^{4}, [SL_{2}(5) ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+4}.Alt_{5}]_{16}^{2}
O_{4} E_{8}^{4}, [SL_{2}(5) ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+4}.Alt_{5}]_{16}^{2}
O_{5} (A_{2} ⊗ E_{8})^{2}, [SL_{2}(5) ⊃ × ⊏_{∞,3}^{2(3)}(C_{3} □^{2} SL_{2}(3))]_{16}^{2}
O_{6} (F_{4} ˜⊗F_{4})^{2}, [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8}^{4}
O_{7} [2_{+}^{1+10}.O_{10}^{+}(2)]_{32}, F_{4} ⊗ [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8}
O_{8} [SL_{2}(5) ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+6}.O_{6}^{−}(2)]_{32}, F_{4} ⊗ E_{8}

\sqrt{15},∞[SL_{2}(5) ⊠^{2} D_{24}]_{4,2} (2^{6} · 3^{2} · 5)

O_{1} (A_{2} ⊗ [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8})^{2}, [(SL_{2}(5) ∘ SL_{2}(5)) : 2 ⊃ × ⊏_{\sqrt{5}}^{2(6)}(C_{3} ⊃ × ⊏^{2(2)}D_{8})]_{32,1}
O_{2} [SL_{2}(5) ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+4}.Alt_{5}]_{16} ⊗ A_{2}, [(Sp_{4}(3)⊗ \sqrt{-3} Sp_{4}(3)) : 2 □^{2} C_{3}]_{32}
O_{3} (F_{4} ˜⊗F_{4})^{2}, [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8}^{4}
O_{4} (F_{4} ˜⊗F_{4})^{2}, [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8}^{4}
O_{5} (A_{2} ⊗ [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8})^{2}, [SL_{2}(5) ⊃ × ⊏_{∞,3}^{2(3)}(Sp_{4}(3) □^{2} C_{3})]_{32}
O_{6} [SL_{2}(5) ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+4}.Alt_{5}]_{16}^{2}, E_{8}^{4}
O_{7} [SL_{2}(5) ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+6}.O_{6}^{−}(2)]_{32}, F_{4} ⊗ E_{8}
O_{8} [2_{+}^{1+10}.O_{10}^{+}(2)]_{32}, F_{4} ⊗ [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8}

\sqrt{15},∞[SL_{2}(5) ⊠^{2} D_{24}]_{4,3} (2^{6} · 3^{2} · 5)

O_{1} F_{4} ⊗ E_{8}, [(SL_{2}(5) ∘ SL_{2}(5)) : 2 ⊃ × ⊏_{\sqrt{5}}^{2(6)}(C_{3} ⊃ × ⊏^{2(2)}D_{8})]_{32,1}
O_{2} [2_{+}^{1+10}.O_{10}^{+}(2)]_{32}, F_{4} ⊗ [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8}
O_{3} (A_{2} ⊗ [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8})^{2}, [(Sp_{4}(3) ∘ C_{3}) ⊠_{\sqrt{-3}}^{2} SL_{2}(3)]_{16}^{2}
O_{4} (A_{2} ⊗ [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8})^{2}, [(Sp_{4}(3) ∘ C_{3}) ⊠_{\sqrt{-3}}^{2} SL_{2}(3)]_{16}^{2}
O_{5} (F_{4} ˜⊗F_{4})^{2}, [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8}^{4}
O_{6} (A_{2} ⊗ E_{8})^{2}, [SL_{2}(5) ⊃ × ⊏_{∞,3}^{2(3)}(SL_{2}(3) □^{2} C_{3})]_{16}^{2}
O_{7} A_{2} ⊗ F_{4} ˜⊗F_{4}, [SL_{2}(5) ⊃ × ⊏_{∞,3}^{2(3)}(Sp_{4}(3) □^{2} C_{3})]_{32}
O_{8} [SL_{2}(5) ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+4}.Alt_{5}]_{16} ⊗ A_{2}, [(Sp_{4}(3)⊗ \sqrt{-3} Sp_{4}(3)) : 2 □^{2} C_{3}]_{32}

\sqrt{15},∞[SL_{2}(5) ⊠^{2} D_{24}]_{4,4} (2^{6} · 3^{2} · 5)

O_{1} F_{4} ⊗ [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8}, [(SL_{2}(5) ∘ SL_{2}(5)) : 2 ⊃ × ⊏_{\sqrt{5}}^{2(6)}(C_{3} ⊃ × ⊏^{2(2)}D_{8})]_{32,1}
O_{2} [SL_{2}(5) ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+6}.O_{6}^{−}(2)]_{32}, F_{4} ⊗ E_{8}
O_{3} (A_{2} ⊗ E_{8})^{2}, [SL_{2}(5) ⊃ × ⊏_{∞,3}^{2(3)}(SL_{2}(3) □^{2} C_{3})]_{16}^{2}
O_{4} (A_{2} ⊗ E_{8})^{2}, [SL_{2}(5) ⊃ × ⊏_{∞,3}^{2(3)}(SL_{2}(3) □^{2} C_{3})]_{16}^{2}
O_{5} E_{8}^{4}, [SL_{2}(5) ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+4}.Alt_{5}]_{16}^{2}
O_{6} (A_{2} ⊗ [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8})^{2}, [(Sp_{4}(3) ∘ C_{3}) ⊠_{\sqrt{-3}}^{2} SL_{2}(3)]_{16}^{2}
O_{7} A_{2} ⊗ [SL_{2}(5) ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+4}.Alt_{5}]_{16}, [(Sp_{4}(3)⊗ \sqrt{-3} Sp_{4}(3)) : 2 □^{2} C_{3}]_{32}
O_{8} A_{2} ⊗ F_{4} ˜⊗F_{4}, [SL_{2}(5) ⊃ × ⊏_{∞,3}^{2(3)}(Sp_{4}(3) □^{2} C_{3})]_{32}

\sqrt{15},∞[D_{10} ⊠^{2} Q_{24}]_{4,1} (2^{5} · 3 · 5)

O_{1} [2_{−}^{1+4}.Alt_{5}⊗_{∞,2} SL_{2}(5) ⊠_{\sqrt{5}}^{2} D_{10}]_{32}, [(SL_{2}(5) ∘ SL_{2}(5)) : 2 ⊠_{\sqrt{5}}^{2} D_{10}]_{16}^{2}
O_{2} [2_{−}^{1+4}.Alt_{5}⊗_{∞,2} SL_{2}(5) ⊠_{\sqrt{5}}^{2} D_{10}]_{32}, A_{4} ⊗ E_{8}
O_{3} [D_{120}.(C_{4} × C_{2})]_{16}^{2}, (A_{2} ⊗ A_{4})^{4}
O_{4} [D_{120}.(C_{4} × C_{2})]_{16}^{2}, (A_{2} ⊗ A_{4})^{4}
O_{5} [2_{−}^{1+4}.Alt_{5}⊗_{∞,2} SL_{2}(5) ⊠_{\sqrt{5}}^{2} D_{10}]_{32}, (A_{4} ⊗ F_{4})^{2}
O_{6} [D_{120}.(C_{4} × C_{2})]_{16}^{2}, (A_{2} ⊗ A_{4})^{4}
O_{7} A_{2} ⊗ A_{4} ⊗ F_{4}, [C_{15} : C_{4} ⊃ × ⊏^{2(2)}F_{4}]_{32}
O_{8} A_{2} ⊗ A_{4} ⊗ F_{4}, [C_{15} : C_{4} ⊃ × ⊏^{2(2)}F_{4}]_{32}

\sqrt{15},∞[D_{10} ⊠^{2} Q_{24}]_{4,2} (2^{5} · 3 · 5)

O_{1} [2_{−}^{1+4}.Alt_{5}⊗_{∞,2} SL_{2}(5) ⊠_{\sqrt{5}}^{2} D_{10}]_{32}, [(SL_{2}(5) ∘ SL_{2}(5)) : 2 ⊠_{\sqrt{5}}^{2} D_{10}]_{16}^{2}
O_{2} A_{2} ⊗ A_{4} ⊗ F_{4}, [C_{15} : C_{4} ⊃ × ⊏^{2(2)}F_{4}]_{32}
O_{3} (F_{4} ⊗ A_{4})^{2}, [(SL_{2}(5) ∘ SL_{2}(5)) : 2 ⊠_{\sqrt{5}}^{2} D_{10}]_{16}^{2}
O_{4} (A_{4} ⊗ F_{4})^{2}, [(SL_{2}(5) ∘ SL_{2}(5)) : 2 ⊠_{\sqrt{5}}^{2} D_{10}]_{16}^{2}
O_{5} [D_{120}.(C_{4} × C_{2})]_{16}^{2}, [(SL_{2}(5) ∘ SL_{2}(5)) : 2 ⊠_{\sqrt{5}}^{2} D_{10}]_{16}^{2}
O_{6} (F_{4} ⊗ A_{4})^{2}, [(SL_{2}(5) ∘ SL_{2}(5)) : 2 ⊠_{\sqrt{5}}^{2} D_{10}]_{16}^{2}
O_{7} [2_{−}^{1+4}.Alt_{5}⊗_{∞,2} SL_{2}(5) ⊠_{\sqrt{5}}^{2} D_{10}]_{32}, A_{4} ⊗ E_{8}
O_{8} [2_{−}^{1+4}.Alt_{5}⊗_{∞,2} SL_{2}(5) ⊠_{\sqrt{5}}^{2} D_{10}]_{32}, A_{4} ⊗ E_{8}

\sqrt{17},∞[SL_{2}(17)]_{4} (2^{5} · 3^{2} · 17)

[SL_{2}(17) ⊃ ⊏^{2(3)} S̃_{3}]_{32,i}(i = 1, 2)

\sqrt{21},∞[2.Alt_{7} ⊠^{2} C_{3}]_{4} (2^{5} · 3^{3} · 5 · 7)

O_{1} (A_{2} ⊗ E_{8})^{2}, [2.Alt_{10}]_{16}^{2}, [2.Alt_{7} ⊃ × ⊏_{\sqrt{-7}}^{2(3)} S̃_{3}]_{16}^{2}
O_{2} [2_{+}^{1+10}.O_{10}^{+}(2)]_{32}, [(2.Alt_{7} ⊠_{\sqrt{-7}}^{2} 2.Alt_{7}) : 2]_{32}

\sqrt{21},∞[SL_{2}(7) ⊠^{2} C_{3}]_{4} (2^{5} · 3^{2} · 7)

O_{1} [SL_{2}(7) ⊃ × ⊏_{\sqrt{-7}}^{2(3)} S̃_{3}]_{16}^{2}
O_{2} [SL_{2}(7) ⊠_{\sqrt{-7}}^{2} 2.Alt_{7}]_{32}

In this table, the symbol ∼ means that the r.i.m.f. supergroups acting on the O_{3}G- and O_{4}G-lattices are the same as the ones for O_{2}.

The proof is split into twelve lemmata which are organized according to the different candidates for quasi-semi-simple normal subgroups and normal p-subgroups. For the rest of this section let Q be a definite quaternion algebra with center K and let G be a primitive a.i.m.f. subgroup of GL_{4}(Q). Assume that 1 ≠ N ⊴ G is a quasi-semi-simple normal subgroup of G. By Table 9.1 and Lemma 7.2 N is one of Alt_{5}, SL_{2}(5), SL_{2}(5) ∘ SL_{2}(5), SL_{2}(5) ∘ SL_{2}(5)⊗ \sqrt{5} SL_{2}(5), SL_{2}(7) (2 groups), SL_{2}(9) (2 groups), SL_{2}(17), 2.Alt_{7}, or Sp_{4}(3) = 2.U_{4}(2).

The first lemma deals with the absolutely irreducible candidates for normal subgroups N.

Lemma 14.15. If G contains a normal subgroup N isomorphic to SL_{2}(9) with character χ_{8a} (or χ_{8b}) resp. SL_{2}(17) with character χ_{8a} (or χ_{8b}), then G is conjugate to _{\sqrt{5},∞}[SL_{2}(9)]_{4} resp. _{\sqrt{17},∞}[SL_{2}(17)]_{4}.

If G contains a normal subgroup N conjugate to SL_{2}(5)∘SL_{2}(5)⊗ \sqrt{5} SL_{2}(5), then G = _{\sqrt{5},∞}[(SL_{2}(5) ∘ SL_{2}(5)⊗ \sqrt{5} SL_{2}(5)) : S_{3}]_{4}.

Proof. In both cases N is already absolutely irreducible. One computes that G = B^{∘}(N) is maximal finite.

The next two lemmata deal with primitively saturated groups.

Lemma 14.16. If G contains a normal subgroup N ≅ Alt_{5} with character χ_{4}, then G is one of _{\sqrt{2},∞}[S̃_{4}]_{1} ⊗ A_{4} or _{\sqrt{3},∞}[Q_{24}]_{1} ⊗ A_{4}.

Proof. By Proposition 7.5 G is of the form A_{4} ⊗ H, where H ≤ GL_{1}(Q) is a primitive a.i.m.f. group. Hence by Theorem 6.1 H is one of _{\sqrt{2},∞}[S̃_{4}]_{1}, _{\sqrt{3},∞}[Q_{24}]_{1}, or _{\sqrt{5},∞}[SL_{2}(5)]_{1}. The lemma follows because G = _{\sqrt{5},∞}[SL_{2}(5)]_{1} ⊗A_{4} is contained in _{\sqrt{5},∞}[(SL_{2}(5) ∘ SL_{2}(5)⊗ \sqrt{5} SL_{2}(5)) : S_{3}]_{4}.

Lemma 14.17. N is not conjugate to SL_{2}(5) ∘ SL_{2}(5).

Proof. Assume that G contains a normal subgroup N = SL_{2}(5) ∘ SL_{2}(5). The enveloping algebra of N is ℚ[\sqrt{5}]^{4×4}. Hence K = ℚ[\sqrt{5}]. Since B := B^{∘}(N) = SL_{2}(5) ∘ SL_{2}(5) : 2 is primitively saturated over K, the group G = BC, where C := C_{G}(N) is a centrally irreducible maximal finite subgroup of GL_{1}(Q). Hence Q = Q_{\sqrt{5},∞} = K ⊗ Q_{∞,2} = K ⊗ Q_{∞,3} and C = SL_{2}(5). But this contradicts Lemma 14.15.

Now we come to the centrally irreducible groups N.

Lemma 14.18. If G contains a normal subgroup N isomorphic to SL_{2}(7) with character χ_{8}, then G is conjugate to _{\sqrt{3},∞}[SL_{2}(7).2]_{4}.

Proof. The group N is a centrally irreducible subgroup of GL_{4}(Q). Therefore C_{G}(N) ⊆ K is ^{±}1. Since G is absolutely irreducible, it contains N of index 2. With [CCNPW 85] one gets G = _{\sqrt{3},∞}[SL_{2}(7).2]_{4}.

There are three candidates N, for which the centralizer C_{G}(N) is contained in the character field K[χ(N)] of a constituent of the natural character of N.

Lemma 14.19. If G contains a normal subgroup N isomorphic to SL_{2}(7), 2.Alt_{7}, resp. Sp_{4}(3), with character χ_{4a} + χ_{4b}, then G is conjugate to one of

\sqrt{7},∞[SL_{2}(7) ⊠^{2} C_{4}]_{4} or _{\sqrt{21},∞}[SL_{2}(7) ⊠^{2} C_{3}]_{4},

\sqrt{7},∞[2.Alt_{7} ⊠^{2} C_{4}]_{4} or _{\sqrt{21},∞}[2.Alt_{7} ⊠^{2} C_{3}]_{4},

resp. _{\sqrt{3},∞}[Sp_{4}(3) ⊠_{\sqrt{-3}}^{2} C_{12}]_{4}.

Proof. In all cases C_{G}(N) =: C is contained in the extension of K by the character values of the natural character of N. Hence in the first two cases C ≤ K[\sqrt{−7}]^{∗} and C ≤ K[\sqrt{−3}]^{∗} in the last case. In all cases G contains the normal subgroup B^{∘}(N)C of index 2. Since the dimension of the enveloping ℚ-algebra of N is 32, Lemma 2.14 implies that C is not contained in N. Hence in the first two cases C is one of C_{4} or C_{3} and K = ℚ[\sqrt{7}] or ℚ[\sqrt{21}]. By Lemma 2.17 there is a unique extension G = (N ⊗ C_{4}).2 or G = (N ⊗ C_{3}).2 with real Schur index 2. The maximality of these four groups is checked with Remark 2.6. In the last case, B^{∘}(N) = Sp_{4}(3)∘C_{3} has a nontrivial normal 3-subgroup. The primitivity of G implies that C = C_{12}

and K = ℚ[\sqrt{3}]. Again G = _{\sqrt{3},∞}[Sp_{4}(3) ⊠_{\sqrt{-3}}^{2} C_{12}]_{4} is unique.

In the next case \overline{KN} is a proper central simple K-subalgebra of Q^{4×4}.

Lemma 14.20. If G contains a normal subgroup N = SL_{2}(9) with character χ_{4}, then G is conjugate to one of _{∞,3}[SL_{2}(9)]_{2} ⊗ \sqrt{2}[D_{16}]_{2}, _{∞,3}[SL_{2}(9)]_{2} ⊗ \sqrt{5}[D_{10}]_{2},

\sqrt{5},∞,3,5[C_{5} ⊃ × ⊏^{2(3)}SL_{2}(9)]_{4}, or _{\sqrt{6},∞}[(SL_{2}(9) ⊗ D_{8}).2]_{4}.

Proof. Since \overline{KN} is central simple, by [Rei 75, 7.11] the algebra \overline{ℚG} = Q^{4×4} is a tensor product Q^{4×4} = \overline{KN}⊗_{K} A, where A is the commuting algebra of N, an indefinite quaternion algebra over K. Let B := B_{K}^{∘}(N) and C := C_{G}(N). Then O_{3}(C) = 1, because B^{∘}(N ∘ C_{3}) = Sp_{4}(3) ∘ C_{3}. Distinguish 2 cases:

a) K = ℚ[\sqrt{3}]. Then B = 2.S_{6} is primitively saturated. By Lemma 7.5, G = B⊗_{K} C for some centrally irreducible maximal finite subgroup C ≤ A. Using the classification of finite subgroups of GL_{2}(ℂ) in [Bli 17], one finds that C = D_{24} (which contains ^{±}S_{3} and D_{8}) contradicting O_{3}(C) = 1.

b) Now let K ≠ ℚ[\sqrt{3}]. Then B = N = SL_{2}(9) and G contains the normal subgroup BC of index ≤ 2. Assume first, that C is a centrally irreducible subgroup of A. Then A = K^{2×2} by Remark 6.2 and C ≤ GL_{2}(K) is a dihedral group with O_{3}(C) = 1. Hence C = ^{±}D_{10}, D_{16}, or D_{8}. In the first two cases C is an absolutely irreducible subgroup of GL_{2}(K) for K = ℚ[\sqrt{5}] resp. ℚ[\sqrt{2}]. Computing the automorphism groups of the NC-lattices one finds that G is _{∞,3}[SL_{2}(9)]_{2} ⊗ \sqrt{5}[D_{10}]_{2} resp. _{∞,3}[SL_{2}(9)]_{2} ⊗ \sqrt{2}[D_{16}]_{2}. In the third case NC = SL_{2}(9) ⊗ D_{8} is not absolutely irreducible. Since Out(NC) = C_{2} × C_{2}, the group G = NC.2 is one of _{\sqrt{3},∞,∞}[2.S_{6}]_{2} ⊗ [D_{8}]_{2} (and imprimitive), _{∞,3}[SL_{2}(9)]_{2} ⊗ \sqrt{2}[D_{16}]_{2} (leading to a bigger C), or the a.i.m.f. group _{\sqrt{6},∞}[(SL_{2}(9) ⊗ D_{8}).2]_{4}, because in each case there is a unique extension with real character field. If C is not centrally irreducible, then C is cyclic. The conditions G = NC.2 and O_{3}(C) = 1 imply that C is one of

C_{8} or C_{5} and G = C_{8} ⊃ × ⊏^{2(3)} SL_{2}(9) = D_{16} ^{|C^{2}} ◇_{|} 2.S_{6} or G = C_{5} ⊃ × ⊏^{2(3)} SL_{2}(9). Since 3

is a norm in ℚ[ζ_{8}]/ℚ[\sqrt{2}] but not in ℚ[ζ_{5}]/ℚ[\sqrt{5}], the algebra Q is Q_{\sqrt{2},∞} in the first case and Q_{\sqrt{5},∞,3,5} in the second case. Whereas the second group is maximal finite, the first one is a subgroup of _{\sqrt{2},∞}[2_{−}^{1+6}.O_{6}^{−}(2).2]_{4}.

The last and most fruitful case is the one where G has a normal subgroup N ≅ SL_{2}(5). This case is split up into two lemmata according to whether N is primitively saturated over K or not.

Lemma 14.21. If K = ℚ[\sqrt{5}] and G contains a normal subgroup N ≅ SL_{2}(5) with character χ_{2a} (or χ_{2b}), then G is _{\sqrt{5},∞}[SL_{2}(5)]_{1}⊗ \sqrt{5} C, where C is one of

F_{4}, \sqrt{5}[D_{10}]_{2} ⊗ A_{2}, _{\sqrt{5},3,5}[C_{5} ⊃ × ⊏^{2(3)} S_{3}]_{2}, _{\sqrt{5},3,5}[C_{5} ⊃ × ⊏^{2(3)} S̃_{3}]_{2}, _{\sqrt{5},2,5}[C_{5} ⊃ × ⊏^{2(2)} D_{8}]_{2}, or

\sqrt{5},2,5[C_{5} ⊃ × ⊏^{2(2)}SL_{2}(3)]_{2}.

Proof. N is primitively saturated over K = ℚ[\sqrt{5}]. Hence by Lemma 7.5 G = NC, for some primitive centrally irreducible maximal finite subgroup C := C_{G}(N) ≤ D^{∗}, where D := C_{Q^{4×4}}(N). By the formula in [Schu 05] (cf. Proposition 2.16), the only primes dividing the order of G are 2, 3, and 5. If C is not absolutely irreducible in D^{∗}, then C is a maximal finite subgroup of GL_{4}(ℚ), because ℚ[\sqrt{5}] splits the possible p-adic Schur indices at p = 2, 3, and 5. Using Lemma 14.16 and the classification of maximal finite subgroups of GL_{4}(ℚ) (cf. e.g. [BBNWZ 78]), one gets that G = _{\sqrt{5},∞}[SL_{2}(5)]_{1} ⊗ F_{4}. Now assume that C is absolutely irreducible. Then the character field of the natural character of C is K = ℚ[\sqrt{5}]. By Lemmas 14.17 and 14.15 C has no normal subgroup SL_{2}(5) or SL_{2}(5) ∘ SL_{2}(5). With [CCNPW 85] one finds that C is soluble. An inspection of the possible normal p-subgroups yields O_{5}(C) = C_{5}. The centralizer D := C_{C}(O_{5}(C)) embeds into C_{D}(C) which is a quaternion algebra over ℚ[ζ_{5}] and C contains D of index 2. Since C is absolutely irreducible, [Bli 17] yields the possibilities D = ^{±}C_{5}E, where E is one of SL_{2}(3), D_{8}, S_{3}, or S̃_{3}.

If C_{C}(E) > ^{±}C_{5}, then C is one of^{±}C_{5}.C_{2}⊗ \sqrt{−1} SL_{2}(3), ^{±}D_{10}⊗D_{8}, ^{±}C_{5}.C_{2}⊗ \sqrt{−1} S̃_{3}, or ^{±}D_{10} ⊗ S_{3}. Now the first and third groups are not maximal finite, but contained in SL_{2}(5)⊗ \sqrt{−1} SL_{2}(3) resp. SL_{2}(5)⊗ \sqrt{−1} S̃_{3} and the second group is imprimitive. So G = _{\sqrt{5},∞}[SL_{2}(5)]_{1}⊗ \sqrt{5} \sqrt{5}[^{±}D_{10} ⊗ S_{3}]_{2} in this case.

If C_{C}(E) = ^{±}C_{5}, then G = ^{±}C_{5} ⊃ × ⊏^{2(p)} E, for some square free p ∈ ℕ_{>1}. Since |Glide(E)| = 2 and the enveloping algebra of E is central simple, the outer automorphism and p are unique. By Lemma 2.17 there is a unique extension G in GL_{4}(ℝ), in each of the four cases.

Lemma 14.22. If K ≠ ℚ[\sqrt{5}] and G contains a normal subgroup N ≅ SL_{2}(5) with character χ_{2a} + χ_{2b}, then G is conjugate to one of

∞,5[SL_{2}(5).2]_{2} ⊗ \sqrt{2}[D_{16}]_{2}, _{\sqrt{2},∞,2,5}[SL_{2}(5) ⊠^{2(2+\sqrt{2})} D_{16}]_{4,1},

\sqrt{2},∞,2,5[SL_{2}(5) ⊠^{2(2+\sqrt{2})} D_{16}]_{4,2}, _{∞,5}[SL_{2}(5).2]_{2} ⊗ \sqrt{3}[D_{24}]_{2},

\sqrt{3},∞[SL_{2}(5) ⊃ × ⊏^{2(2+\sqrt{3})}D_{24}]_{4}, _{\sqrt{10},∞}[SL_{2}(5) ⊠^{2} D_{16}]_{4},

or _{\sqrt{15},∞}[SL_{2}(5) ⊠^{2} D_{24}]_{4,i} (1 ≤ i ≤ 4).

Proof. Let A := C_{Q^{4×4}}(N) be the commuting algebra of N. Then A is an indefinite quaternion algebra with center K[\sqrt{5}]. The centralizer C := C_{G}(N) embeds into A with \overline{ℚ[\sqrt{5}]C} = A. By [Bli 17] this implies that C is one of D_{16} or D_{24}. In both cases, the outer automorphism group of C is isomorphic to C_{2} × C_{2}.

Assume first that C ≅ D_{16}. Let D_{16} = ⟨x, y | x^{8}, y^{2}, (xy)^{2}⟩. Then Out(D_{16}) = ⟨α, β⟩, with α(x) = x^{−1}, α(y) = x^{3}y and β(x) = x^{3}, β(y) = y (cf. Lemma 8.5). Then α^{2} = β^{2} = id, but (αβ)^{2} is the conjugation by x. Since αβ does not fix x, this implies that there is no extension D_{16}.2, where αβ is an inner automorphism. Note that the action of α on the epimorphic image C is induced by conjugation with y(1 − x) and hence by an inner automorphism of \overline{ℚC}. If γ denotes the outer automorphism of SL_{2}(5), then G/NC induces one of γ, γα, or γβ on the central product NC. In all cases there are 2 = |H^{2}(C_{2}, Z(NC) ≅ C_{2})| extensions NC.2. Only in the last case do they lead to isomorphic groups, because there is an element of norm −1 in ℚ[\sqrt{2}]. Since the group _{∞,5}[SL_{2}(5) : 2]_{2} ⊗ \sqrt{2}[D_{16}]_{2} is contained in \sqrt{2},∞[2_{−}^{1+6}.O_{6}^{−}(2).2]_{4} the group G is one of the a.i.m.f. groups

∞,5[SL_{2}(5).2]_{2} ⊗ \sqrt{2}[D_{16}]_{2}, _{\sqrt{2},∞,2,5}[SL_{2}(5) ⊠^{2(2+\sqrt{2})} D_{16}]_{4,1},

\sqrt{2},∞,2,5[SL_{2}(5) ⊠^{2(2+\sqrt{2})} D_{16}]_{4,2}, or _{\sqrt{10},∞}[SL_{2}(5) ⊠^{2} D_{16}]_{4}.

The case C ≅ D_{24} is similar. Here all the groups C.2 exist and one has eight different groups NC.2, Since _{∞,5}[SL_{2}(5) : 2]_{2} ⊗ \sqrt{3}[D_{24}]_{2} is contained in

\sqrt{3},∞[Sp_{4}(3) ⊠_{\sqrt{-3}}^{2} C_{12}]_{4} and one extension SL_{2}(5) ⊃ × ⊏^{2(2+\sqrt{3})} D_{24} a proper subgroup

of _{\sqrt{3},∞}[D_{8} ⊗ D_{8} ⊗ C_{4}.S_{6} ⊠^{2} C_{3}]_{4} the group G is one of the six a.i.m.f. groups

∞,5[SL_{2}(5).2]_{2} ⊗ \sqrt{3}[D_{24}]_{2}, _{\sqrt{3},∞}[SL_{2}(5) ⊃ × ⊏^{2(2+\sqrt{3})} D_{24}]_{4}, or _{\sqrt{15},∞}[SL_{2}(5) ⊠^{2} D_{24}]_{4,i}

(1 ≤ i ≤ 4).

For the rest of this section we assume that G does not contain a quasi-semi-simple normal subgroup. By Lemma 11.2 and Corollary 2.4 one has O_{p}(G) = 1 for p > 5 and O_{p}(G) ≤ C_{p} for p = 3, 5.

Lemma 14.23. If G does not contain a quasi-semi-simple normal subgroup and O_{p}(G) = 1 for all odd primes p, then O_{2}(G) = 2_{−}^{1+6} = D_{8} ⊗ D_{8} ⊗ Q_{8} and G = \sqrt{2},∞[2_{−}^{1+6}.O_{6}^{−}(2).2]_{4}.

Proof. By Proposition 8.9 O_{2}(G) is one of 2_{−}^{1+6} or Q_{8} ∘ Q_{8} ⊗ Q_{16}. In the first case G = _{\sqrt{2},∞}[2_{−}^{1+6}.O_{6}^{−}(2).2]_{4} is maximal finite. In the other case N is already irreducible. The Bravais group on a normal critical N-lattice (cf. Definition 2.7) is \sqrt{2},∞[2_{−}^{1+4}.S_{5}]_{2}^{2} contradicting the primitivity of G.

Lemma 14.24. If G does not contain a quasi-semi-simple normal subgroup, O_{5}(G) = 1, and O_{3}(G) = C_{3}, then G is one of

\sqrt{2},∞[2_{−}^{1+4}.S_{5}]_{1} ⊗ A_{2}, _{\sqrt{2},∞,2,3}[C_{3} ⊃ ⊏^{2(2+\sqrt{2})}(S̃_{4}⊗ \sqrt{2} D_{16})]_{4},

\sqrt{2},∞,2,3[C_{3} ⊃ × ⊏^{2(2+\sqrt{2})}(S̃_{4} ∘ Q_{16})]_{4} ,_{\sqrt{3},∞}[D_{8} ⊗ D_{8} ⊗ C_{4}.S_{6} ⊠^{2} C_{3}]_{4},

\sqrt{6},∞[(S_{3} ⊗ 2_{−}^{1+4}.Alt_{5}).2]_{4}, or _{\sqrt{6},∞}[(S_{3} ⊗ F_{4}).2]_{4}.

Proof. The centralizer C := C_{G}(O_{3}(G)) is an absolutely irreducible subgroup of (ℚ[ζ_{3}] ⊗ Q^{2×2})^{∗} and G/C ≅ C_{2}. Table 8.7 gives that O_{2}(G) = O_{2}(C) is one of Q_{8} ∘ Q_{8} ⊗ C_{4}, D_{8} ⊗ Q_{8}, Q_{8} ∘ Q_{8}, Q_{8} ⊗ D_{16}, Q_{8}⊗ \sqrt{-2} QD_{16}, or Q_{8} ∘ Q_{16}.

In the first case K = ℚ[\sqrt{3}] and G contains B := B^{∘}(C) = C_{3} ⊗(C_{4} ⊗Q_{8} ∘Q_{8}).S_{6}

of index 2. Hence G is conjugate to _{\sqrt{3},∞}[D_{8} ⊗ D_{8} ⊗ C_{4}.S_{6} ⊠^{2} C_{3}]_{4} in this case.

If O_{2}(C) = Q_{8} ⊗ D_{8}, then G contains B = C_{3} ∘ 2_{−}^{1+4}.Alt_{5} of index 2^{2}. Hence C_{G}(O_{2}(C)) = C_{G}(B^{∘}(O_{2}(G)) = ^{±}S_{3}, and G is one of the two groups

\sqrt{6},∞[(S_{3} ⊗ 2_{−}^{1+4}.Alt_{5}).2]_{4} or _{\sqrt{2},∞}[2_{−}^{1+4}.S_{5}]_{1} ⊗ A_{2}.

Note that in both cases there is a unique extension with real character field.

In the case O_{2}(G) = Q_{8} ∘ Q_{8} one similarly finds that G contains S̃_{3} ⊗ F_{4} of index 2. Since the group F_{4}.2 ⊗ S̃_{3} is contained in _{\sqrt{2},∞}[2_{−}^{1+6}.O_{6}^{−}(2).2]_{4}, G is conjugate to _{\sqrt{6},∞}[(S_{3} ⊗ F_{4}).2]_{4} in this case.

If O_{2}(C) = Q_{8}⊗D_{16}, then K = ℚ[\sqrt{2}] and G contains B = S̃_{4}⊗ \sqrt{2} D_{16}∘C_{3} of index 2. If the elements in G − B induce an outer automorphism of B^{∘}(O_{2}(G)), then G is

conjugate to_{\sqrt{2},∞,2,3}[C_{3} ⊃ ⊏^{2(2+\sqrt{2})}(S̃_{4}⊗ \sqrt{2} D_{16})]_{4}. If they don’t, then C_{G}(O_{2}(G)) ≅ ^{±}S_{3}

and G is contained in _{\sqrt{2},∞}[2_{−}^{1+4}.S_{5}]_{2} ⊗ A_{2}.

Assume now that O_{2}(G) = Q_{8}⊗ \sqrt{-2} QD_{16}. Then K = ℚ[\sqrt{6}] and G/B ≅ C_{2}. Hence there is a unique group G = B.2 with real Schur index 2. This group is not maximal finite but contained in _{\sqrt{6},∞}[(S_{3} ⊗ 2_{−}^{1+4}.Alt_{5}).2]_{4}.

In the last case O_{2}(G) = Q_{8} ∘ Q_{16}. Now K = ℚ[\sqrt{2}] and G = B.2 is conjugate

to _{\sqrt{2},∞,2,3}[C_{3} ⊃ × ⊏^{2(2+\sqrt{2})}(S̃_{4} ∘ Q_{16})]_{4}, because S̃_{4} ∘ Q_{16} ⊗ S̃_{3} is contained in \sqrt{2}[F_{4}.2]_{4} ⊗

∞,3[S̃_{3}]_{1}.

The last and most complicated case is the case O_{5}(G) > 1. In this case O_{5}(G) ≅ C_{5}. Recall that we assume for the rest of this chapter, that G is a primitive a.i.m.f. group of GL_{4}(Q), K = Z(Q) a real quadratic field and that G does not contain a quasi-semi-simple normal subgroup. As for the case SL_{2}(5) ⊴ G, there are two essentially different situations: K = ℚ[\sqrt{5}] and K ≠ Q[\sqrt{5}] which are treated separately.

Lemma 14.25. If K = ℚ[\sqrt{5}] and O_{5}(G) = C_{5}, then G is one of

\sqrt{5},∞,2,5[C_{5} ⊃ × ⊏^{2(2)}F_{4}]_{4}, _{\sqrt{5},∞,2,5}[C_{5} ⊃ × ⊏_{\sqrt{5}′}^{2(2)}2_{−}^{1+4}.Alt_{5}]_{4},

∞,2[2_{−}^{1+4}.Alt_{5}]_{2} ⊗ \sqrt{5}[^{±}D_{10}]_{2}, _{∞,3}[SL_{2}(3) □^{2} C_{3}]_{2} ⊗ \sqrt{5}[^{±}D_{10}]_{2},

A_{2} ⊗_{\sqrt{5},∞,2,5}[C_{5} ⊃ × ⊏^{2(2)}D_{8}]_{2}, _{\sqrt{5},∞,2,3}[C_{5} ⊃ × ⊏^{2(6)}S_{3} ⊗ D_{8}]_{32},

\sqrt{5},∞,2,3[C_{5} ⊃ × ⊏_{\sqrt{5}′}^{2(6)} S̃_{3} ⊗ D_{8}]_{32}, _{\sqrt{5},∞,2,5}[C_{5} ⊃ × ⊏_{\sqrt{5}}^{2(2)}(C_{3} ⊃ × ⊏^{2(2)}D_{8})]_{32},

\sqrt{5},∞,5,3[C_{5} ⊃ × ⊏_{\sqrt{5}}^{2(3)}(C_{3} ⊃ × ⊏^{2(2)}D_{8})]_{32}, _{\sqrt{5},∞,2,3}[C_{5} ⊃ × ⊏_{\sqrt{5}}^{2(6)}(C_{3} ⊃ × ⊏^{2(2)}D_{8})]_{32},

\sqrt{5},∞,5,3[C_{5} ⊃ × ⊏_{\sqrt{5}′}^{2(3)}(C_{3} ⊃ × ⊏^{2(2)}D_{8})]_{32}, _{\sqrt{5},∞,2,3}[C_{5} ⊃ × ⊏_{\sqrt{5}′}^{2(6)}(C_{3} ⊃ × ⊏^{2(2)}D_{8})]_{32},

\sqrt{5},∞,5,3[C_{5} ⊃ × ⊏^{2(3)}(SL_{2}(3) □^{2} C_{3})]_{4}, _{\sqrt{5},∞,2,3}[C_{5} ⊃ × ⊏^{2(6)}(SL_{2}(3) □^{2} C_{3})]_{4},

\sqrt{5},∞,2,5[C_{5} ⊃ × ⊏_{\sqrt{5}′}^{2(2)}(SL_{2}(3) □^{2} C_{3})]_{4}, _{\sqrt{5},∞,5,3}[C_{5} ⊃ × ⊏_{\sqrt{5}′}^{2(3)}(SL_{2}(3) □^{2} C_{3})]_{4},

\sqrt{5},∞,2,3[C_{5} ⊃ × ⊏_{\sqrt{5}′}^{2(6)}(SL_{2}(3) □^{2} C_{3})]_{4}, _{\sqrt{5},∞,2,3}[C_{5} ⊃ × ⊏_{\sqrt{5}}^{2(6)} S̃_{3}⊗ \sqrt{-3} SL_{2}(3)]_{4},

A_{2} ⊗_{\sqrt{5},∞,2,5}[C_{5} ⊃ × ⊏_{\sqrt{5}′}^{2(2)}SL_{2}(3)]_{2} or _{\sqrt{5},∞,2,3}[C_{5} ⊃ × ⊏_{\sqrt{5}′}^{2(6)}S_{3} ⊗ SL_{2}(3)]_{4}.

Proof. The centralizer C := C_{G}(O_{5}(G)) is an absolutely irreducible subgroup of (ℚ[ζ_{5}] ⊗_{K} Q^{2×2})^{∗} and G/C ≅ C_{2}.

Assume first that O_{3}(C) = 1. Using Table 8.7 one finds that O_{2}(G) = O_{2}(C) is one of Q_{8} ∘ Q_{8} or Q_{8} ⊗ D_{8} and G contains C of index 2. Since Q_{20} ⊗ F_{4}

is a subgroup of _{\sqrt{5},∞}[SL_{2}(5)]_{1} ⊗ F_{4}, G is _{\sqrt{5},∞,2,5}[C_{5} ⊃ × ⊏^{2(2)} F_{4}]_{4} in the first case.

The second case leads to the two a.i.m.f. groups _{\sqrt{5},∞,2,5}[C_{5} ⊃ × ⊏_{\sqrt{5}′}^{2(2)} 2_{−}^{1+4}.Alt_{5}]_{4} and

\sqrt{5}[D_{10}]_{2} ⊗_{∞,2}[2_{−}^{1+4}.Alt_{5}]_{2}.

Now assume that O_{3}(C) ≠ 1. Then O_{3}(C) ≅ C_{3} and C ≅ C_{5} × H. Since O_{5}(H) = 1 and H does not contain a quasi-semi-simple normal subgroup and 5 does not divide the order of the automorphism groups of the possible normal 2subgroups, \overline{ℚH} is a central simple ℚ-algebra of dimension 16. Table 10.4 yields

that H is one of S_{3} ⊗D_{8}, S̃_{3} ⊗D_{8}, C_{3} ⊃ × ⊏^{2(2)}D_{8} (2 groups), S_{3} ⊗SL_{2}(3), S̃_{3}⊗ \sqrt{-3} SL_{2}(3),

or C_{3} ⊃ × ⊏^{2(2)} SL_{2}(3) (2 groups). In all cases Glide(H) is isomorphic to C_{2} × C_{2}. If G = C_{G}(H)H, then C_{G}(H) is one of ^{±}D_{10} or Q_{20} according to the real Schur index of an absolutely irreducible constituent of the natural character of H. In the second case G is not maximal finite but contained in SL_{2}(5)H. In the first case H is a maximal finite subgroup of its enveloping algebra. Therefore the only possibility for

H is H = _{∞,3}[SL_{2}(3) □^{2} C_{3}]_{2}. One checks that G = _{∞,3}[SL_{2}(3) □^{2} C_{3}]_{2}⊗\sqrt{5}[^{±}D_{10}]_{2}

is maximal finite.

If C_{G}(H) = ^{±}C_{5}, then one has for each group H three possible automorphisms. By Lemma 2.17 there is at most one extension G = C.2 in each case. Constructing the twenty-four groups G = H.2 one finds that G is one of the groups in the Lemma.

More precisely, the three nontrivial “outer” elements in N\overline{ℚH}^{∗}(H) may be distinguished via their norms, which are 2, 3, respectively 6. If one considers of the isoclinic pairs the group H with real Schur index 1 first, one finds from Table 10.4 that in the first and second case, the automorphism with norm 3 yields imprimitive groups. In the fourth and fifth case the normalizer element of norm 2 yields a

proper subgroup of _{\sqrt{5},∞,2,5}[C_{5} ⊃ × ⊏_{\sqrt{5}′}^{2(2)}2_{−}^{1+4}.Alt_{5}]_{4} resp. _{\sqrt{5},∞,2,5}[C_{5} ⊃ × ⊏^{2(2)}F_{4}]_{4}.

In some cases, G is not maximal finite due to the fact, that an outer element normalizes SL_{2}(5) ≤ (ℚ[\sqrt{5}] ⊗ \overline{ℚH})^{∗}. These cannot be read off directly from Table 10.4 and are the following: In the second case, additionally the automorphism with

norm 2 gives rise to a group contained in_{\sqrt{5},∞}[SL_{2}(5)]_{1}⊗ \sqrt{5} _{\sqrt{5},2,5}[C_{5} ⊃ × ⊏^{2(2)}D_{8}]_{2}. In the seventh case, the automorphisms of prime norm 2 resp. 3 yield proper subgroups of

\sqrt{5},∞[SL_{2}(5)]_{1}⊗ \sqrt{5} _{\sqrt{5},2,5}[C_{5} ⊃ × ⊏_{\sqrt{5}′}^{2(2)}SL_{2}(3)]_{2} resp. _{\sqrt{5},∞}[SL_{2}(5)]_{1}⊗ \sqrt{5} _{\sqrt{5},5,3}[C_{5} ⊃ × ⊏_{\sqrt{5}′}^{2(3)} S̃_{3}]_{2}.

Finally the group SL_{2}(3) ⊗ C_{5} ⊃ × ⊏_{\sqrt{5}′}^{2(3)} S_{3} is not maximal finite but contained in

\sqrt{5},∞[SL_{2}(5)]_{1}⊗ \sqrt{5} _{\sqrt{5},3,5}[C_{5} ⊃ × ⊏_{\sqrt{5}′}^{2(3)} S̃_{3}]_{2}.

Lemma 14.26. If K ≠ ℚ[\sqrt{5}] and O_{5}(G) = C_{5}, then G is one of

\sqrt{2},∞,2,5[D_{10} ⊠^{2(2+\sqrt{2})} Q_{16}]_{4}, _{\sqrt{3},∞}[D_{10} ⊃ × ⊏^{2}Q_{24}]_{4},

\sqrt{10},∞[D_{10} ⊠^{2} S̃_{4}]_{4}, _{\sqrt{15},∞}[D_{10} ⊠^{2} Q_{24}]_{4,1}, _{\sqrt{15},∞}[D_{10} ⊠^{2} Q_{24}]_{4,2}.

Proof. The centralizer C := C_{G}(O_{5}(G)) is an absolutely irreducible subgroup of (ℚ[ζ_{5}] ⊗ Q)^{∗} ≅ GL_{2}(K[ζ_{5}]) and G/C ≅ C_{4}. From the classification of finite subgroups of P GL_{2}(ℂ) one concludes that C = C_{5}×H, where H is one of D_{16}, Q_{16}, S̃_{4}, Q_{24}, or D_{24}. In all cases the exponent of Out(H) is 2. So C_{G}(H) > ^{±}C_{5} contains one of ^{±}D_{10} or Q_{20}, according to the real Schur index of an absolutely irreducible constituent of the natural character of H. Since Glide(H) does not contain an element of norm 5, one concludes that G is not maximal in the second case, but an additional quasi-semi-simple normal subgroup SL_{2}(5) arises. This excludes the first and last case. In the other three cases, G is clearly not of the form HC_{G}(H), since otherwise G = C_{5} : C_{4} ⊗ H is contained in A_{4} ⊗ H. Hence HC_{G}(H) = D_{10}H and G = HC_{G}(H).2. Fixing the outer automorphism one has two possible extensions in each case. They lead to isomorphic groups G.

If H = Q_{16}, then Out(H) ≅ C_{2} × C_{2}. Since one outer automorphism does not give rise to an extension H.2 (cf. proof of Lemma 14.22), two groups G need to be

considered. The group D_{10} ⊠^{2}Q_{16} is contained in _{\sqrt{10},∞}[D_{10} ⊠^{2} S̃_{4}]_{4} so G is conjugate

to _{\sqrt{2},∞,2,5}[D_{10} ⊠^{2(2+\sqrt{2})} Q_{16}]_{4} in this case.

If H = S̃_{4} and K = ℚ[\sqrt{2}], then H is primitively saturated over K. Hence

G = C_{G}(H)H is not maximal finite. One finds that G is _{\sqrt{10},∞}[D_{10} ⊠^{2} S̃_{4}]_{4} in this case.

Finally, if H = Q_{24}, then Out(H) ≅ C_{2}×C_{2}. Here three different automorphisms

have to be considered. They yield the three a.i.m.f. groups _{\sqrt{3},∞}[D_{10} ⊃ × ⊏^{2} Q_{24}]_{4},

\sqrt{15},∞[D_{10} ⊠^{2} Q_{24}]_{4,1}, and _{\sqrt{15},∞}[D_{10} ⊠^{2} Q_{24}]_{4,2}.

15. The a.i.m.f. subgroups of GL_{5}(Q)

Z(Q) = ℚ

Theorem 15.1. Let Q be a definite quaternion algebra with center ℚ and let G be a primitive a.i.m.f. subgroup of GL_{5}(Q). Then G is conjugate to one of the groups in the following table.

List of the primitive a.i.m.f. subgroups of GL_{5}(Q).

lattice L  |Aut(L)|  r.i.m.f. supergroups
∞,2[^{±}U_{5}(2)]_{5}  2^{11} · 3^{5} · 5 · 11 [U_{5}(2)^{2(2)} ∘ SL_{2}(3)]_{20}
∞,2[SL_{2}(11)]_{5}  2^{3} · 3 · 5 · 11 [SL_{2}(11)^{2(2)} ∘ SL_{2}(3)]_{20}
A_{5} ⊗_{∞,2}[SL_{2}(3)]_{1}  2^{7} · 3^{3} · 5  A_{5} ⊗ F_{4}
∞,3[^{±}U_{4}(2) □^{2} C_{3}]_{5}  2^{8} · 3^{5} · 5  [^{±}U_{4}(2) □^{2} C_{3}]_{10}^{2}
∞,11[^{±}L_{2}(11).2]_{5}  2^{4} · 3 · 5 · 11 (A_{10}^{(2)})^{2}

[L_{2}(11) ⊃ × ⊏^{2(3)}D_{12}]_{20}

Proof. Let Q be a definite quaternion algebra with center ℚ and let G be a primitive a.i.m.f. subgroup of GL_{5}(Q). Assume that 1 ≠ N ⊴ G is a quasi-semi-simple normal subgroup of G. By Table 9.1 N is one of Alt_{6}, L_{2}(11), SL_{2}(11), U_{4}(2), or U_{5}(2). The centralizer C := C_{G}(N) in G of N embeds into the commuting algebra C_{Q^{5×5}}(N), which is isomorphic to Q, Q, ℚ[\sqrt{-11}], ℚ, ℚ[\sqrt{-3}], resp. ℚ in the respective cases. In the first case G = B^{∘}(N)C is one of A_{5} ⊗ _{∞,2}[SL_{2}(3)]_{1} or A_{5} ⊗_{∞,3}[S̃_{3}]_{1} by Corollary 7.6 and Theorem 6.1. Whereas the first group is a maximal finite subgroup of GL_{5}(Q_{∞,2}), the second group is a proper subgroup of

∞,3[^{±}U_{4}(2) □^{2} C_{3}]_{5}.

In cases 3 and 5, N is already absolutely irreducible and lattice sparse. Its unique a.i.m.f. supergroup is _{∞,2}[^{±}U_{5}(2)]_{5}, resp. _{∞,2}[SL_{2}(11)]_{5}.

In the other two cases, C is contained in B^{∘}(N), which is a normal subgroup of index 2 = |Out(N)| in G. Since the commuting algebra C_{Q^{5×5}}(N) is an imaginary quadratic field, there is in both cases only one absolutely irreducible subgroup G = B^{∘}(N).2 ≤ GL_{5}(Q). One computes G = _{∞,11}[^{±}L_{2}(11).2]_{5} resp. G =

∞,3[^{±}U_{4}(2) □^{2} C_{3}]_{5}.

Now assume that G has no quasi-semi-simple normal subgroup. Since the possible normal p-subgroups of G, which embed into GL_{1}(Q) do not admit an automorphism of order 5, one has O_{11}(G) ≅ C_{11}, contradicting Lemma 11.2.

Theorem 15.2. Let Q be a definite quaternion algebra with center ℚ and let G be an a.i.m.f. subgroup of GL_{5}(Q). Then Q is one of Q_{∞,2}, Q_{∞,3}, or Q_{∞,11}. The simplicial complexes M_{5}^{irr}(Q) are as follows:

[figure: _{∞,2}[SL_{2}(11)]_{5}; A_{5} ⊗_{∞,2}[SL_{2}(3)]_{1}; _{∞,2}[SL_{2}(3)]_{1}^{5}; _{∞,2}[^{±}U_{5}(2)]_{5}]

[figure: _{∞,3}[^{±}U_{4}(2) □^{2} C_{3}]_{5}; _{∞,3}[S̃_{3}]_{1}^{5}]

{ \atop _{∞,11}[(^{±}L_{2}(11)).2]_{5}}

List of maximal simplices in M_{5}^{irr}(Q_{∞,2}):

simplex  a common subgroup
(_{∞,2}[^{±}U_{5}(2)]_{5}, _{∞,2}[SL_{2}(3)]_{1}^{5}) 2^{4+4} : (Alt_{5} × C_{3})

List of maximal simplices in M_{5}^{irr}(Q_{∞,3}):

simplex  a common subgroup
(_{∞,3}[^{±}U_{4}(2) □^{2} C_{3}]_{5}, _{∞,3}[S̃_{3}]_{1}^{5}) 2^{4} : Alt_{5} ⊃ ⊏^{2}C_{3}

Proof. Theorems 15.1 and 6.1 prove the completeness of the list of quaternion algebras Q and of a.i.m.f. subgroups of GL_{5}(Q). One only has to show the completeness of the list of maximal simplices in M_{5}^{irr}(Q_{∞,2}), because for the other two quaternion algebras Q, the simplicial complex M_{5}^{irr}(Q) consists of a single simplex: Let 𝔐_{2} denote a maximal order of Q_{∞,2}. Since the group_{∞,2}[SL_{2}(11)]_{5} fixes a lattice of determinant divisible by 11, the minimal absolutely irreducible subgroups of the group ∞,2[SL_{2}(11)]_{5} are of order divisible by 11 (cf. Lemma 2.13). Going through the list of maximal subgroups of L_{2}(11) in [CCNPW 85] one sees that_{∞,2}[SL_{2}(11)]_{5} is minimal absolutely irreducible. Hence _{∞,2}[SL_{2}(11)]_{5} forms a 0-simplex in M_{5}^{irr}(Q_{∞,2}). The minimal absolutely irreducible subgroups of A_{5} ⊗_{∞,2}[SL_{2}(3)]_{1} are Alt_{5} ⊗ Q_{8}

and C_{4} ⊃ × ⊏^{2} Alt_{5}. Both groups do not embed into any other a.i.m.f. subgroup of

GL_{5}(Q_{∞,2}), since they do not fix any 3-unimodular 𝔐_{2}-lattice. Therefore the list of maximal simplices in M_{5}^{irr}(Q_{∞,2}) is complete.

Z(Q) real quadratic.

Theorem 15.3. Let G be a primitive absolutely irreducible maximal finite subgroup of GL_{5}(Q), where Q is a totally definite quaternion algebra with center K and [K : ℚ] = 2. Then Q is isomorphic to Q_{\sqrt{5},∞}, Q_{\sqrt{2},∞}, Q_{\sqrt{3},∞}, Q_{\sqrt{11},∞}, or Q_{\sqrt{33},∞} and G is conjugate to one of the groups given in the table below, which is built up as Table 12.7:

Proof. Let Q be a definite quaternion algebra with center K := Z(Q) a real quadratic field. Let G be a primitive absolutely irreducible subgroup of GL_{5}(Q), and p be a prime such that O_{p}(G) ≠ 1. Then either O_{p}(G) is a subgroup of GL_{1}(Q) or p = 5 and O_{5}(G) ≅ C_{25} or 5_{+}^{1+2}, or p = 11 and O_{11}(G) ≅ C_{11}.

If O_{5}(G) ≅ C_{25}, then C_{G}(O_{5}(G)) = ^{±}O_{5}(G). But 5 divides the index of the abelian normal subgroup O_{5}(G) in G, because G is absolutely irreducible. This contradicts the assumption O_{5}(G) ≅ C_{25} (and also the primitivity of G).

Now let O_{5}(G) ≅ 5_{+}^{1+2}. Then K = ℚ[\sqrt{5}]. The inclusion 5_{+}^{1+2} ⊴ B := ^{±}5_{+}^{1+2} : SL_{2}(5) ≤ GL_{5}(ℚ[ζ_{5}]) together with Out(5_{+}^{1+2}) ≅ GL_{2}(5) then shows that G contains a normal subgroup B of index 2. There is a unique extension B.2 with real Schur index 2. Hence G = _{\sqrt{5},∞}[^{±}5_{+}^{1+2} : SL_{2}(5).2]_{5} in this case.

If O_{11}(G) ≅ C_{11}, then C_{G}(O_{11}(G)) is isomorphic to one of C_{22}, C_{44}, or C_{66}. In the first case, G is not absolutely irreducible and in the other two cases, G is a

proper subgroup of _{\sqrt{11},∞}[C_{4} □^{2} L_{2}(11)]_{5} resp. _{\sqrt{33},∞}[^{±}C_{3} □^{2} L_{2}(11)]_{5}.

List of the primitive a.i.m.f. subgroups of GL_{5}(Q) where Q is a totally definite

quaternion algebra over a real quadratic number field Z(Q).

lattice L  |Aut(L)|  some r.i.m.f. supergroups
\sqrt{5},∞[SL_{2}(5)]_{1} ⊗ A_{5}  2^{7} · 3^{3} · 5^{2}  A_{5} ⊗ E_{8}, A_{5} ⊗ H_{4}
\sqrt{5},∞[^{±}5_{+}^{1+2} : SL_{2}(5).2]_{5}  2^{5} · 3 · 5^{4}  [^{±}5_{+}^{1+2} : SL_{2}(5).2 □^{2} SL_{2}(5)]_{40}
\sqrt{2},∞[SL_{2}(11).2]_{5}  2^{4} · 3 · 5 · 11 [SL_{2}(11)^{2(2)} ∘ SL_{2}(3)]_{20}^{2},

[SL_{2}(11) ⊗_{∞,2}^{2(2)} 2_{−}^{1+4}.Alt_{5}]_{40}

\sqrt{2},∞[^{±}U_{5}(2).2]_{5}  2^{12} · 3^{5} · 5 · 11 [^{±}U_{5}(2)^{2(2)} ∘ SL_{2}(3)]_{20}^{2},

[U_{5}(2) ⊗_{∞,2}^{2(2)} 2_{−}^{1+4}.Alt_{5}]_{40}

\sqrt{2},∞[SL_{2}(9)]_{5}  2^{4} · 3^{2} · 5  [2.U_{4}(2)^{2(2)} ∘ SL_{2}(3)]_{40}
\sqrt{2},∞[S̃_{4}]_{1} ⊗ A_{5}  2^{8} · 3^{3} · 5  A_{5} ⊗ E_{8}, (A_{5} ⊗ F_{4})^{2}
\sqrt{3},∞[SL_{2}(11)]_{5}  2^{3} · 3 · 5 · 11 [SL_{2}(11) ⊃ ⊏^{2(3)}C_{12}.C_{2}]_{40}

[SL_{2}(11) ⊃ ⊏^{2(2)}SL_{2}(3)]_{40}

\sqrt{3},∞[C_{12} ⊠_{\sqrt{-3}}^{2} U_{4}(2)]_{5}  2^{9} · 3^{5} · 5  [^{±}C_{3} □^{2} U_{4}(2)]_{10}^{4}, [^{±}U_{5}(2)^{2(2)} ∘ SL_{2}(3)]_{20}^{2}

[^{±}C_{3} □^{2} U_{4}(2)]_{10} ⊗ F_{4},

[U_{5}(2) ⊗_{∞,2}^{2(2)} 2_{−}^{1+4}.Alt_{5}]_{40}

\sqrt{11},∞[C_{4} □^{2} L_{2}(11)]_{5}  2^{5} · 3 · 5 · 11 (A_{10}^{(3)})^{4} [^{±}U_{5}(2)^{2(2)} ∘ SL_{2}(3)]_{20}^{2}

A_{10}^{(3)} ⊗ F_{4} , [U_{5}(2) ⊗_{∞,2}^{2(2)} 2_{−}^{1+4}.Alt_{5}]_{40}

\sqrt{33},∞[^{±}C_{3} □^{2} L_{2}(11)]_{5}  2^{4} · 3^{2} · 5 · 11 (A_{10}^{(3)} ⊗ A_{2})^{2}, [^{±}U_{5}(2)^{2(2)} ∘ SL_{2}(3)]_{20}^{2},

[L_{2}(11) ⊃ × ⊏^{2(3)}D_{12}]_{20}^{2}

[(L_{2}(11)⊗ \sqrt{-11} SL_{2}(3) ⊗ S_{3}).2]_{40}

Assume now, that for all primes p, O_{p}(G) is a subgroup of GL_{1}(Q). Then the automorphism group of O_{p}(G) is soluble and its order is not divisible by 5. Since G is absolutely irreducible, the last term of the derived series of G^{(∞)} is a quasi-semi-simple group. Table 9.1 implies that G^{(∞)} is one of Alt_{6}, SL_{2}(9), L_{2}(11), SL_{2}(11) (2 matrix groups), U_{4}(2), or U_{5}(2).

The two groups _{\sqrt{2},∞}[SL_{2}(9)]_{5} and _{\sqrt{3},∞}[SL_{2}(11)]_{5} are a.i.m.f. groups.

In the first case, B^{∘}(Alt_{6}) ≅ ^{±}S_{6} = A_{5} is a primitively saturated absolutely irreducible subgroup of GL_{5}(ℚ). Therefore Corollary 7.6 says that G is a tensor product ^{±}S_{6} ⊗ U, where U is a maximal finite subgroup of GL_{1}(Q). By Theorem 6.1 U is one of _{\sqrt{5},∞}[SL_{2}(5)]_{1}, _{\sqrt{2},∞}[S̃_{4}]_{1}, or _{\sqrt{3},∞}[C_{12}.C_{2}]_{1}. Whereas in the first two cases, G is maximal finite, the last group _{\sqrt{3},∞}[C_{12}.C_{2}]_{1} ⊗A_{5} is contained in the

maximal finite group_{\sqrt{3},∞}[C_{12} ⊠_{\sqrt{-3}}^{2}U_{4}(2)]_{5}. (Note that^{±}C_{3}∘U_{4}(2) = B^{∘}(Alt_{6}⊗C_{3}).)

Now assume that G^{(∞)} ≅ L_{2}(11). Then the centralizer C := C_{G}(G^{(∞)}) embeds into GL_{1}(K[\sqrt{-11}]), and G : G^{(∞)}C = 2. If C = ^{±}1, then the center of the enveloping algebra of G is ℚ and therefore G is not absolutely irreducible in GL_{5}(Q). Hence the biquadratic field K[\sqrt{-11}] contains a root of unity. Therefore K = ℚ[\sqrt{11}] and C ≅ C_{4} or K = ℚ[\sqrt{33}] and C ≅ ^{±}C_{3}. By Lemma 2.17 in both cases, there is a unique extension G = G^{(∞)}C.C_{2} with real Schur index. Hence G is one of

\sqrt{11},∞[C_{4} ⊠^{2} L_{2}(11)]_{5} or _{\sqrt{33},∞}[^{±}C_{3} ⊠^{2} L_{2}(11)]_{5}.

Next let G^{(∞)} be SL_{2}(11), where the restriction of the natural character of G to G^{(∞)} is χ_{10}. Then the centralizer C := C_{G}(G^{(∞)}) embeds into GL_{1}(K), hence is trivial, and G : G^{(∞)} = 2. There is a unique extension G = SL_{2}(11).2 with real character field. Therefore G = _{\sqrt{2},∞}[SL_{2}(11).2]_{5}.

The case G^{(∞)} = U_{5}(2) is completely analogous.

The remaining case is G^{(∞)} ≅ U_{4}(2). Then G contains the normal subgroup B := B^{∘}(G^{(∞)}) = ^{±}C_{3}∘U_{4}(2). Moreover C := C_{G}(B) = C_{G}(G^{(∞)}) ≤ GL_{1}(K[\sqrt{-3}]) and G contains BC of index 2. If C ≤ B, then the character field of the natural character of G is ℚ, contradicting the fact that G is absolutely irreducible in GL_{5}(Q). Hence C ≅ C_{12} and K = ℚ[\sqrt{3}]. The unique extension G = BC.2 in GL_{5}(Q) is

\sqrt{3},∞[C_{12} ⊠_{\sqrt{-3}}^{2} U_{4}(2)]_{5}.

Theorem 15.4. M_{5}^{irr}(Q_{\sqrt{2},∞}) is as follows.

[figure: _{\sqrt{2},∞}[SL_{2}(11).2]_{5}; _{\sqrt{2},∞}[^{±}U_{5}(2).2]_{5}; _{\sqrt{2},∞}[SL_{2}(9)]_{5}; _{\sqrt{2},∞}[S̃_{4}]_{1}^{5}; _{\sqrt{2},∞}[S̃_{4}]_{1} ⊗ A_{5}]
List of the maximal simplices in M_{5}^{irr}(Q_{\sqrt{2},∞})
simplex  a common subgroup
(_{\sqrt{2},∞}[^{±}U_{5}(2).2]_{5},_{\sqrt{2},∞}[S̃_{4}]_{1}^{5}) (^{±}2^{4+4}.(C_{3} × Alt_{5})).2

Proof. The completeness of the list of a.i.m.f. subgroups in GL_{5}(Q_{\sqrt{2},∞}) follows from Theorems 6.1 and 15.3. So we only have to show that the list of maximal simplices in M_{5}^{irr}(Q_{\sqrt{2},∞}) is complete. The two a.i.m.f. groups _{\sqrt{2},∞}[SL_{2}(9)]_{5} and \sqrt{2},∞[SL_{2}(11).2]_{5} are minimal absolutely irreducible.

So we only have to deal with G := _{\sqrt{2},∞}[S̃_{4}]_{1} ⊗ A_{5}. The minimal absolutely irreducible subgroups V of G contain a normal subgroup N of index ≤ 2 of the form N := U ⊗ Alt_{5}, where U is of index ≤ 2 in one of the two absolutely irreducible subgroups Q_{16} or S̃_{4} of S̃_{4}. The minimality of V implies that U is of order 8 or 16. Hence the 3-modular defect of V is one. Let 𝔐 be a maximal order of Q_{\sqrt{2},∞}. Then the 𝔐V -lattices are fixed under the group H := Q_{16} ∘ V, where Q_{16} is the Sylow 2-subgroup of the unit group of 𝔐. The group H is an absolutely irreducible subgroup of GL_{20}(ℚ[\sqrt{2}]). Since 3 is inert in ℚ[\sqrt{2}], one concludes that the 3-modular constituents of H are of degree 8 and 32. Therefore V does not fix a 3-unimodular lattice.

Theorem 15.5. M_{5}^{irr}(Q_{\sqrt{3},∞}) is as follows.

[figure: _{\sqrt{3},∞}[C_{12} ⊠_{\sqrt{-3}}^{2} U_{4}(2)]_{5}; _{\sqrt{3},∞}[C_{12}.C_{2}]_{1}^{5}; _{\sqrt{3},∞}[SL_{2}(11)]_{5}]
List of the maximal simplices in M_{5}^{irr}(Q_{\sqrt{3},∞})
simplex  a common subgroup
(_{\sqrt{3},∞}[C_{12} ⊠_{\sqrt{-3}}^{2} U_{4}(2)]_{5}, _{\sqrt{3},∞}[C_{12}.C_{2}]_{1}^{5}) C_{12} ⊠_{\sqrt{-3}}^{2} 2^{4}.Alt_{5}

Proof. The completeness of the list of a.i.m.f. subgroups in GL_{5}(Q_{\sqrt{3},∞}) follows from Theorems 6.1 and 15.3. The list of maximal simplices in M_{5}^{irr}(Q_{\sqrt{3},∞}) is complete, because _{\sqrt{3},∞}[SL_{2}(11)]_{5} is minimal absolutely irreducible.

Theorem 15.6. M_{5}^{irr}(Q_{\sqrt{5},∞}) is as follows.

[figure: _{\sqrt{5},∞}[(^{±}5_{+}^{1+2} : SL_{2}(5)).2]_{5}; _{\sqrt{5},∞}[SL_{2}(5)]_{1}^{5}; _{\sqrt{5},∞}[SL_{2}(5)]_{1} ⊗ A_{5}]
List of the maximal simplices in M_{5}^{irr}(Q_{\sqrt{5},∞})
simplex  a common subgroup
(_{\sqrt{5},∞}[(^{±}5_{+}^{1+2} : SL_{2}(5)).2]_{5}, _{\sqrt{5},∞}[SL_{2}(5)]_{1}^{5}) (^{±}5_{+}^{1+2}).2

Proof. To see the completeness of the list of maximal simplices in M_{5}^{irr}(Q_{\sqrt{5},∞}) it suffices to show that there is no common absolutely irreducible subgroup of G := _{\sqrt{5},∞}[SL_{2}(5)]_{1} ⊗ A_{5} and one of the other two maximal finite subgroups of GL_{5}(Q_{\sqrt{5},∞}). The minimal absolutely irreducible subgroups U of G are ^{±}C_{5}.C_{2} ⊗

Alt_{5} and ^{±}C_{5} ⊃ × ⊏^{2} Alt_{5}. If 𝔐 is a maximal order of Q_{\sqrt{5},∞}, then the 3-modular constituents of the natural representation of U𝔐 are of degree 8 and 32. So both groups U do not embed into _{\sqrt{5},∞}[(^{±}5_{+}^{1+2} : SL_{2}(5)).2]_{5} or _{\sqrt{5},∞}[SL_{2}(5)]_{1}^{5}.

Theorem 15.7. The two simplicial complexes M_{5}(Q_{\sqrt{11},∞})^{irr} and M_{5}(Q_{\sqrt{33},∞})^{irr} consists of one 0-simplex each.

16. The a.i.m.f. subgroups of GL_{6}(Q)

Theorem 16.1. Let Q be a definite quaternion algebra with center ℚ and let G be a primitive a.i.m.f. subgroup of GL_{6}(Q). Then G is conjugate to one of the groups in the table below, which is built up as Table 12.7.

List of the primitive a.i.m.f. subgroups of GL_{6}(Q).

lattice L  |Aut(L)|  r.i.m.f. supergroups
∞,2[2.G_{2}(4)]_{6}  2^{13} · 3^{3} · 5^{2} · 7 · 13 [2.Co_{1}]_{24}
∞,2[(^{±}3).P GL_{2}(9)]_{6}  2^{5} · 3^{3} · 5  [(^{±}3).P GL_{2}(9)^{2(2)} ∘ SL_{2}(3)]_{24}
∞,2[3_{+}^{1+2} : SL_{2}(3) ⊃ × ⊏^{2(2)}D_{8}]_{6}  2^{7} · 3^{4}  [Sp_{4}(3) ⊠_{\sqrt{-3}}^{2} (3_{+}^{1+2} : SL_{2}(3))]_{24}
∞,2[SL_{2}(5) ⊗^{2(2)} D_{8}]_{6}  2^{6} · 3 · 5  [SL_{2}(5) ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+4}.Alt_{5}]_{24}
∞,2[L_{2}(7) ⊗^{2(2)} SL_{2}(3)]_{6}  2^{7} · 3^{2} · 7  [L_{2}(7) ⊗^{2(2)} F_{4}]_{24}
∞,2[L_{2}(7) ⊃ × ⊏^{2(2)}SL_{2}(3)]_{6}  2^{7} · 3^{2} · 7  [L_{2}(7) ⊃ × ⊏^{2(2)}F_{4}]_{24}
∞,2[C_{4} ⊃ ⊏^{2(3)}U_{3}(3)]_{6}  2^{8} · 3^{3} · 7  [(SL_{2}(3) ∘ C_{4}) ⊃ × ⊏_{\sqrt{-1}}^{2(3)}U_{3}(3)]_{24}
∞,2[SL_{2}(5)]_{3} ⊗ A_{2}  2^{4} · 3^{2} · 5  A_{2} ⊗ [SL_{2}(5)^{2(2)} ∘ SL_{2}(3)]_{12}
∞,2[SL_{2}(3)]_{1} ⊗ E_{6}  2^{10} · 3^{4} · 5  E_{6} ⊗ F_{4}
∞,2[SL_{2}(3)]_{1} ⊗ A_{6}  2^{7} · 3^{3} · 5 · 7  A_{6} ⊗ F_{4}
∞,2[SL_{2}(3)]_{1} ⊗ A_{6}^{(2)}  2^{7} · 3^{2} · 7  A_{6}^{(2)} ⊗ F_{4}
∞,2[SL_{2}(3)]_{1} ⊗ M_{6,2}  2^{6} · 3^{2} · 5  M_{6,2} ⊗ F_{4}
∞,3[6.U_{4}(3).2^{2}]_{6}  2^{10} · 3^{7} · 5 · 7  [6.U_{4}(3).2^{2}]_{12}^{2}
∞,3[C_{3} ⊃ ⊏^{2(2)}SL_{2}(5)]_{6}  2^{4} · 3^{2} · 5  [SL_{2}(5)^{2(2)} ∘ SL_{2}(3)]_{12}^{2}
∞,3[3_{+}^{1+2} : SL_{2}(3) ⊃ × ⊏^{2(2)}SL_{2}(3)]_{6}  2^{7} · 3^{5}  [3_{+}^{1+2} : SL_{2}(3) ⊃ × ⊏^{2(2)}SL_{2}(3)]_{12}^{2}
∞,3[S̃_{3}]_{1} ⊗ A_{6}  2^{6} · 3^{3} · 5 · 7  (A_{6} ⊗ A_{2})^{2}
∞,3[S̃_{3}]_{1} ⊗ A_{6}^{(2)}  2^{6} · 3^{2} · 7  (A_{6}^{(2)} ⊗ A_{2})^{2}
∞,3[S̃_{3}]_{1} ⊗ M_{6,2}  2^{5} · 3^{2} · 5  (M_{6,2} ⊗ A_{2})^{2}
∞,5[SL_{2}(25)]_{6}  2^{4} · 3 · 5^{2} · 13  [2.Co_{1}]_{24}
∞,5[2.J_{2}.2]_{6}  2^{9} · 3^{3} · 5^{2} · 7  [2.Co_{1}]_{24}
∞,5[2.J_{2} : 2]_{6}  2^{9} · 3^{3} · 5^{2} · 7  [2.J_{2} □^{2} SL_{2}(5)]_{24}
∞,5[Alt_{5} ⊠_{\sqrt{5}}^{2} SL_{2}(5)]_{6,1}  2^{6} · 3^{2} · 5^{2}  [(SL_{2}(5) ∘ SL_{2}(5)) : 2 ⊠_{\sqrt{5}}^{2} Alt_{5}]_{24,1}
∞,5[Alt_{5} ⊠_{\sqrt{5}}^{2} SL_{2}(5)]_{6,2}  2^{6} · 3^{2} · 5^{2}  [(SL_{2}(5) ∘ SL_{2}(5)) : 2 ⊠_{\sqrt{5}}^{2} Alt_{5}]_{24,2}
∞,7[L_{2}(7) ⊃ × ⊏^{2(2)}D_{8}]_{6}  2^{7} · 3 · 7  [L_{2}(7) ⊃ × ⊏^{2(2)}D_{8}]_{12}^{2}
∞,7[L_{2}(7) ⊃ × ⊏^{2(3)} S̃_{3}]_{6}  2^{6} · 3^{2} · 7  [6.U_{4}(3).2^{2}]_{12}^{2}
∞,7[^{±}L_{2}(7).2]_{3} ⊗ A_{2}  2^{6} · 3^{2} · 7  (A_{2} ⊗ A_{6}^{(2)})^{2}
∞,11[SL_{2}(11).2]_{6}  2^{4} · 3 · 5 · 11  (B_{24})
[2.Co_{1}]_{24}
∞,13[SL_{2}(13).2]_{6}  2^{4} · 3 · 7 · 13  [2.Co_{1}]_{24}
∞,13[SL_{2}(13) : 2]_{6}  2^{4} · 3 · 7 · 13  [SL_{2}(13) ⊃ ⊏^{2(2)}SL_{2}(3)]_{24}

The proof is split into seventeen lemmata. For the rest of this section let Q be a definite quaternion algebra with center ℚ and G be a primitive a.i.m.f. subgroup of GL_{6}(Q).

Lemma 16.2. The last term G^{(∞)} of the derived series is either 1 or a quasi-

semi-simple group. If G is soluble, then G is one of _{∞,2}[3_{+}^{1+2} : SL_{2}(3) ⊃ × ⊏^{2(2)} D_{8}]_{6} or

∞,3[3_{+}^{1+2} : SL_{2}(3) ⊃ × ⊏^{2(2)}SL_{2}(3)]_{6}.

Proof. Since the possible normal p-subgroups of G embed into GL_{12}(ℚ) and GL_{6}(Q), Theorem 8.1 together with Lemma 2.18 leave the following possibilities for O_{p}(G): C_{13}, C_{7}, C_{3}, C_{9}, 3_{+}^{1+2}, C_{2}, C_{4}, D_{8}, Q_{8}, C_{8}, or QD_{16}. The automorphism groups of these groups are soluble, so the first part of the lemma follows.

The first case is excluded by Lemma 11.2 and the case O_{7}(G) ≅ C_{7} is excluded with the help of Lemma 8.12.

If O_{3}(G) = C_{9}, then C := C_{G}(O_{3}(G)) embeds into GL_{2}(ℚ[ζ_{9}]), because ℚ[ζ_{9}] splits all possible quaternion algebras. With [Bli 17] one finds that C is one of C_{9} ⊗ D_{8} or C_{9}⊗ \sqrt{-3} B^{∘}(Q_{8}) = C_{9}⊗ \sqrt{-3} SL_{2}(3), and G contains C of index 6. In both cases 3 does not divide the order of the outer automorphism group of C/O_{3}(G), so one concludes that O_{3}(G) > C_{9} which is a contradiction.

Now assume that O_{3}(G) = 3_{+}^{1+2}. Then G contains a normal subgroup B := B^{∘}(O_{3}(G)) ≅ ^{±}3_{+}^{1+2} : SL_{2}(3). Similarly as above one has that C := C_{G}(O_{3}(G)) is one of C_{3} ⊗ D_{8} or C_{3} ∘ SL_{2}(3). Moreover G contains BC of index 2. Let α ∈ G − BC. In each case one has two possibilities, either α induces a nontrivial outer automorphism on B^{∘}(O_{2}(G)) or not. For each possibility there is a unique group G with real Schur index 2. For the first possibility one computes that G

is _{∞,2}[3_{+}^{1+2} : SL_{2}(3) ⊃ × ⊏^{2(2)} D_{8}]_{6} or _{∞,3}[3_{+}^{1+2} : SL_{2}(3) ⊃ × ⊏^{2(2)} SL_{2}(3)]_{6}. For the second

possibility, one computes that G is either a proper subgroup of the imprimitive a.i.m.f. group _{∞,3}[^{±}3_{+}^{1+2}.GL_{2}(3)]_{3}^{2} or a proper subgroup of E_{6} ⊗_{∞,2}[SL_{2}(3)]_{1}.

In the other cases, B^{∘}(O_{p}(G)) does not admit an outer automorphism of order 3. One concludes that G being absolutely irreducible, has to contain a quasi-semi-simple normal subgroup (cf. Lemma 8.11).

Lemma 16.3. If G contains a subgroup U conjugate to 3.Alt_{6}, where the restriction of the natural character of G to U is χ_{6} + χ′_{6}, then G is one of _{∞,2}[2.G_{2}(4)]_{6}, ∞,2[(^{±}3).P GL_{2}(9)]_{6}, or _{∞,3}[6.U_{4}(3).2^{2}]_{6}.

Proof. The last term of the derived series G^{(∞)} has to contain 3.Alt_{6}, hence is one of 3.Alt_{6}, 6.L_{3}(4), 6.U_{4}(3), or 2.G_{2}(4).

First we prove that Q is either Q_{∞,3} or Q_{∞,2}.

If G^{(∞)} = 2.G_{2}(4), then G^{(∞)} is already an absolutely irreducible subgroup of GL_{6}(Q_{∞,2}), hence in this case it is clear that Q = Q_{∞,2}. In the other three cases, one has O_{3}(G) = C_{3} and the enveloping algebra of U coincides with the one of C_{G}(O_{3}(G)) =: C. The discriminant of the enveloping ℤ-order of U is 2^{18} · 3^{11+36}. Therefore, 2 and 3 are the only primes, which may divide the discriminant of the enveloping ℤ-order of C. Since C is a normal subgroup of G of index 2, Lemma 2.15 together with the fact that the number of ramified primes is even and Q is ramified at ∞, implies that Q is either Q_{∞,2} or Q_{∞,3}.

Let 𝔐_{2} (resp. 𝔐_{3}) denote a maximal order of Q_{∞,2} (resp. Q_{∞,3}).

The group U is a uniform subgroup of GL_{6}(Q) and fixes up to isomorphism 6·6 = 36 𝔐_{2}-lattices. Computing the automorphism groups of the relevant lattices one finds that _{∞,2}[2.G_{2}(4)]_{6} and _{∞,2}[(^{±}3).P GL_{2}(9)]_{6} are the only primitive a.i.m.f. supergroups of U in GL_{6}(Q_{∞,2}).

Similarly U fixes up to isomorphism 2 · 6 = 12 𝔐_{3}-lattices and one finds ∞,3[6.U_{4}(3).2^{2}]_{6} as the only primitive a.i.m.f. supergroup of U in GL_{6}(Q_{∞,3}).

The next lemma follows also from Theorem 11.2.

Lemma 16.4. If G^{(∞)} is SL_{2}(13), then G is one of _{∞,13}[SL_{2}(13).2]_{6} or ∞,13[SL_{2}(13) : 2]_{6}.

Proof. The centralizer C_{G}(G^{(∞)}) embeds into ℚ[\sqrt{13}], and is therefore ^{±}1. Hence G contains G^{(∞)} of index 2 = |Out(G^{(∞)})|. One computes that both groups G are a.i.m.f. groups in GL_{6}(Q_{∞,13}).

Similarly one finds

Lemma 16.5. If G^{(∞)} is 2.J_{2}, then G is one of _{∞,5}[2.J_{2}.2]_{6} or _{∞,5}[2.J_{2} : 2]_{6}.

Lemma 16.6. If G^{(∞)} is SL_{2}(11), then G is conjugate to _{∞,11}[SL_{2}(11).2]_{6}.

Proof. The centralizer C_{G}(G^{(∞)}) embeds into ℚ[\sqrt{-11}], and is therefore ^{±}1. Hence G contains G^{(∞)} of index 2 = |Out(G^{(∞)})|. By Lemma 2.17 there is at most one extension G^{(∞)}.2 which is an a.i.m.f. group of GL_{6}(Q). One computes that G is conjugate to _{∞,11}[SL_{2}(11).2]_{6}.

Lemma 16.7. G^{(∞)} is not isomorphic to SL_{2}(7).

Proof. Assume that G^{(∞)} is isomorphic to SL_{2}(7). Then the restriction of the natural character of G to G^{(∞)} is 2(χ_{6a} + χ_{6b}). The centralizer C_{G}(G^{(∞)}) embeds into ℚ[\sqrt{2}] hence is ^{±}1 and G = G^{(∞)} or G = G^{(∞)}.2. The lemma follows, since the second group is no subgroup of GL_{6}(Q), because the character field is of degree 4 over ℚ, and G^{(∞)} is not absolutely irreducible.

Lemma 16.8. If G^{(∞)} is SL_{2}(25), then G = G^{(∞)} is conjugate to _{∞,5}[SL_{2}(25)]_{6}.

Proof. The group SL_{2}(25) has two characters of degree 12. The corresponding representations lead to conjugate groups in GL_{6}(Q_{∞,5}), since the characters are interchanged by an outer automorphism of G^{(∞)}. The absolutely irreducible subgroup of GL_{6}(Q_{∞,5}) fixes up to isomorphism 1 lattice. Since it is the full automorphism group of this lattice, the lemma follows.

Lemma 16.9. If G^{(∞)} is U_{4}(2), then G is conjugate to E_{6} ⊗_{∞,2}[SL_{2}(3)]_{1}.

Proof. Let B := B^{∘}(G^{(∞)}) = Aut(E_{6}). By Corollary 7.6 G = B ⊗ C_{G}(B) is a tensor product and C_{G}(B) is an a.i.m.f. subgroup of GL_{1}(Q). From Proposition 6.1 one gets that G is one of E_{6} ⊗_{∞,2}[SL_{2}(3)]_{1} or E_{6} ⊗_{∞,3}[S̃_{3}]_{1}. In the last case, G is not maximal finite, but a proper subgroup of _{∞,3}[6.U_{4}(3).2^{2}]_{6}. (Note that B^{∘}(C_{3} ⊗ U_{4}(2)) = 6.U_{4}(3).2.)

Similarly one finds

Lemma 16.10. If G^{(∞)} is Alt_{7}, then G is conjugate to one of A_{6} ⊗_{∞,2}[SL_{2}(3)]_{1} or A_{6} ⊗_{∞,3}[S̃_{3}]_{1}.

Lemma 16.11. G^{(∞)} is not isomorphic to SL_{2}(9).

Proof. Assume that G^{(∞)} = SL_{2}(9). Lemma 2.18 implies that Q = Q_{∞,3}. The centralizer C_{G}(G^{(∞)}) embeds into GL_{3}(ℚ). The primitivity of G implies that C = ±1. But then G is not irreducible, because 3 | |̸Out(SL_{2}(9))|.

Lemma 16.12. If G contains a normal subgroup N conjugate to SL_{2}(5), where the restriction of the natural character of G to N is 3(χ_{2a} + χ_{2b}), then G is one of

the two isoclinic groups _{∞,5}[Alt_{5} ⊠_{\sqrt{5}}^{2} SL_{2}(5)]_{6,1} or _{∞,5}[Alt_{5} ⊠_{\sqrt{5}}^{2} SL_{2}(5)]_{6,2}.

Proof. The centralizer C := C_{G}(N) is a centrally irreducible subgroup of GL_{3}(ℚ[\sqrt{5}]) hence C = Alt_{5}. Moreover G contains CN of index 2. Computing the two possible extensions one finds that G is one of the two groups in the lemma.

Lemma 16.13. If G contains a normal subgroup N isomorphic to Alt_{5}, then G is one of

M_{6,2} ⊗_{∞,2}[SL_{2}(3)]_{1}, M_{6,2} ⊗_{∞,3}[S̃_{3}]_{1},

∞,5[Alt_{5} ⊠_{\sqrt{5}}^{2} SL_{2}(5)]_{6,1} or _{∞,5}[Alt_{5} ⊠_{\sqrt{5}}^{2} SL_{2}(5)]_{6,2}.

Proof. The centralizer C := C_{G}(N) is a centrally irreducible subgroup of GL_{1}(ℚ[\sqrt{5}] ⊗ Q) and G contains CN of index 2. Assume first that C_{G}(C) > ^{±}N. Then C_{G}(C) ≅ ^{±}N.2 and G is a tensor product G = C_{G}(C) ⊗ C, where C is an a.i.m.f. subgroup of GL_{1}(Q). Since the nonsplit extension S_{5} ∧^{|}C_{4} is a monomial subgroup of GL_{6}(ℚ) (cf. [PlN 95, (V.3)]), one finds that G is one of M_{6,2} ⊗_{∞,2}[SL_{2}(3)]_{1} or M_{6,2} ⊗_{∞,3}[S̃_{3}]_{1}. Now let C_{G}(C) = ^{±}N. Using the classification of a.i.m.f. subgroups of GL_{1}(Q′) for definite quaternion algebras Q′ with center ℚ or ℚ[\sqrt{5}], one finds that C = B^{∘}(C) is one of SL_{2}(3), S̃_{3}, Q_{20}, or SL_{2}(5). In the last case, the lemma follows from Lemma 16.12, whereas in the first three cases both extensions G = NC_{G}(N).2 are proper subgroups of one of the two groups of Lemma 16.12.

Lemma 16.14. If N := G^{(∞)} = U_{3}(3), then G is conjugate to _{∞,2}[C_{4} ⊃ ⊏^{2(3)}U_{3}(3)]_{6}.

Proof. The centralizer C := C_{G}(N) embeds into an indefinite quaternion algebra with center ℚ. Since B^{∘}(N ∘ C_{3}) = 6.U_{4}(3).2, one concludes that O_{3}(C) = 1. One has the following possibilities for O_{2}(G) = O_{2}(C): ^{±}1, C_{4}, or D_{8}. The first possibility immediately yields a contradiction, then G contains N of index 2 and Lemma 2.14 implies that dim(\overline{ℚG}) ≤ 2dim(\overline{ℚN}) = 2 · 36 < 144. Therefore G is not absolutely irreducible in this case. In the other two cases G contains a normal subgroup N ∘ C_{4}. The discriminant of the enveloping ℤ-order of N ∘ C_{4} is 3^{12} · 2^{36}. Therefore Lemma 2.15 implies that Q is one of Q_{∞,2} or Q_{∞,3}, if G contains N ∘ C_{4} of index two. If [G : (N ∘ C_{4})] > 2, then O_{2}(C) = D_{8} and NO_{2}(C) is already an absolutely irreducible subgroup of GL_{6}(Q_{∞,3}). So in this case Q = Q_{∞,3}.

Let 𝔐_{2} (resp. 𝔐_{3}) denote a maximal order of Q_{∞,2} (resp. Q_{∞,3}).

Then N ∘C_{4} fixes only one 𝔐_{2}-lattice. The automorphism group of this lattice is

∞,2[C_{4} ⊃ ⊏^{2(3)}U_{3}(3)]_{6}. Hence G is conjugate to this a.i.m.f. group, if Q = Q_{∞,2}. If Q = Q_{∞,3}, then N ∘C_{4} fixes up to isomorphism 6 𝔐_{3}-lattices. The automorphism group of the normalized lattices is conjugate to _{∞,3}[^{±}U_{3}(3)]_{3}^{2} contradicting the primitivity of G.

Lemma 16.15. If G^{(∞)} is conjugate to SL_{2}(5), where the restriction of the nat-

ural character of G to G^{(∞)} is 2χ_{6}, then G is one of _{∞,2}[SL_{2}(5) ⊗^{2(2)} D_{8}]_{6}, A_{2} ⊗

∞,2[SL_{2}(5)]_{3}, or _{∞,3}[C_{3} ⊃ ⊏^{2(2)}SL_{2}(5)]_{6}.

Proof. Let G^{(∞)} be conjugate to SL_{2}(5) as described in the lemma. Then the centralizer C := C_{G}(G^{(∞)}) embeds into GL_{1}(Q′), where Q′ is an indefinite quaternion algebra with center ℚ. Hence C is soluble. Moreover G contains CG^{(∞)} of index ≤ 2 = |Out(G^{(∞)})|. By Lemma 2.14 this implies that C ≠ ^{±}1. Therefore one either has O_{3}(C) = C_{3} or O_{3}(C) = 1 and C = C_{4} or D_{8}. The discriminant of the enveloping ℤ-order of G^{(∞)} is (5^{8} · 2^{5+9})^{2}. Using Lemma 2.15 one excludes in all cases that Q is ramified at 5. Hence Q is one of Q_{∞,2} or Q_{∞,3}, where the latter possibility only occurs if O_{3}(G) = C_{3}. Let 𝔐_{2} (resp. 𝔐_{3}) denote a maximal order of Q_{∞,2} (resp. Q_{∞,3}). In the first case N := SL_{2}(5) ∘ C_{3} is a normal subgroup of G. The Bravais group of a normal critical 𝔐_{2}N-lattice (cf. Definition 2.7) is conjugate to A_{2} ⊗ _{∞,2}[SL_{2}(5)]_{3}. So G is conjugate to this group, if Q = Q_{∞,2}. If Q = Q_{∞,3}, then every 𝔐_{3}N-lattice is normal critical. One concludes that G is

conjugate to _{∞,3}[C_{3} ⊃ ⊏^{2(2)}SL_{2}(5)]_{6} in this case. If O_{3}(C) = 1, then G contains a normal subgroup N := SL_{2}(5) ∘ C_{4}. Moreover Q = Q_{∞,2}. The automorphism group

of a normal critical 𝔐_{2}N-lattice is conjugate to _{∞,2}[SL_{2}(5) ⊗^{2(2)} D_{8}]_{6}. Therefore

G = _{∞,2}[SL_{2}(5) ⊗^{2(2)} D_{8}]_{6}.

Similarly one finds

Lemma 16.16. If G^{(∞)} is conjugate to L_{2}(7), where the restriction of the natural

character of G to G^{(∞)} contains χ_{6}, then G is conjugate to _{∞,2}[L_{2}(7) ⊗^{2(2)} SL_{2}(3)]_{6}.

Lemma 16.17. If G^{(∞)} is conjugate to L_{2}(7), with character χ_{3a} + χ_{3b}, then G is

conjugate to one of _{∞,2}[L_{2}(7) ⊃ × ⊏^{2(2)}SL_{2}(3)]_{6}, _{∞,2}[SL_{2}(3)]_{1} ⊗ A_{6}^{(2)}, _{∞,3}[L_{2}(7) ⊃ × ⊏^{2(3)}S_{3}]_{6},

∞,3[S̃_{3}]_{1} ⊗ A_{6}^{(2)}, _{∞,7}[L_{2}(7) ⊃ × ⊏^{2(2)}D_{8}]_{6}, _{∞,7}[L_{2}(7) ⊃ × ⊏^{2(3)} S̃_{3}]_{6}, or _{∞,7}[^{±}L_{2}(7).2]_{3} ⊗ A_{2}.

Proof. The centralizer C := C_{G}(G^{(∞)}) embeds into a quaternion algebra Q′ with center ℚ[\sqrt{-7}], more precisely Q′ = ℚ[\sqrt{-7}]^{2×2} if 2 is not ramified in Q and Q′ = Q_{\sqrt{-7},2,2}, if 2 ramifies in Q. Because G is absolutely irreducible, G contains CG^{(∞)} of index 2 and C is a centrally irreducible subgroup of GL_{1}(Q′). The classification of the finite subgroup of P GL_{2}(ℂ) in [Bli 17] implies that C is one of SL_{2}(3), D_{8}, S_{3}, or S̃_{3}. Since the enveloping algebra of C is central simple, one has 2=|N\overline{ℚC}^{∗}(C)/C\overline{ℚC}^{∗}(C)| possible automorphisms. By Lemma 2.17 there is for each automorphism a unique extension with real Schur index 2. Since D_{8}⊗_{∞,7}[^{±}L_{2}(7).2]_{3} is imprimitive, G is one of the seven groups in the Lemma.

Lemma 16.18. G^{(∞)} is not conjugate to 3.Alt_{6}, where the restriction of the natural representation of G to G^{(∞)} is χ_{3a} + χ′_{3a} + χ_{3b} + χ′_{3b}.

Proof. Assume that G^{(∞)} is conjugate to 3.Alt_{6}. Then C_{G}(G^{(∞)}) = ^{±}C_{3} is contained in ^{±}G^{(∞)} and G contains ^{±}G^{(∞)} of index 2^{2} = |Out(G^{(∞)})|. Since Q is positive definite, G contains the unique extension N := ^{±}3.P GL_{2}(9) with real Schur index 2 (cf. Lemma 2.17) of index 2. The Bravais group of a normal critical ℤN-lattice is 2.J_{2} contradicting the assumption that G^{(∞)} = 3.Alt_{6}.

Theorem 16.19. Let Q be a definite quaternion algebra with center ℚ and G be an a.i.m.f. subgroup of GL_{6}(Q). Then Q is one of Q_{∞,2}, Q_{∞,3}, Q_{∞,5}, Q_{∞,7}, Q_{∞,11}, or Q_{∞,13}. The simplicial complexes M_{6}^{irr}(Q) are as follows:

[figure: _{∞,2}[2.G_{2}(4)]_{6}; _{∞,2}[2_{−}^{1+4}.Alt_{5}]_{2}^{3}; _{∞,2}[3_{+}^{1+2} : SL_{2}(3) ⊃ × ⊏^{2(2)}D_{8}]_{6}; _{∞,2}[L_{2}(7) ⊃ × ⊏^{2(2)}SL_{2}(3)]_{6}; _{∞,2}[L_{2}(7) ⊗^{2(2)} SL_{2}(3)]_{6}; _{∞,2}[SL_{2}(3)]_{1} ⊗ A_{6}; _{∞,2}[SL_{2}(3)]_{1} ⊗ A_{6}^{(2)}; _{∞,2}[SL_{2}(3)]_{1}^{6}; _{∞,2}[(^{±}3).P GL_{2}(9)]_{6}; _{∞,2}[C_{4} ⊃ ⊏^{2(3)}U_{3}(3)]_{6}; _{∞,2}[SL_{2}(5)]_{3} ⊗ A_{2}; _{∞,2}[SL_{2}(3)]_{1} ⊗ M_{6,2}; _{∞,2}[SL_{2}(3)]_{1} ⊗ E_{6}; (_{∞,2}[SL_{2}(3)]_{1} ⊗ A_{2})^{3}; _{∞,2}[SL_{2}(5)]_{3}^{2}; _{∞,2}[SL_{2}(5) ⊗^{2(2)} D_{8}]_{6}]

[figure: _{∞,3}[U_{3}(3)]_{3}^{2}; _{∞,3}[6.U_{4}(3).2^{2}]_{6}; _{∞,3}[S̃_{3}]_{1}^{6}; _{∞,3}[SL_{2}(9)]_{2}^{3}; _{∞,3}[SL_{2}(3) □^{2} C_{3}]_{2}^{3}; _{∞,3}[3_{+}^{1+2}.SL_{2}(3) ⊃ × ⊏^{2(2)}SL_{2}(3)]_{6}; _{∞,3}[^{±}3_{+}^{1+2}.GL_{2}(3)]_{3}^{2}; _{∞,3}[C_{3} ⊃ ⊏^{2(2)}SL_{2}(5)]_{6}; _{∞,3}[S̃_{3}]_{1} ⊗ M_{6,2}; _{∞,3}[S̃_{3}]_{1} ⊗ A_{6}^{(2)}; _{∞,3}[S̃_{3}]_{1} ⊗ A_{6}]

[figure: _{∞,5}[SL_{2}(25)]_{6}; _{∞,5}[SL_{2}(5).2]_{2}^{3}; _{∞,5}[SL_{2}(5) : 2]_{2}^{3}; _{∞,5}[2.J_{2}.2]_{6}; _{∞,5}[2.J_{2} : 2]_{6}; _{∞,5}[Alt_{5} ⊠_{\sqrt{5}}^{2} SL_{2}(5)]_{6,1}; _{∞,5}[Alt_{5} ⊠_{\sqrt{5}}^{2} SL_{2}(5)]_{6,2}]

[figure: _{∞,7}[L_{2}(7) ⊃ × ⊏^{2(2)}D_{8}]_{6}; _{∞,7}[L_{2}(7) ⊃ × ⊏^{2(3)} S̃_{3}]_{6}; _{∞,7}[^{±}L_{2}(7).2]_{3} ⊗ A_{2}; _{∞,7}[^{±}L_{2}(7).2]_{3}^{2}]

{ \atop _{∞,11}[SL_{2}(11).2]_{6}}

[figure: _{∞,13}[SL_{2}(13) : 2]_{6}; _{∞,13}[SL_{2}(13).2]_{6}]

List of maximal simplices in M_{6}^{irr}(Q_{∞,2}):
simplex  a common subgroup
(_{∞,2}[2.G_{2}(4)]_{6}, _{∞,2}[2_{−}^{1+4}.Alt_{5}]_{2}^{3}, _{∞,2}[SL_{2}(3)]_{1}^{6}) Alt_{4} ⊗ D_{8} ⊗ Q_{8}
(_{∞,2}[3_{+}^{1+2} : SL_{2}(3) ⊃ × ⊏^{2(2)}D_{8}]_{6}, _{∞,2}[2_{−}^{1+4}.Alt_{5}]_{2}^{3})  3_{+}^{1+2} ⊃ × ⊏^{2(2)}D_{8}
(_{∞,2}[SL_{2}(5) ⊗^{2(2)} D_{8}]_{6}, _{∞,2}[SL_{2}(5)]_{3}^{2})  SL_{2}(5) ⊗ D_{8}
((A_{2} ⊗_{∞,2}[SL_{2}(3)]_{1})^{3}, E_{6} ⊗_{∞,2}[SL_{2}(3)]_{1})  3_{+}^{1+2} : 2 ⊗ Q_{8}
(_{∞,2}[L_{2}(7) ⊃ × ⊏^{2(2)}SL_{2}(3)]_{6}, _{∞,2}[L_{2}(7) ⊗^{2(2)} SL_{2}(3)]_{6}) C_{7} : C_{3} ⊃ × ⊏^{2(2)}SL_{2}(3)
(A_{6} ⊗_{∞,2}[SL_{2}(3)]_{1}, _{∞,2}[L_{2}(7) ⊗^{2(2)} SL_{2}(3)]_{6})  L_{2}(7) ⊗ Q_{8}
(A_{6} ⊗_{∞,2}[SL_{2}(3)]_{1}, A_{6}^{(2)} ⊗_{∞,2}[SL_{2}(3)]_{1})  C_{7} : C_{6} ⊗ SL_{2}(3)

List of maximal simplices in M_{6}^{irr}(Q_{∞,3}):
simplex  a common subgroup
(_{∞,3}[U_{3}(3)]_{3}^{2}, _{∞,3}[6.U_{4}(3).2^{2}]_{6}, _{∞,3}[^{±}3_{+}^{1+2}.GL_{2}(3)]_{3}^{2}) 3_{+}^{1+2} : C_{8} ⊗ S_{3}
(_{∞,3}[6.U_{4}(3).2^{2}]_{6}, _{∞,3}[^{±}3_{+}^{1+2}.GL_{2}(3)]_{3}^{2}, _{∞,3}[S̃_{3}]_{1}^{6})  ^{±}3_{+}^{1+2}.C_{2} ⊗ S_{3}
(_{∞,3}[S̃_{3}]_{1}^{6}, _{∞,3}[^{±}3_{+}^{1+2}.GL_{2}(3)]_{3}^{2}, _{∞,3}[SL_{2}(9)]_{2}^{3})  (^{±}C_{3}^{4}).C_{12}
(_{∞,3}[SL_{2}(3) □^{2} C_{3}]_{2}^{3}, _{∞,3}[SL_{2}(9)]_{2}^{3})  S̃_{4} ⊗ Alt_{4}
(_{∞,3}[3_{+}^{1+2} : SL_{2}(3) ⊃ × ⊏^{2(2)}SL_{2}(3)]_{6}, _{∞,3}[SL_{2}(3) □^{2} C_{3}]_{2}^{3}) 3_{+}^{1+2} ⊃ × ⊏^{2(2)}Q_{8}
(A_{6} ⊗_{∞,3}[S̃_{3}]_{1}, A_{6}^{(2)} ⊗_{∞,3}[S̃_{3}]_{1})  C_{7} : C_{6} ⊗ S̃_{3}

List of maximal simplices in M_{6}^{irr}(Q_{∞,5}):
simplex  a common subgroup
\begin{matrix}(_{∞,5}[SL_{2}(5).2]_{2}^{3},_{∞,5}[SL_{2}(5) : 2]_{2}^{3}, \\ _{∞,5}[2.J_{2}.2]_{6},_{∞,5}[2.J_{2} : 2]_{6},_{∞,5}[SL_{2}(25)]_{6}) \end{matrix}  (^{±}C_{5} × C_{5}).C_{12}

\begin{matrix}(_{∞,5}[SL_{2}(5).2]_{2}^{3},_{∞,5}[SL_{2}(5) : 2]_{2}^{3}, \\ _{∞,5}[Alt_{5} ⊠_{\sqrt{5}}^{2} SL_{2}(5)]_{6,1},_{∞,5}[Alt_{5} ⊠_{\sqrt{5}}^{2} SL_{2}(5)]_{6,2}) \end{matrix} Q_{20} ⊃ × ⊏^{2} Alt_{4}

(_{∞,5}[2.J_{2}.2]_{6},_{∞,5}[Alt_{5} ⊠_{\sqrt{5}}^{2} SL_{2}(5)]_{6,1})  Alt_{5} ⊃ ⊏^{2}Q_{8}
(_{∞,5}[2.J_{2} : 2]_{6},_{∞,5}[Alt_{5} ⊠_{\sqrt{5}}^{2} SL_{2}(5)]_{6,2})  Alt_{5} ⊃ ⊏^{2}Q_{8}

List of maximal simplices in M_{6}^{irr}(Q_{∞,13}):
simplex  a common subgroup
(_{∞,13}[SL_{2}(13) : 2]_{6}, _{∞,13}[SL_{2}(13).2]_{6}) ^{±}C_{13}.C_{12}

Theorems 16.1, 13.1, 12.1, and 6.1 prove the completeness of the list of quaternion algebras Q and of a.i.m.f. subgroups of GL_{6}(Q). So it remains to prove the completeness of the list of maximal simplices in M_{6}^{irr}(Q). That the simplices listed do exist, can be easily seen by computing the automorphism groups of the invariant lattices of the groups listed in the column “a common subgroup”.

To make the formulations of the proofs not so lengthy, we introduce some notation for imprimitive groups.

Notation 16.20. Let G = H ≀ S_{d} = (H_{1} × … × H_{d}) : S_{d} be an imprimitive subgroup of GL_{n}(D) and Δ its natural representation.

For a subgroup U ≤ G the restriction of Δ to the stabilizer

S_{1}(U) := {u ∈ U | h_{1}u ∈ H_{1} for all h_{1} ∈ H_{1}}

of the first component has a summand Δ_{1} : S_{1}(U) → H_{1}. Define

π_{1}(U) := Δ_{1}(S_{1}(U)) ≤ H_{1}.

Then Δ_{|U} is induced up from Δ_{1}, hence Frobenius reciprocity implies that if U is absolutely irreducible, then π_{1}(U) is an absolutely irreducible subgroup of GL\frac{n}{d} (D).

The base group of U is defined as the intersection of U with H_{1} × … × H_{d} ⊴ G. Clearly this is a normal subgroup of U.

The proof is split up into several lemmata according to the different quaternion algebras Q_{∞,p}. Since M_{6}^{irr}(Q_{∞,13}) and M_{6}^{irr}(Q_{∞,11}) consist of one simplex, it suffices to consider p ≤ 7.

Lemma 16.21. M_{6}^{irr}(Q_{∞,7}) consists of four 0-simplices.

Proof. The minimal absolutely irreducible subgroups of the three primitive a.i.m.f.

groups _{∞,7}[L_{2}(7) ⊃ × ⊏^{2(2)}D_{8}]_{6}, _{∞,7}[L_{2}(7) ⊃ × ⊏^{2(3)} S̃_{3}]_{6}, and A_{2} ⊗_{∞,7}[^{±}L_{2}(7).2]_{3} are the nor-

malizers in these groups of the Sylow 7-subgroups. The lemma follows by computing the invariant lattices of these three groups.

Lemma 16.22. The list of maximal simplices in M_{6}^{irr}(Q_{∞,5}) given in Theorem 16.19 is complete.

Proof. Since the other five a.i.m.f. groups in GL_{6}(Q_{∞,5}) form a 4-simplex, it suffices to consider the minimal absolutely irreducible subgroups U of one of the two groups

G _{∞,5}[Alt_{5} ⊠_{\sqrt{5}}^{2} SL_{2}(5)]_{6,1} and _{∞,5}[Alt_{5} ⊠_{\sqrt{5}}^{2} SL_{2}(5)]_{6,2}. Let N ⊴ U be the intersection

of U with the normal subgroup Alt_{5}⊴G and M := U ∩SL_{2}(5). Then the restriction of the natural representation Δ of U to N is of degree 1 or 3. If it is of degree 1, the index of NM in U is divisible by 3. Since 5 divides the order of U and subgroups of Alt_{5} and SL_{2}(5), of which the order is a multiple of 5 have no nontrivial factor group of order divisible by 3, one concludes that U ≅ SL_{2}(5).2 is a full subdirect product. But this contradicts the fact that U is absolutely irreducible. Hence Δ_{|N} is of degree 3, and N is one of Alt_{4} or Alt_{5}. In the first case, 5 divides the order of

the centralizer C := C_{U}(N) and the minimality of U implies that U = Q_{20} ⊃ × ⊏^{2} Alt_{4}.

In the second case, C ≤ SL_{2}(5) is a centrally irreducible subgroup of Q_{\sqrt{5},∞}. This leaves the possibilities C = Q_{20}, Q_{8}, SL_{2}(3), and S̃_{3}. Moreover U contains

NC of index 2. In the first case U contains Q_{20} ⊃ × ⊏^{2} Alt_{4} from above and the

third possibility C = SL_{2}(3) gives groups containing NQ_{8}.2. An inspection of the lattices of the remaining groups yields the lemma.

Proposition 16.23. The list of maximal simplices in M_{6}^{irr}(Q_{∞,3}) given in Theorem 16.19 is complete.

The proof is divided into seven lemmata, which are organized according to the largest primes dividing the determinant of an invariant primitive lattice of the a.i.m.f. group.

Let S denote a maximal simplex of M_{6}^{irr}(Q_{∞,3}) not listed in Theorem 16.19.

Lemma 16.24. S contains no vertex A_{6} ⊗_{∞,3}[S̃_{3}]_{1} or A_{6}^{(2)} ⊗_{∞,3}[S̃_{3}]_{1}.

Proof. Let U be a minimal absolutely irreducible subgroup of one of the two a.i.m.f. groups of the lemma. Then by Lemma 2.13 7 divides the order of U. If U contains a normal subgroup of order 7, then U = C_{7} : C_{6} ⊗ _{∞,3}[S̃_{3}]_{1} is a common subgroup of the 2 groups. If the Sylow 7-subgroup of U is not normal in U, then the minimality of U implies that U^{(∞)} = L_{2}(7), where the restriction of the natural character of U to U^{(∞)} is 4χ_{6}, where χ_{6} + 1 is a permutation character of L_{2}(7). Since the corresponding permutation representation does not extend to L_{2}(7).2, one concludes that U = L_{2}(7) ⊗_{∞,3}[S̃_{3}]_{1}, which has only _{∞,3}[S̃_{3}]_{1} ⊗ A_{6} as an a.i.m.f. supergroup.

Lemma 16.25. S contains no vertex _{∞,3}[C_{3} ⊃ ⊏^{2(2)}SL_{2}(5)]_{6} or M_{6,2} ⊗_{∞,3}[S̃_{3}]_{1}.

Proof. Let U be a minimal absolutely irreducible subgroup of one of the two groups G. Then by Lemma 2.13 5 divides |U|. Since for both groups, the normalizer of a Sylow 5-subgroup is reducible, one concludes that U^{(∞)} = G^{(∞)}. In both cases one finds that U = G is minimal absolutely irreducible.

Lemma 16.26. The irreducible subgroups V ≤ B^{∘}(3_{+}^{1+2}) = 3_{+}^{1+2} : SL_{2}(3) ≤ GL_{3}(ℚ[ζ_{3}]) satisfy 3_{+}^{1+2} or 3_{−}^{1+2} ≤ O_{3}(V).

Proof. Let V be an irreducible subgroup of G := 3_{+}^{1+2} : SL_{2}(3). If N := V ∩ O_{3}(G) = 1, then V is isomorphic to a subgroup of SL_{2}(3) and reducible. Therefore N, being nontrivial, contains the center of G and is one of O_{3}(G), C_{3} × C_{3} or C_{3}. In the first case, 3_{+}^{1+2} ≤ O_{3}(V). In the second case V/N stabilizes a flag in 𝔽_{3}^{2} = O_{3}(G)/Z(G), hence is an abelian subgroup of SL_{2}(3). Since the degree of the natural character of V is 3 and N is an abelian normal subgroup of V, one has O_{3}(V) = 3_{−}^{1+2} in this case. The last case again contradicts the irreducibility of V, being contained in C_{3} × SL_{2}(3).

Lemma 16.27. S contains no vertex G := _{∞,3}[3_{+}^{1+2} : SL_{2}(3) ⊃ × ⊏^{2(2)}SL_{2}(3)]_{6}.

Proof. Let 𝔐 be a maximal order in Q_{∞,3}. Let U be a minimal absolutely irreducible subgroup of G. The natural representation of U is of the form Δ_{1} ⊗ Δ_{2}, where Δ_{1}(U) has a subgroup V of index 2 such that V ≤ ^{±}3_{+}^{1+2}.SL_{2}(3) is an irreducible subgroup of GL_{3}(ℚ[ζ_{3}]). By Lemma 16.26 O_{3}(V) contains an extraspecial 3-group. In particular the 2-modular constituents of the natural representation of V 𝔐 are of degree 12. Comparing the determinants of the invariant integral lattices one sees that the only other a.i.m.f. group, into which U might embed, is

∞,3[SL_{2}(3) □^{2} C_{3}]_{3}^{2}.

Lemma 16.28. S contains no vertex G := _{∞,3}[SL_{2}(3) □^{2} C_{3}]_{2}^{3}.

Proof. Let 𝔐 be a maximal order in Q_{∞,3}. Let U be a minimal absolutely irreducible subgroup of G. With the notation introduced in 16.20 the group π_{1}(U) ≤

∞,3[SL_{2}(3) □^{2} C_{3}]_{2} is an absolutely irreducible subgroup of GL_{2}(Q_{∞,3}). Hence

π_{1}(U) contains one of the two minimal absolutely irreducible subgroups Q_{8} □^{2} C_{3}

or S̃_{4} of_{∞,3}[SL_{2}(3) □^{2} C_{3}]_{2} (cf. proof of Theorem 12.3). In particular the 3-modular constituents of 𝔐π_{1}(U) are of degree 4.

The intersection N of U with the base group of G is a normal subgroup N ⊴ U with U/N ≅ C_{3} or S_{3}. In the first case, π_{1}(U) = π_{1}(N) and the Clifford theory implies that the degrees of the 3-modular constituents of U are divisible by 4. With Lemma 16.27 one sees that U does not define a new simplex. In the second case, π_{1}(N) is a normal subgroup of index 2 in π_{1}(U). Assuming that the 3-modular constituents of 𝔐π_{1}(N) are not all of degree 4, one only finds the possibility that π_{1}(N) = C_{12}.C_{2} = S_{3} ^{|}∧^{C_{2}}C_{8}. But then the 2-modular constituents of 𝔐π_{1}(N) are of degree 4 which implies that U does not define a new simplex.

Lemma 16.29. There is no common absolutely irreducible subgroup of G := ∞,3[U_{3}(3)]_{3}^{2} and one of _{∞,3}[S̃_{3}]_{1}^{6} or _{∞,3}[SL_{2}(9)]_{2}^{3}.

Proof. Let U be an absolutely irreducible subgroup of G. Then U contains a normal subgroup N ⊴ U of index 2, such that the restriction of the natural representation Δ of U to N is the sum of two inequivalent absolutely irreducible representations Δ_{|N} = Δ_{1} + Δ_{2}. The groups Δ_{1}(N) ≅ Δ_{2}(N) are absolutely irreducible subgroups of _{∞,3}[U_{3}(3)]_{3}. Hence Δ_{1}(N) is either ≅ U_{3}(3) or 3_{+}^{1+2} : C_{8}. Both groups have no subgroup of index 3 or 6. Therefore U is not a subgroup of_{∞,3}[S̃_{3}]_{1}^{6} or_{∞,3}[SL_{2}(9)]_{2}^{3}.

Proposition 16.23 now follows from the next lemma:

Lemma 16.30. There is no common absolutely irreducible subgroup of G := ∞,3[SL_{2}(9)]_{2}^{3} and H := _{∞,3}[6.U_{4}(3).2^{2}]_{6}.

Proof. Let U be an absolutely irreducible subgroup of G. Then π_{1}(U) is one of the four absolutely irreducible subgroups ^{±}3^{2}.C_{4}, S̃_{4}, SL_{2}(5), or SL_{2}(9) of π_{1}(G) = ∞,3[SL_{2}(9)]_{2}.

If π_{1}(U) = S̃_{4}, then π_{1}(U) is a subgroup of _{∞,3}[SL_{2}(3) □^{2} C_{3}]_{2} and the lemma follows from Lemma 16.28.

Since U is a subgroup of H the centralizer N := C_{U}(O_{3}(H)) ⊴ U in U of O_{3}(H) is a subgroup of U of index 2 with commuting algebra C_{Q_{∞,3}^{6×6}}(N) ≅ ℚ[ζ_{3}]. Since N is normal, π_{1}(N) ⊴ π_{1}(U) is a subgroup of index 2 with C_{Q_{∞,3}^{2×2}}(π_{1}(N)) ≅ ℚ[ζ_{3}].

Since in the first case the unique subgroup V of index 2 in π_{1}(U) has commuting algebra C_{Q_{∞,3}^{2×2}}(V) = ℚ ⊕ ℚ and the last two groups π_{1}(U) are perfect, this is a contradiction.

Proposition 16.31. The list of maximal simplices in M_{6}^{irr}(Q_{∞,2}) given in Theorem 16.19 is complete.

This proposition is proved in the rest of this chapter which concludes the proof of Theorem 16.19.

The first lemma is easily checked with help of Lemma 2.13 and [CCNPW 85].

Lemma 16.32. (i) The minimal absolutely irreducible subgroup of _{∞,2}[SL_{2}(3)]_{1}

⊗ M_{6,2} is S_{5} ⊗ Q_{8}. (ii) The minimal absolutely irreducible subgroups of _{∞,2}[SL_{2}(3)]_{1} ⊗ A_{6} are C_{7} :

C_{6} ⊗ Q_{8}, C_{7} : C_{6} ^{|}∧^{C_{3}}SL_{2}(3), and L_{2}(7) ⊗ Q_{8}. (iii) The minimal absolutely irreducible subgroups of _{∞,2}[SL_{2}(3)]_{1} ⊗ A_{6}^{(2)} embed

into _{∞,2}[SL_{2}(3)]_{1} ⊗ A_{6}. (iv) The group _{∞,2}[SL_{2}(5)]_{3} ⊗ A_{2} is minimal absolutely irreducible.

(v) The group _{∞,2}[(^{±}3).P GL_{2}(9)]_{6} is minimal absolutely irreducible.

(vi) The minimal absolutely irreducible subgroup of _{∞,2}[C_{4} ⊃ ⊏^{2(3)} U_{3}(3)]_{6} is the nor-

malizer of the Sylow 3-subgroup C_{4} ⊃ ⊏^{2(3)}3_{+}^{1+2} : C_{8}.

Corollary 16.33. The restriction of M_{6}^{irr}(Q_{∞,2}) to the set {_{∞,2}[L_{2}(7) ⊗^{2(2)}SL_{2}(3)]_{6},

∞,2[L_{2}(7) ⊃ × ⊏^{2(2)}SL_{2}(3)]_{6}, A_{6}^{(2)}⊗_{∞,2}[SL_{2}(3)]_{1}, A_{6}⊗_{∞,2}[SL_{2}(3)]_{1}, _{∞,2}[SL_{2}(3)]_{1}⊗M_{6,2},

A_{2}⊗_{∞,2}[SL_{2}(5)]_{3}, _{∞,2}[(^{±}3).P GL_{2}(9)]_{6}, _{∞,2}[C_{4} ⊃ ⊏^{2(3)}U_{3}(3)]_{6}} consists of full simplices and is as given in Theorem 16.19.

Proof. After computing the a.i.m.f. supergroups of the minimal absolutely irreducible subgroups given in Lemma 16.32, it suffices to show, that there is no common absolutely irreducible group of one of the first two groups and one further a.i.m.f. group not mentioned in the corollary. Let U be such an absolutely irreducible group. Then by Lemma 2.13 the order of U is divisible by 7. Hence the only other a.i.m.f. group into which U may embed is _{∞,2}[2.G_{2}(4)]_{6}. Therefore C_{U}(C_{7}) ≤ ^{±}C_{3} and U is clearly not an absolutely irreducible subgroup of one of the first two groups.

Lemma 16.34. The restriction of M_{6}^{irr}(Q_{∞,2}) to the set {_{∞,2}[SL_{2}(5) ⊗^{2(2)} D_{8}]_{6}, ∞,2[SL_{2}(5)]_{3}^{2}} consists of full simplices and is a one dimensional simplex.

Proof. Let U be an absolutely irreducible subgroup of one of the two a.i.m.f. groups G in the lemma and one other a.i.m.f. group. Then U^{(∞)} = SL_{2}(5) (or SL_{2}(5) × SL_{2}(5)), where the restriction of the natural character of G to U is 2χ_{6} (or χ_{6}+χ′_{6}). The second possibility is clearly impossible, hence U^{(∞)} = SL_{2}(5). The only other a.i.m.f. groups into which U may embed are _{∞,2}[2.G_{2}(4)]_{6} and _{∞,2}[SL_{2}(3)]_{1}^{6}. Since U is absolutely irreducible, the centralizer of U^{(∞)} in U is a nontrivial 2-group. One concludes that the 5-modular constituents of the absolutely irreducible group U ∘ SL_{2}(3) ≤ GL_{24}(ℚ) are of degree 8 and 16. So U ∘ SL_{2}(3) does not fix a 5-unimodular ℤ-lattice.

Lemma 16.35. The only maximal simplex in M_{6}^{irr}(Q_{∞,2}) with vertex G :=

∞,2[3_{+}^{1+2} : SL_{2}(3) ⊃ × ⊏^{2(2)}D_{8}]_{6} is (G,_{∞,2}[2_{−}^{1+4}.Alt_{5}]_{2}^{3}).

Proof. Let U be an absolutely irreducible subgroup of G. Then U = V ⊃ × ⊏^{2(2)} D_{8}, where V ≤ 3_{+}^{1+2} : SL_{2}(3) is an absolutely irreducible subgroup of GL_{3}(ℚ[ζ_{3}]). By Lemma 16.26 O_{3}(V) either contains O_{3}(G) or is 3_{−}^{1+2}. Let 𝔐 be a maximal order in Q_{∞,2}, and SL_{2}(3) its unit group. The natural representation of O_{3}(V)⊗ \sqrt{−3} SL_{2}(3) has two different 2-modular constituents of degree 6. These are interchanged by the outer automorphism of V ⊗ D_{8} inducing the Galois automorphism on the center of O_{3}(V). Therefore the 2-modular constituents of U ∘ SL_{2}(3) are of degree 12. So the only a.i.m.f. groups into which U may embed are G,_{∞,2}[2_{−}^{1+4}.Alt_{5}]_{2}^{3}, and ∞,2[2.G_{2}(4)]_{6}. Since the order of the latter group is not divisible by 3^{4} and the normalizer of its Sylow 3-subgroup is not absolutely irreducible, the lemma follows.

Lemma 16.36. There is no common absolutely irreducible subgroup of_{∞,2}[SL_{2}(3)]_{1} ⊗ E_{6} or (_{∞,2}[SL_{2}(3)]_{1} ⊗ A_{2})^{3} and one of _{∞,2}[2.G_{2}(4)]_{6}, _{∞,2}[2_{−}^{1+4}.Alt_{5}]_{2}^{3}, or ∞,2[SL_{2}(3)]_{1}^{6}.

Proof. Let V be an absolutely irreducible subgroup of E_{6} ⊗_{∞,2}[SL_{2}(3)]_{1} or (A_{2} ⊗ ∞,2[SL_{2}(3)]_{1})^{3}, 𝔐 a maximal order of Q_{∞,2} and U ≅ SL_{2}(3) the unit group of 𝔐. Assume that V embeds into one of the last three groups of the lemma. Then the degrees of the 3-modular constituents of the natural representation of V ∘U are not all divisible by 4. We claim that

(⋆) (U ∘ V)/(O_{3}(U ∘ V)) contains a normal 2-subgroup ≥ C_{4} ∘ Q_{8}.

Assume first that V ≤ E_{6} ⊗_{∞,2}[SL_{2}(3)]_{1}. Then the natural representation of V is a tensor product Δ_{1} ⊗ Δ_{2} with Δ_{1}(V) ≤ E_{6} ≤ GL_{6}(ℚ) and Δ_{2}(V) ≤ SL_{2}(3) ≤ GL_{1}(Q_{∞,2}) absolutely irreducible. Clearly, Δ_{1}(V) does not contain U_{4}(2), hence is soluble and contained in one of the two absolutely irreducible maximal subgroups ±3^{1+2}.2.S_{4} or ^{±}3^{3} : (S_{4} × C_{2}) ([CCNPW 85]) of E_{6}. Moreover Δ_{2}(V), being absolutely irreducible, contains the normal two subgroup Q_{8} ≤ SL_{2}(3). One concludes that O_{2}(V/O_{3}(V)) contains a subgroup C_{4}, hence (⋆).

Assume now that V ≤ (_{∞,2}[SL_{2}(3)]_{1} ⊗ A_{2})^{3}. Then π_{1}(V) is an absolutely irreducible subgroup of_{∞,2}[SL_{2}(3)]_{1}⊗A_{2}, hence contains S_{3}⊗Q_{8}. The first component of the base group of V is a normal subgroup of index ≤ 2 in π_{1}(U), hence contains C_{4}. Again one sees that O_{2}(V/O_{3}(V)) contains a subgroup C_{4}, hence (⋆).

Since C_{4} ∘ Q_{8} is an irreducible subgroup of GL_{4}(𝔽_{3}) the 3-modular constituents of V ∘ U have degree divisible by 4, which gives a contradiction.

Since the first two and the last three groups of the lemma above form a full simplex in M_{6}^{irr}(Q_{∞,2}), one now gets Proposition 16.31

17. The a.i.m.f. subgroups of GL_{7}(Q)

Theorem 17.1. Let Q be a definite quaternion algebra with center ℚ and G be a primitive a.i.m.f. subgroup of GL_{7}(Q). Then G is conjugate to one of the groups in the following table.

List of the primitive a.i.m.f. subgroups of GL_{7}(Q).

lattice L  |Aut(L)| r.i.m.f. supergroups
∞,2[^{±}U_{3}(3) □^{2} C_{4}]_{7} 2^{8} · 3^{3} · 7 [U_{3}(3) ⊠_{\sqrt{-1}}^{2} (Q_{8} ∘ C_{4}).S_{3}]_{28}
∞,2[SL_{2}(13)]_{7}  2^{3} · 3 · 7 · 13 [SL_{2}(13)^{2(2)} ∘ SL_{2}(3)]_{28}
∞,2[2.J_{2}]_{7}  2^{8} · 3^{3} · 5^{2} · 7 [2.J_{2}^{2(2)} ∘ SL_{2}(3)]_{28}
∞,2[SL_{2}(3)]_{1} ⊗ E_{7} 2^{12} · 3^{5} · 5 · 7 F_{4} ⊗ E_{7}
∞,3[S̃_{3}]_{1} ⊗ E_{7}  2^{11} · 3^{5} · 5 · 7 (A_{2} ⊗ E_{7})^{2}

Proof. Let Q be a definite quaternion algebra with center ℚ and G be a primitive a.i.m.f. subgroup of GL_{7}(Q). Assume that 1 ≠ N ⊴ G is a quasi-semi-simple normal subgroup of G. With Table 9.1 one finds that B := B^{∘}(N) is one of SL_{2}(13), U_{3}(3) ∘ C_{4}, ^{±}S_{6}(2), or 2.J_{2}. The centralizer C := C_{G}(N) = C_{G}(B) in G of N embeds into the commuting algebra C_{Q^{7×7}}(N), which is isomorphic to ℚ, ℚ[\sqrt{-1}], Q, or ℚ in the respective cases. If B = SL_{2}(13) or B = 2.J_{2}, the group B is already absolutely irreducible and one computes and concludes that G = B is ∞,2[SL_{2}(13)]_{7} or _{∞,2}[2.J_{2}]_{7}.

If B = U_{3}(3) ∘ C_{4}, then C ≅ C_{4} is contained in B and G contains B of index 2=|Out(N)|. Since the commuting algebra of B is isomorphic to an imaginary quadratic field, one finds only one group G = B.2 in GL_{7}(Q). Hence G is

∞,2[^{±}U_{3}(3) □^{2} C_{4}]_{7} in this case.

If N = S_{6}(2), then B = ^{±}S_{6}(2) = Aut(E_{7}) is tensor decomposing. With Theorem 6.1 one finds that G is conjugate to one of E_{7} ⊗_{∞,2}[SL_{2}(3)]_{1} or E_{7} ⊗_{∞,3}[S̃_{3}]_{1}.

Now assume that G does not contain a quasi-semi-simple normal subgroup. Since there are no nilpotent groups having a character of degree 7 or 14, the largest nilpotent normal subgroup of G embeds into GL_{1}(Q), hence is one of ^{±}C_{3} or Q_{8}.

This leads to a contradiction, because both groups clearly have no automorphism of order 7.

Theorem 17.2. Let Q be a definite quaternion algebra with center ℚ and G be an a.i.m.f. subgroup of GL_{7}(Q). Then Q is one of Q_{∞,2} or Q_{∞,3}. The simplicial complexes M_{7}^{irr}(Q) are as follows:

[figure: _{∞,2}[SL_{2}(13)]_{7}; _{∞,2}[SL_{2}(3)]_{1} ⊗ E_{7}; _{∞,2}[SL_{2}(3)]_{1}^{7}; _{∞,2}[2.J_{2}]_{7}; _{∞,2}[U_{3}(3) □^{2} C_{4}]_{7}]

[figure: _{∞,3}[S̃_{3}]_{1} ⊗ E_{7}; _{∞,3}[S̃_{3}]_{1}^{7}]
List of maximal simplices in M_{7}^{irr}(Q_{∞,2}):
simplex  a common subgroup
(_{∞,2}[SL_{2}(3)]_{1} ⊗ E_{7}, _{∞,2}[SL_{2}(3)]_{1}^{7}) L_{2}(7) ⊗ SL_{2}(3)

List of maximal simplices in M_{7}^{irr}(Q_{∞,3}):
simplex  a common subgroup
(_{∞,3}[S̃_{3}]_{1} ⊗ E_{7}, _{∞,3}[S̃_{3}]_{1}^{7}) L_{2}(7) ⊗ S̃_{3}

Proof. Theorems 17.1 and 6.1 prove the completeness of the list of quaternion algebras Q and of a.i.m.f. subgroups of GL_{7}(Q). One only has to show the completeness of the list of maximal simplices in M_{7}^{irr}(Q_{∞,2}), because the simplicial complex M_{7}^{irr}(Q_{∞,3}) consists of a single simplex: The group _{∞,2}[SL_{2}(13)]_{7} fixes a ℤ-lattice of determinant divisible by 13. So the minimal absolutely irreducible subgroups of the group _{∞,2}[SL_{2}(13)]_{7} are of order divisible by 13 (Lemma 2.13). Since the orders of the maximal subgroups of L_{2}(13) are not divisible by 7 · 13, one concludes that _{∞,2}[SL_{2}(13)]_{7} is minimal absolutely irreducible. Hence _{∞,2}[SL_{2}(13)]_{7} forms a 0-simplex in M_{7}^{irr}(Q_{∞,2}). By [CCNPW 85] the maximal subgroups of 2.J_{2} of order divisible by 7 are C_{2} × U_{3}(3) and (C_{2} × L_{2}(7)).2. Since the last group has no irreducible character of degree 14, and the unique irreducible character of U_{3}(3) of degree 14 belongs to an orthogonal representation, one sees that _{∞,2}[2.J_{2}]_{7} is minimal absolutely irreducible. Similarly, the restriction of the characters χ_{7a} and χ_{7b} of U_{3}(3) to a maximal subgroup of U_{3}(3) become reducible, because these characters are constituents of the permutation character of U_{3}(3) associated to its unique subgroup of order divisible by 7. One concludes that (^{±}U_{3}(3)).2 is the unique

minimal absolutely irreducible subgroup of _{∞,2}[U_{3}(3) □^{2} C_{4}]_{7}. Since this subgroup

does not embed into one of the other a.i.m.f. groups, _{∞,2}[U_{3}(3) □^{2} C_{4}]_{7} forms a component on its own in M_{7}^{irr}(Q_{∞,2}). The remaining two a.i.m.f. groups form a 1-simplex in M_{7}^{irr}(Q_{∞,2}), so the proof is complete.

18. The a.i.m.f. subgroups of GL_{8}(Q)

Theorem 18.1. Let Q be a definite quaternion algebra with center ℚ and G a primitive a.i.m.f. subgroup of GL_{8}(Q). Then G is conjugate to one of the groups listed in the following table:

List of the primitive a.i.m.f. subgroups of GL_{8}(Q).

lattice L  |Aut(L)|  r.i.m.f. supergroups
∞,2[2_{−}^{1+8}.O_{8}^{−}(2)]_{8}  2^{21} · 3^{4} · 5 · 7 · 17[2_{+}^{1+10}.O_{10}^{+}(2)]_{32}
∞,2[2_{−}^{1+4}.Alt_{5}]_{2} ⊗ A_{4}  2^{10} · 3^{2} · 5^{2} A_{4} ⊗ E_{8}
∞,2[SL_{2}(3)]_{1} ⊗ A_{4} ⊗ A_{2}  2^{7} · 3^{3} · 5 A_{2} ⊗ A_{4} ⊗ F_{4}
∞,2[2_{−}^{1+6}.O_{6}^{−}(2)]_{4} ⊗ A_{2}  2^{14} · 3^{5} · 5 A_{2} ⊗ F_{4} ˜⊗F_{4}
∞,2[SL_{2}(5) ⊃ × ⊏^{2(2)}D_{8}]_{4} ⊗ A_{2}  2^{7} · 3^{2} · 5 A_{2} ⊗ [SL_{2}(5) ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+4′}.Alt_{5}]_{16}
∞,2[SL_{2}(3)]_{1} ⊗ E_{8}  2^{16} · 3^{6} · 5^{2} · 7 F_{4} ⊗ E_{8}
∞,2[SL_{2}(3)]_{1} ⊗ [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8} 2^{8} · 3^{3} · 5^{2} F_{4} ⊗ [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8}
∞,2[SL_{2}(3)]_{1} ⊗ M_{8,3}  2^{7} · 3^{2} · 7 F_{4} ⊗ M_{8,3}
∞,2[C_{4} ⊃ ⊏^{2(3)}SL_{2}(7)]_{8}  2^{6} · 3 · 7  [SL_{2}(3) ∘ C_{4}.2 ⊃ × ⊏_{\sqrt{-1}}^{2(3)}SL_{2}(7)]_{32}
∞,2[Sp_{4}(3) ∘ C_{3} ⊃ × ⊏^{2(2)}D_{8}]_{8}  2^{10} · 3^{6} · 5 [(Sp_{4}(3)⊗ \sqrt{-3} Sp_{4}(3)) : 2 □^{2} C_{3}]_{32}
∞,2[SL_{2}(5) ⊃ × ⊏^{2(2)}F_{4}]_{8}  2^{10} · 3^{3} · 5 [SL_{2}(5) ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+6}.O_{6}^{−}(2)]_{32}
∞,2[SL_{2}(5) ⊃ × ⊏^{2(3)}(C_{3} ⊃ × ⊏^{2(2)}D_{8})]_{8}  2^{7} · 3^{2} · 5 [SL_{2}(5) ⊃ × ⊏_{∞,3}^{2(3)}(Sp_{4}(3) □^{2} C_{3})]_{32}
∞,2[SL_{2}(5) ⊠_{\sqrt{5}}^{2(2)} (D_{10} ⊗ D_{8})]_{8}  2^{7} · 3 · 5^{2} [2_{−}^{1+4}.Alt_{5}⊗_{∞,2} SL_{2}(5) ⊠_{\sqrt{5}}^{2} D_{10}]_{32}
∞,2[D_{10} ⊃ × ⊏^{2(6)}(C_{3} □^{2} SL_{2}(3))]_{8}  2^{6} · 3^{2} · 5 [C_{15} : C_{4} ⊃ × ⊏^{2(2)}F_{4}]_{32}
∞,3[S̃_{3}]_{1} ⊗ E_{8}  2^{15} · 3^{6} · 5^{2} · 7 (A_{2} ⊗ E_{8})^{2}
∞,3[S̃_{3}]_{1} ⊗ [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8}  2^{7} · 3^{3} · 5^{2} (A_{2} ⊗ [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8})^{2}
∞,3[S̃_{3}]_{1} ⊗ M_{8,3}  2^{6} · 3^{2} · 7 (A_{2} ⊗ M_{8,3})^{2}
∞,3[SL_{2}(9)]_{2} ⊗ A_{4}  2^{7} · 3^{3} · 5^{2} A_{4} ⊗ E_{8}
∞,3[SL_{2}(9)]_{2} ⊗ F_{4}  2^{11} · 3^{4} · 5 F_{4} ⊗ E_{8}
∞,3[SL_{2}(3) □^{2} C_{3}]_{2} ⊗ A_{4}  2^{7} · 3^{3} · 5 (A_{4} ⊗ F_{4})^{2}
∞,3[Sp_{4}(3) ∘ C_{3} ⊃ × ⊏_{\sqrt{-3}}^{2(2)}SL_{2}(3)]_{8}  2^{10} · 3^{6} · 5 [Sp_{4}(3) ∘ C_{3} ⊃ × ⊏_{\sqrt{-3}}^{2(2)}SL_{2}(3)]_{16}^{2}
∞,3[SL_{2}(7) ⊗^{2(3)} S_{3}]_{8}  2^{6} · 3^{2} · 7 [SL_{2}(7)^{2(3)} ∘ S̃_{3}]_{16}^{2}
∞,3[SL_{2}(17)]_{8}  2^{5} · 3^{2} · 17 [SL_{2}(17)^{2(3)} ∘ S̃_{3}]_{16}
∞,3[SL_{2}(7) ⊃ × ⊏^{2(3)}S_{3}]_{8}  2^{6} · 3^{2} · 7 [SL_{2}(7) ⊃ × ⊏_{\sqrt{-7}}^{2(3)} S̃_{3}]_{16}^{2}
∞,3[2.Alt_{7} ⊃ × ⊏^{2(3)}S_{3}]_{8}  2^{6} · 3^{3} · 5 · 7 [2.Alt_{7} ⊃ × ⊏_{\sqrt{-7}}^{2(3)} S̃_{3}]_{16}^{2}
∞,3[SL_{2}(5) ⊃ × ⊏^{2(3)}(SL_{2}(3) □^{2} C_{3})]_{8}  2^{7} · 3^{3} · 5 [SL_{2}(5) ⊃ × ⊏_{∞,3}^{2(3)}(SL_{2}(3) □^{2} C_{3})]_{16}^{2}
∞,3[SL_{2}(5) ⊃ × ⊏^{2(2)}(C_{3} ⊃ × ⊏^{2(2)}D_{8})]_{8}  2^{7} · 3^{2} · 5 [SL_{2}(5) ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+4}.Alt_{5}]_{16}^{2}
∞,3[C_{3} ⊃ ⊏^{2(2)}2_{−}^{1+6}.O_{6}^{−}(2)]_{8}  2^{14} · 3^{5} · 5 (F_{4} ˜⊗F_{4})^{2}
∞,3[SL_{2}(5) ⊠_{\sqrt{5}}^{2(3)} (D_{10} ⊗ S_{3})]_{8}  2^{6} · 3^{2} · 5^{2} [(SL_{2}(5) ∘ SL_{2}(5)) : 2 ⊠_{\sqrt{5}}^{2} D_{10}]_{16}^{2}
∞,3[D_{10} ⊃ × ⊏^{2(6)}(C_{3} ⊃ × ⊏^{2(2)}D_{8})]_{8}  2^{6} · 3 · 5  [D_{120}.(C_{4} × C_{2})]_{16}^{2}

lattice L  |Aut(L)| r.i.m.f. supergroups
∞,5[2.S_{6}]_{8}  2^{5} · 3^{2} · 5 [SL_{2}(9) □^{2} SL_{2}(5)]_{32}
∞,5[SL_{2}(5).2]_{2} ⊗ F_{4}  2^{10} · 3^{3} · 5F_{4} ⊗ E_{8}
∞,5[SL_{2}(5) : 2]_{2} ⊗ F_{4}  2^{10} · 3^{3} · 5F_{4} ⊗ [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8}
∞,5[D_{10} ⊃ × ⊏^{2(3)}SL_{2}(9)]_{8}  2^{6} · 3^{2} · 5^{2}[SL_{2}(9) ⊗ D_{10} □^{2} SL_{2}(5)]_{32}
∞,5[SL_{2}(5) ⊠_{\sqrt{5}}^{2} D_{10}]_{4} ⊗ A_{2}  2^{6} · 3^{2} · 5^{2}A_{2} ⊗ [(SL_{2}(5) ∘ SL_{2}(5)) : 2 ⊠_{\sqrt{5}}^{2} D_{10}]_{16}
∞,5[((SL_{2}(5) ∘ SL_{2}(5)) ⊠_{\sqrt{5}}^{2} SL_{2}(5)) : S_{3}]_{8,1}  2^{9} · 3^{4} · 5^{3}G_{1}
∞,5[((SL_{2}(5) ∘ SL_{2}(5)) ⊠_{\sqrt{5}}^{2} SL_{2}(5)) : S_{3}]_{8,2}  2^{9} · 3^{4} · 5^{3}G_{2}
∞,5[SL_{2}(5) ⊃ × ⊏^{2(6)}(C_{3} ⊃ × ⊏^{2(2)}D_{8})]_{8,1}  2^{7} · 3^{2} · 5 [(SL_{2}(5) ∘ SL_{2}(5)) : 2 ⊃ × ⊏_{\sqrt{5}}^{2(6)}(C_{3} ⊃ × ⊏^{2(2)}D_{8})]_{32,1}
∞,5[SL_{2}(5) ⊃ × ⊏^{2(6)}(C_{3} ⊃ × ⊏^{2(2)}D_{8})]_{8,2}  2^{7} · 3^{2} · 5 [(SL_{2}(5) ∘ SL_{2}(5)) : 2 ⊃ × ⊏_{\sqrt{5}}^{2(6)}(C_{3} ⊃ × ⊏^{2(2)}D_{8})]_{32,2}
∞,5[D_{10} ⊃ × ⊏^{2(2)}2_{−}^{1+4}.Alt_{5}]_{8}  2^{9} · 3 · 5^{2} [(2_{−}^{1+4}.Alt_{5}⊗_{∞,2} SL_{2}(5)) ⊠_{\sqrt{5}}^{2(2)} D_{10}]_{32}
∞,5[D_{10} ⊃ × ⊏^{2(3)}(C_{3} □^{2} SL_{2}(3))]_{8}  2^{6} · 3^{2} · 5 [(SL_{2}(5)⊗ \sqrt{5} D_{10}) ⊃ × ⊏_{∞,3}^{2(3)}(SL_{2}(3) □^{2} C_{3})]_{32}
∞,5[SL_{2}(5).2]_{2}⊗ \sqrt{-3} _{∞,3}[S̃_{3}]_{1}⊗ \sqrt{-3} _{∞,2}[SL_{2}(3)]_{1} 2^{7} · 3^{3} · 5 [Sp_{4}(3) ∘ C_{3} ⊃ × ⊏_{\sqrt{-3}}^{2(2)}SL_{2}(3)]_{16}^{2}
∞,5[SL_{2}(5) : 2]_{2}⊗ \sqrt{-3} _{∞,3}[S̃_{3}]_{1}⊗ \sqrt{-3} _{∞,2}[SL_{2}(3)]_{1} 2^{7} · 3^{3} · 5 [SL_{2}(5) ⊃ × ⊏_{∞,3}^{2(3)}(SL_{2}(3) □^{2} C_{3})]_{16}^{2}
∞,5[SL_{2}(5).2]_{2}⊗ \sqrt{-3} _{2,3}[C_{3} ⊃ × ⊏^{2(2)}D_{8}]_{2}  2^{7} · 3^{2} · 5 E_{8}^{4}
∞,5[SL_{2}(5) : 2]_{2}⊗ \sqrt{-3} _{2,3}[C_{3} ⊃ × ⊏^{2(2)}D_{8}]_{2}  2^{7} · 3^{2} · 5 [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8}^{4}
∞,2,3,5[SL_{2}(5) ⊃ × ⊏^{2(6)}(S_{3} ⊗ D_{8})]_{8,1}  2^{7} · 3^{2} · 5 (F_{4} ˜⊗F_{4})^{2}
∞,2,3,5[SL_{2}(5) ⊃ × ⊏^{2(6)}(S_{3} ⊗ D_{8})]_{8,2}  2^{7} · 3^{2} · 5 [SL_{2}(5) ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+4}.Alt_{5}]_{16}^{2}
∞,2,3,5[SL_{2}(5) ⊃ × ⊏^{2(6)}(SL_{2}(3) □^{2} C_{3})]_{8,1}  2^{7} · 3^{3} · 5 (A_{2} ⊗ E_{8})^{2}
∞,2,3,5[SL_{2}(5) ⊃ × ⊏^{2(6)}(SL_{2}(3) □^{2} C_{3})]_{8,2}  2^{7} · 3^{3} · 5 (A_{2} ⊗ [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8})^{2}
3,5[^{±}D_{10} ⊃ × ⊏^{2(3)}S_{3}]_{4}⊗ \sqrt{-3} _{∞,2}[SL_{2}(3)]_{1}  2^{6} · 3^{2} · 5 (A_{4} ⊗ F_{4})^{2}
∞,2,3,5[D_{10} ⊃ × ⊏^{2(2)}(C_{3} □^{2} SL_{2}(3))]_{8}  2^{6} · 3^{2} · 5 [(SL_{2}(5) ∘ SL_{2}(5)) : 2 ⊠_{\sqrt{5}}^{2} D_{10}]_{16}^{2}
2,5[D_{10} ⊃ × ⊏^{2(2)}D_{8}]_{4}⊗ \sqrt{-3} _{∞,3}[S̃_{3}]_{1}  2^{6} · 3 · 5 [D_{120}.(C_{4} × C_{2})]_{16}^{2}
∞,2,3,5[D_{10} ⊃ × ⊏^{2(3)}(C_{3} ⊃ × ⊏^{2(2)}D_{8})]_{8}  2^{6} · 3 · 5 (A_{2} ⊗ A_{4})^{2}

Here for i = 1, 2 G_{i} := [((SL_{2}(5) ∘ SL_{2}(5)) ⊠_{\sqrt{5}}^{2} (SL_{2}(5) ∘ SL_{2}(5))) : S_{4}]_{32,i}.

lattice L  |Aut(L)| r.i.m.f. supergroups
∞,7[SL_{2}(7).2]_{4} ⊗ A_{2} 2^{6} · 3^{2} · 7 (A_{2} ⊗ E_{8})^{2}
∞,7[SL_{2}(7) ⊃ × ⊏^{2(2)}D_{8}]_{8} 2^{7} · 3 · 7 (F_{4} ˜⊗F_{4})^{2}
∞,7[SL_{2}(7) ⊃ × ⊏_{\sqrt{-7}}^{2(3)} S̃_{3}]_{8} 2^{6} · 3^{2} · 7 [SL_{2}(7) ⊃ × ⊏_{\sqrt{-7}}^{2(3)} S̃_{3}]_{16}^{2}
∞,7[2.S_{7}]_{4} ⊗ A_{2}  2^{6} · 3^{3} · 5 · 7(A_{2} ⊗ E_{8})^{2}
∞,7[Alt_{7} ⊃ × ⊏_{\sqrt{-7}}^{2(3)} S̃_{3}]_{8} 2^{6} · 3^{3} · 5 · 7[Alt_{7} ⊃ × ⊏_{\sqrt{-7}}^{2(3)} S̃_{3}]_{16}^{2}
∞,7[Alt_{7} ⊃ × ⊏^{2(3)}D_{8}]_{8} 2^{7} · 3^{2} · 5 · 7(F_{4} ˜⊗F_{4})^{2}
∞,17[SL_{2}(17).2]_{8} 2^{6} · 3^{2} · 17 [SL_{2}(17) ⊃ ⊏^{2(3)} S̃_{3}]_{32,1}

[SL_{2}(17) ⊃ ⊏^{2(3)} S̃_{3}]_{32,1}

∞,17[SL_{2}(17) : 2]_{8} 2^{6} · 3^{2} · 17 [SL_{2}(17) ⊃ ⊏^{2(3)} S̃_{3}]_{32,2}

[SL_{2}(17) ⊃ ⊏^{2(3)} S̃_{3}]_{32,2}

The proof is split into several lemmata. For the rest of this chapter let Q be a definite quaternion algebra with center ℚ and G be a primitive a.i.m.f. subgroup of GL_{8}(Q).

By Table 9.1 and Lemma 7.2 the possibilities for quasi-semi-simple normal subgroups N of G are SL_{2}(5), SL_{2}(5)∘SL_{2}(5), SL_{2}(5)∘SL_{2}(5)⊗ \sqrt{5} SL_{2}(5), Alt_{5}, L_{2}(7), SL_{2}(7) (2 matrix groups), SL_{2}(9) (2 matrix groups), SL_{2}(17) (2 matrix groups), 2.Alt_{7}, Sp_{4}(3) = 2.U_{4}(2), and 2.O_{8}^{+}(2).

First we treat the tensor decomposing normal subgroups N.

Lemma 18.2. If G contains a normal subgroup N ≅ Alt_{5}, then G is one of

∞,2[2_{−}^{1+4}.Alt_{5}]_{2} ⊗ A_{4}, _{∞,2}[SL_{2}(3)]_{1} ⊗ A_{2} ⊗ A_{4},

∞,3[SL_{2}(3) □^{2} C_{3}]_{2} ⊗ A_{4}, or _{∞,3}[SL_{2}(9)]_{2} ⊗ A_{4}.

Proof. By Corollary 7.6 G is of the form A_{4} ⊗H, where H ≤ GL_{2}(Q) is a primitive a.i.m.f. group. Hence by Theorem 12.1 H is one of

∞,2[(D_{8} ⊗ Q_{8}).Alt_{5}]_{2}, _{∞,2}[SL_{2}(3)]_{1} ⊗ A_{2}, _{∞,3}[SL_{2}(9)]_{2},

∞,3[SL_{2}(3) □^{2} C_{3}]_{2}, _{∞,5}[SL_{2}(5).2]_{2} or _{∞,5}[SL_{2}(5):2]_{2}.

In the last two cases G is a proper subgroup of

∞,5[((SL_{2}(5) ∘ SL_{2}(5)) ⊠_{\sqrt{5}}^{2} SL_{2}(5)) : S_{3}]_{8,1}

resp. _{∞,5}[((SL_{2}(5) ∘ SL_{2}(5)) ⊠_{\sqrt{5}}^{2} SL_{2}(5)) : S_{3}]_{8,2}

Similarly one gets the next two lemmata:

Lemma 18.3. If G contains a normal subgroup N ≅ L_{2}(7), then G is one of ∞,2[SL_{2}(3)]_{1} ⊗ M_{8,3} or _{∞,3}[S̃_{3}]_{1} ⊗ M_{8,3}.

Lemma 18.4. If G contains a normal subgroup N ≅ 2.O_{8}^{+}(2), then G is one of ∞,2[SL_{2}(3)]_{1} ⊗ E_{8} or _{∞,3}[S̃_{3}]_{1} ⊗ E_{8}.

The next lemma deals with the absolutely irreducible quasi-semi-simple normal subgroups N:

Lemma 18.5. If G contains a normal subgroup N isomorphic to SL_{2}(17) with character χ_{16}, then G is conjugate to _{∞,3}[SL_{2}(17)]_{8}.

Proof. N is already absolutely irreducible. One finds G = B^{∘}(N).

Next we treat those candidates for normal subgroups N in G, such that C_{G}(N) has to be contained in B^{∘}(N).

Lemma 18.6. If G has a normal subgroup N isomorphic to one of SL_{2}(9), SL_{2}(5)∘ SL_{2}(5)⊗ \sqrt{5} SL_{2}(5), or SL_{2}(17) with character χ_{8a} + χ_{8b}, then G is conjugate to one of

∞,5[2.S_{6}], _{∞,5}[((SL_{2}(5) ∘ SL_{2}(5)) ⊠_{\sqrt{5}}^{2} SL_{2}(5)) : S_{3}]_{8,1},

∞,5[((SL_{2}(5) ∘ SL_{2}(5)) ⊠_{\sqrt{5}}^{2} SL_{2}(5)) : S_{3}]_{8,2}, _{∞,17}[SL_{2}(17).2]_{8} or _{∞,17}[SL_{2}(17) : 2]_{8}.

Proof. In all cases the centralizer C_{G}(N) embeds into the enveloping algebra of N and hence is contained in B^{∘}(N). Assume first, that N is isomorphic to SL_{2}(9). Since the character field of the extension of the character χ_{8a} to 2.P GL_{2}(9) is of degree 4 over ℚ (cf. [CCNPW 85]), G is isomorphic to 2.S_{6}. (Note that the outer automorphism of S_{6} interchanges the two isoclinism classes of groups 2.S_{6}, so there is only one group to be considered.) Hence G = _{∞,5}[2.S_{6}].

If N = SL_{2}(5) ∘ SL_{2}(5)⊗ \sqrt{5} SL_{2}(5), then B^{∘}(N) = N : S_{3} and one computes that G is one of the two extensions (N : S_{3}).2. In the last case, G = N.2 is one of the two extensions of SL_{2}(17) by Out(SL_{2}(17)) ≅ C_{2}.

Lemma 18.7. If G contains a normal subgroup N ≅ SL_{2}(7) with character χ_{8},

then G is one of _{∞,3}[SL_{2}(7) ⊗^{2(3)} S_{3}]_{8} or _{∞,2}[C_{4} ⊃ ⊏^{2(3)}SL_{2}(7)]_{8}.

Proof. By Table 9.1 the group N is nearly tensor decomposing over ℚ with parameter 3. Since B^{∘}(N) = N by 10.1 G is either G = NC where C := C_{G}(N) is an a.i.m.f. subgroup of GL_{1}(D), where D is an indefinite quaternion algebra

with center ℚ such that (C, 3, D) is not a maximal triple or of the form B ⊗^{2(3)} C

or B ⊃ × ⊏^{2(3)} C where (C, 3, D) is a maximal triple. Since the group SL_{2}(7) ⊗ D_{8}

is imprimitive using Table 10.2 one finds that G is one of _{∞,3}[SL_{2}(7) ⊗^{2(3)} S_{3}]_{8} or

∞,2[C_{4} ⊃ ⊏^{2(3)}SL_{2}(7)]_{8}.

Lemma 18.8. If G contains a normal subgroup N ≅ SL_{2}(9) with character χ_{4},

then G is one of _{∞,3}[SL_{2}(9)]_{2} ⊗ A_{4}, _{∞,3}[SL_{2}(9)]_{2} ⊗ F_{4}, or _{∞,5}[D_{10} ⊃ × ⊏^{2(3)}SL_{2}(9)]_{8}.

Proof. As in the last lemma N is nearly tensor decomposing over ℚ with parameter 3. Hence G is either G = NC where C := C_{G}(N) is an a.i.m.f. subgroup of GL_{2}(D), where D is an indefinite quaternion algebra with center ℚ such that (C, 3, D) is not a maximal triple or of the form B ⊗^{2(3)}C or B ⊃ × ⊏^{2(3)}C where (C, 3, D) is a maximal triple. Moreover B^{∘}(SL_{2}(9) ∘ C_{3}) = Sp_{4}(3) ∘ C_{3} implies that in both cases O_{3}(C) = 1.

Since the group _{∞,2}[(C_{4} ∘ SL_{2}(3)).2 ⊃ × ⊏_{\sqrt{-1}}^{2(3)}SL_{2}(9)]_{8} is contained in _{∞,2}[2_{−}^{1+8}.O_{8}^{−}(2)]_{8},

Table 10.4 implies that G is one of the three a.i.m.f. groups in the Lemma.

In the next three cases, the center of the enveloping algebra of N is an imaginary quadratic field.

Lemma 18.9. If G has a normal subgroup N ≅ SL_{2}(7) with character χ_{4a} +

χ_{4b}, then G is conjugate to one of _{∞,3}[SL_{2}(7) ⊃ × ⊏^{2(3)} S_{3}]_{8}, _{∞,7}[SL_{2}(7).2]_{4} ⊗ A_{2},

∞,7[SL_{2}(7) ⊃ × ⊏^{2(2)}D_{8}]_{8}, or _{∞,7}[SL_{2}(7) ⊃ × ⊏_{\sqrt{-7}}^{2(3)} S̃_{3}]_{8}.

Proof. The centralizer C := C_{G}(N) is a centrally irreducible subgroup of GL_{1}(D) where D is a quaternion algebra over ℚ[\sqrt{-7}] and G contains NC of index 2. Using the classification of finite subgroups of GL_{2}(ℂ) in [Bli 17], one finds that C is one of D_{8}, ^{±}S_{3}, SL_{2}(3), or S̃_{3}. Distinguish 2 cases:

a) C_{G}(C) > ^{±}N. Then G = (N.2)C is one of _{∞,7}[SL_{2}(7).2]_{4} ⊗ [D_{8}]_{2} or ∞,7[SL_{2}(7).2]_{4} ⊗A_{2}. In the first case, G is imprimitive contained in _{∞,7}[SL_{2}(7).2]_{4}^{2}.

b) C_{G}(C) = ^{±}N. The groups C are nearly tensor decomposing over ℚ with parameter 2, 3, 2, resp. 3. Since C = C_{G}(N) Lemma 10.1 implies that G is

one of _{∞,7}[SL_{2}(7) ⊃ × ⊏^{2(2)} D_{8}]_{8}, _{∞,2}[SL_{2}(7) ⊃ × ⊏^{2(2)} SL_{2}(3)]_{8}, _{∞,3}[SL_{2}(7) ⊃ × ⊏^{2(3)} S_{3}]_{8}, or

∞,7[SL_{2}(7) ⊃ × ⊏_{\sqrt{-7}}^{2(3)} S̃_{3}]_{8}. The second group fixes a 32-dimensional extremal unimod-

ular lattice with maximal order as an endomorphism ring. By [BaN 97] this yields that the second group is contained in _{∞,2}[2_{−}^{1+8}.O_{8}^{−}(2)]_{8}.

Completely analogous one finds:

Lemma 18.10. If G has a normal subgroup N ≅ 2.Alt_{7} with character χ_{4a} + χ_{4b}, then G is conjugate to one of

∞,3[2.Alt_{7} ⊃ × ⊏^{2(3)}S_{3}]_{8}, _{∞,7}[2.S_{7}]_{4} ⊗ A_{2},

∞,7[2.Alt_{7} ⊃ × ⊏^{2(2)}D_{8}]_{8} or _{∞,7}[2.Alt_{7} ⊃ × ⊏_{\sqrt{-7}}^{2(3)} S̃_{3}]_{8}.

Lemma 18.11. If G has a normal subgroup N ≅ 2.U_{4}(2) = Sp_{4}(3) with character χ_{4a} + χ_{4b}, then G is conjugate to

∞,2[Sp_{4}(3) ∘ C_{3} ⊃ × ⊏^{2(2)}D_{8}]_{8} or _{∞,3}[Sp_{4}(3) ∘ C_{3} ⊃ × ⊏_{\sqrt{-3}}^{2(2)}SL_{2}(3)]_{8}.

Proof. Let B := B^{∘}(N) ≅ ^{±}C_{3} ∘ N. The centralizer C := C_{G}(N) is a centrally irreducible subgroup of GL_{1}(D) where D is a quaternion algebra over ℚ[\sqrt{-3}] and G contains BC of index 2. Moreover O_{3}(B) = C_{3} implies O_{3}(C) = 1. Hence C is either D_{8} or SL_{2}(3) and the lemma follows as above.

Lemma 18.12. If N = SL_{2}(5) ∘ SL_{2}(5) is a normal subgroup of G, then G is one

of _{∞,2}[SL_{2}(3)]_{1} ⊗ [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8} or _{∞,3}[S̃_{3}]_{1} ⊗ [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8}.

Proof. G contains the normal subgroup B := B^{∘}(N) = N : 2. The centralizer C := C_{G}(N) embeds into ℚ[\sqrt{5}] ⊗ Q. Since the primes dividing |G| are ≤ 5, ℚ[\sqrt{5}] splits all possible Schur indices of Q at a finite prime. Moreover G contains CB of index 2 = [ℚ[\sqrt{5}] : ℚ] and C is a centrally irreducible subgroup of GL_{1}(Q_{\sqrt{5},∞}). Hence C is one of SL_{2}(5), SL_{2}(3), S̃_{3}, or Q_{20}. The first case contradicts Lemma 18.6.

The lemma follows since the groups (SL_{2}(5)∘SL_{2}(5)) : 2 ⊃ × ⊏^{2(2)}SL_{2}(3) (2 extensions),

(SL_{2}(5)∘SL_{2}(5)) : 2 ⊃ × ⊏^{2(3)} S̃_{3} (2 extensions), (SL_{2}(5)∘SL_{2}(5)) : 2 ⊠_{\sqrt{5}}^{2}Q_{20} are contained

in one of the groups _{∞,5}[((SL_{2}(5) ∘ SL_{2}(5)) ⊠_{\sqrt{5}}^{2} SL_{2}(5)) : S_{3}]_{8,i} (i = 1, 2).

Lemma 18.13. If N := SL_{2}(5) is the only quasi-semi-simple normal subgroup of G, then G is one of the following twenty a.i.m.f. groups

∞,5[SL_{2}(5).2]_{2} ⊗ F_{4}, _{∞,5}[SL_{2}(5) : 2]_{2} ⊗ F_{4},
∞,5[SL_{2}(5).2]_{2}⊗ \sqrt{-3} _{∞,3}[S̃_{3}]_{1}⊗ \sqrt{-3} _{∞,2}[SL_{2}(3)]_{1},
∞,5[SL_{2}(5) : 2]_{2}⊗ \sqrt{-3} _{∞,3}[S̃_{3}]_{1}⊗ \sqrt{-3} _{∞,2}[SL_{2}(3)]_{1},

∞,5[SL_{2}(5).2]_{2}⊗ \sqrt{-3} _{2,3}[C_{3} ⊃ × ⊏^{2(2)}D_{8}]_{2}, _{∞,5}[SL_{2}(5) : 2]_{2}⊗ \sqrt{-3} _{2,3}[C_{3} ⊃ × ⊏^{2(2)}D_{8}]_{2},

∞,2[SL_{2}(5) ⊃ × ⊏^{2(2)}F_{4}]_{8}, _{∞,2,3,5}[SL_{2}(5) ⊃ × ⊏^{2(6)}(SL_{2}(3) □^{2} C_{3})]_{8,1},

∞,2,3,5[SL_{2}(5) ⊃ × ⊏^{2(6)}(SL_{2}(3) □^{2} C_{3})]_{8,2}, _{∞,2}[SL_{2}(5) ⊃ × ⊏^{2(2)}D_{8}]_{4} ⊗ A_{2},

∞,3[SL_{2}(5) ⊃ × ⊏^{2(3)}(SL_{2}(3) □^{2} C_{3})]_{8}, _{∞,2,3,5}[SL_{2}(5) ⊃ × ⊏^{2(6)}(S_{3} ⊗ D_{8})]_{8,1},

∞,2,3,5[SL_{2}(5) ⊃ × ⊏^{2(6)}(S_{3} ⊗ D_{8})]_{8,2}, _{∞,3}[SL_{2}(5) ⊃ × ⊏^{2(2)}(C_{3} ⊃ × ⊏^{2(2)}D_{8})]_{8},

∞,2[SL_{2}(5) ⊃ × ⊏^{2(3)}(C_{3} ⊃ × ⊏^{2(2)}D_{8})]_{8}, _{∞,5}[SL_{2}(5) ⊃ × ⊏^{2(6)}(C_{3} ⊃ × ⊏^{2(2)}D_{8})]_{8,1},

∞,5[SL_{2}(5) ⊃ × ⊏^{2(6)}(C_{3} ⊃ × ⊏^{2(2)}D_{8})]_{8,2}, _{∞,5}[SL_{2}(5) ⊠_{\sqrt{5}}^{2} D_{10}]_{4} ⊗ A_{2},

∞,3[SL_{2}(5) ⊠_{\sqrt{5}}^{2(3)} (D_{10} ⊗ S_{3})]_{8} or _{∞,2}[SL_{2}(5) ⊠_{\sqrt{5}}^{2(2)} (D_{10} ⊗ D_{8})]_{8}.

Proof. Let C := C_{G}(N). Then C embeds into ℚ[\sqrt{5}]^{4×4} (again since ℚ[\sqrt{5}] splits all possible finite Schur indices) and G contains the group NC of index 2. By 2.14 C is a centrally irreducible subgroup of GL_{4}(ℚ[\sqrt{5}]). Distinguish two cases:

a) C_{G}(C) > N. Then C_{G}(C) = N.2 is one of the two extensions of N by Out(N) and G = N.2 ⊗ C, where C is an a.i.m.f. subgroup of Q_{∞,5} ⊗ Q. Hence C is either a r.i.m.f. subgroup of GL_{4}(ℚ), thus C = F_{4} by Lemma 18.2 or a 3-parametric irreducible Bravais group in GL_{8}(ℚ). By [Sou 94] C is one of SL_{2}(3)⊗ \sqrt{-3} S̃_{3} (B_{21})

or C_{3} ⊃ × ⊏^{2(2)}D_{8} (B_{19} ∼ B_{20}). Hence G is one of the first six groups of the lemma.

b) C_{G}(C) = N. The groups C will be constructed according to their possible normal p-subgroups:

(i) Assume first that O_{3}(C) = O_{5}(C) = 1. Then Table 8.7 together with the central irreducibility of C implies that O_{2}(C) = Q_{8} ∘ Q_{8} and G contains N ⊗ F_{4} of index 2. Moreover the elements in G − CN induce an outer automorphism of

F_{4}. Since one of the two extensions SL_{2}(5) ⊃ × ⊏^{2(2)} F_{4} embeds into _{∞,2}[2_{−}^{1+8}.O_{8}^{−}(2)]_{8}

the group G is _{∞,2}[SL_{2}(5) ⊃ × ⊏^{2(2)}F_{4}]_{8} in this case.

(ii) Now assume that O_{3}(C) > 1 and O_{5}(C) = 1. Then O_{3}(C) ≅ C_{3} and C_{C}(O_{3}(C)) is a centrally irreducible subgroup of GL_{2}(ℚ[\sqrt{5}, ζ_{3}]). Hence O_{2}(C)

is one of Q_{8} or D_{8} and C is one of SL_{2}(3)⊗ \sqrt{-3} S̃_{3}, SL_{2}(3) □^{2} C_{3}, D_{8} ⊗ S_{3}, or

C_{3} ⊃ × ⊏^{2(2)} D_{8}. Note that by Lemma 2.17 in each case there is a unique extension C = C_{C}(O_{3}(C)).2 with real Schur index 1. For all four candidates for C, the outer automorphism group Out(C) is isomorphic to C_{2} × C_{2}, hence one has to consider three nontrivial outer automorphisms which can be distinguished via the determinants of the elements in \overline{ℚC} inducing the automorphism by conjugation (cf. Corollary 7.12). In each case there are two extensions G = NC.2. Hence one has to construct twenty-four candidates for a.i.m.f. groups G. The groups

SL_{2}(5) ⊃ × ⊏_{∞,2}^{2(2)} SL_{2}(3) ⊗ S̃_{3} (both extensions) and SL_{2}(5) ⊃ × ⊏_{∞,3}^{2(3)} S̃_{3} ⊗ SL_{2}(3) (both ex-

tensions) clearly embed into _{∞,3}[S̃_{3}]_{1} ⊗ E_{8}, _{∞,3}[S̃_{3}]_{1} ⊗ [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8},

∞,2[SL_{2}(3)]_{1} ⊗ E_{8}, respectively _{∞,2}[SL_{2}(3)]_{1} ⊗ [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8}. Also the

two groups SL_{2}(5) ⊃ × ⊏_{\sqrt{5}}^{2(6)}(SL_{2}(3)⊗ \sqrt{-3} S̃_{3}) are not maximal finite but contained in the

respective groups _{∞,5}[((SL_{2}(5) ∘ SL_{2}(5)) ⊠_{\sqrt{5}}^{2} SL_{2}(5)) : S_{3}]_{8,i} (i = 1, 2).

The two groups SL_{2}(5) ⊃ × ⊏^{2(2)} (SL_{2}(3) □^{2} C_{3}) may be enlarged to the respective

groups SL_{2}(5) ⊃ × ⊏^{2(2)}F_{4}. One of the extensions SL_{2}(5) ⊃ × ⊏^{2(3)}(SL_{2}(3) □^{2} C_{3}) is contained

in _{∞,3}[Sp_{4}(3) ∘ C_{3} ⊃ × ⊏_{\sqrt{-3}}^{2(2)}SL_{2}(3)]_{8}.

The two groups _{∞,3}[SL_{2}(5) ⊃ × ⊏^{2(3)}S_{3}]_{4} ⊗ D_{8} are imprimitive and one of the groups

SL_{2}(5) ⊃ × ⊏^{2(2)} D_{8} ⊗ A_{2} is contained in _{∞,2}[2_{−}^{1+6}.O_{6}^{−}(2)]_{4} ⊗ A_{2}. One of the extensions

SL_{2}(5) ⊃ × ⊏_{\sqrt{-1}}^{2(2)}(C_{3} ⊃ × ⊏^{2(2)}D_{8}) is contained in_{∞,3}[C_{3} ⊃ ⊏^{2(2)}2_{−}^{1+6}.O_{6}^{−}(2)]_{8} and one of the groups

SL_{2}(5) ⊃ × ⊏_{\sqrt{-1}}^{2(3)}(C_{3} ⊃ × ⊏^{2(2)}D_{8}) is a proper subgroup of _{∞,2}[Sp_{4}(3) ∘ C_{3} ⊃ × ⊏^{2(2)}D_{8}]_{8}. Hence G is

one of the ten groups numbered 8–17 of the lemma.

(iii) Now assume that O_{5}(C) > 1. Then O_{5}(C) ≅ C_{5} and C_{C}(O_{5}(C)) is a centrally irreducible subgroup of GL_{2}(ℚ[ζ_{5}]) and hence of the form H ⊗ C_{5}, where H is one of S_{3}, S̃_{3}, D_{8}, or SL_{2}(3). Moreover C contains C_{C}(O_{5}(C)) of index 2. Since the outer automorphism group of H is C_{2} in all cases, C is one of ^{±}D_{10} ⊗ S_{3},

±C_{5} ⊃ × ⊏^{2(3)}S_{3}, Q_{20} ∘S̃_{3}, C_{5} ⊃ × ⊏_{\sqrt{5}′}^{2(3)} S̃_{3}, D_{10} ⊗D_{8}, C_{5} ⊃ × ⊏^{2(2)}D_{8}, Q_{20} ∘SL_{2}(3), or C_{5} ⊃ × ⊏_{\sqrt{5}′}^{2(2)}SL_{2}(3).

In the four cases where C_{C}(H) = ^{±}C_{5} the outer automorphism group of C is cyclic of order 4 yielding no possibilities for primitive groups G = NC.2 ≤ GL_{8}(Q). In the other four cases Out_{stab}(C) ≅ C_{2} and one has two possible extensions G = NC.2. But now they lead to isomorphic groups. In all cases where a normal subgroup Q_{20} is involved, one may enlarge NC.2 by replacing Q_{20} by SL_{2}(5). Since the group

∞,5[SL_{2}(5) ⊠_{\sqrt{5}}^{2} D_{10}]_{4} ⊗ D_{8} is imprimitive, G is one of the last three groups of the

Lemma.

For the rest of this chapter we assume that G does not contain a quasi-semi-simple normal subgroup. By Lemma 11.2 O_{17}(G) = 1.

Immediately from Proposition 8.9 one finds

Lemma 18.14. If O_{3}(G) = O_{5}(G) = 1, then G = _{∞,2}[2_{−}^{1+8}.O_{8}^{−}(2)]_{8}.

Lemma 18.15. If O_{5}(G) = 1 and O_{3}(G) > 1, then G is one of

∞,2[2_{−}^{1+6}.O_{6}^{−}(2)]_{4} ⊗ A_{2} or _{∞,3}[C_{3} ⊃ ⊏^{2(2)}2_{−}^{1+6}.O_{6}^{−}(2)]_{8}.

Proof. Then O_{3}(G) ≅ C_{3} and C := C_{G}(O_{3}(G)) is a normal subgroup of index 2 in G. Moreover C is a centrally irreducible normal subgroup of GL_{4}(Q ⊗ ℚ[ζ_{3}]), whence O_{2}(C) = 2_{−}^{1+6} or 2_{+}^{1+6}. Let B := B^{∘}(O_{2}(G)). Then G contains the normal subgroup O_{3}(G)B of index two. The enveloping algebra of B is a central simple ℚalgebra and B fixes up to isomorphism 2 lattices. With Corollary 7.12 one finds that Glide(B) is (at most) C_{2}. The group 2_{+}^{1+6}.Alt_{8} ⊗ S̃_{3} is contained in _{∞,3}[S̃_{3}]_{1} ⊗ E_{8}

and C_{3} ⊃ × ⊏^{2(2)} 2_{+}^{1+6}.Alt_{8} has the a.i.m.f. supergroup _{∞,2}[2_{−}^{1+8}.O_{8}^{−}(2)]_{8}. So G is one of the two groups in the lemma.

Lemma 18.16. If O_{5}(G) > 1 and O_{3}(G) = 1, then G is conjugate to

∞,5[D_{10} ⊃ × ⊏^{2(2)}2_{−}^{1+4}.Alt_{5}]_{8}.

Proof. If O_{5}(G) > 1, then O_{5}(G) ≅ C_{5} and C := C_{G}(O_{5}(G)) is a centrally irreducible subgroup of GL_{4}(ℚ[ζ_{5}]). Moreover G/C ≅ C_{4} ≅ Out(C_{5}). Let B := B^{∘}(O_{2}(G)). Table 8.7 gives that C = C_{5}B, since B is one of F_{4} or 2_{−}^{1+4}.Alt_{5}. In both cases Glide(B) ≅ C_{2}, hence G contains the normal subgroup Q_{20} ⊗ F_{4} resp. D_{10} ⊗ 2_{−}^{1+4}.Alt_{5} of index 2. The first possibility leads to groups contained in

∞,5[SL_{2}(5).2]_{2}⊗F_{4} or_{∞,2}[SL_{2}(5) ⊃ × ⊏^{2(2)}F_{4}]_{8}. In the second case G is the a.i.m.f. group of the lemma, since (C_{5} : C_{4}) ⊗ 2_{−}^{1+4}.Alt_{5} is contained in _{∞,2}[2_{−}^{1+4}.Alt_{5}]_{2} ⊗ A_{4}.

Lemma 18.17. If O_{5}(G) > 1 and O_{3}(G) > 1, then G is conjugate to one of

2,5[D_{10} ⊃ × ⊏^{2(2)}D_{8}]_{4}⊗ \sqrt{-3} _{∞,3}[S̃_{3}]_{1}, _{3,5}[^{±}D_{10} ⊃ × ⊏^{2(3)}S_{3}]_{4}⊗ \sqrt{-3} _{∞,2}[SL_{2}(3)]_{1},

∞,2,3,5[D_{10} ⊃ × ⊏^{2(2)}(C_{3} □^{2} SL_{2}(3))]_{8}, _{∞,5}[D_{10} ⊃ × ⊏^{2(3)}(C_{3} □^{2} SL_{2}(3))]_{8},

∞,2[D_{10} ⊃ × ⊏^{2(6)}(C_{3} □^{2} SL_{2}(3))]_{8}, _{∞,2,3,5}[D_{10} ⊃ × ⊏^{2(3)}(C_{3} ⊃ × ⊏^{2(2)}D_{8})]_{8},

or _{∞,3}[D_{10} ⊃ × ⊏^{2(6)}(C_{3} ⊃ × ⊏^{2(2)}D_{8})]_{8}.

Proof. As in the previous lemma O_{5}(G) = C_{5} and C := C_{G}(O_{5}(G)) is a centrally irreducible subgroup of GL_{4}(ℚ[ζ_{5}]). One finds that C is of the form C_{5} × H where H does not admit an outer automorphism of order 4. Hence G contains a normal subgroup Q_{20}H or D_{10}H of index 2. In the first case, one has the same candidates for H as in the proof of the Lemma 18.13 b) (ii). In all four cases the enveloping ℚalgebra of H is central simple and Glide(H) does not contain an element of norm 5. One concludes that G is not maximal but contained in one of the groups of

Lemma 18.13. In the second case H is one of SL_{2}(3) ⊗ S_{3}, SL_{2}(3) □^{2} C_{3}, D_{8} ⊗ S̃_{3},

or C_{3} ⊃ × ⊏^{2(2)} D_{8}. As in part b) (ii) of Lemma 18.13 Out(H) ≅ C_{2} × C_{2} in all cases. Since the groups C_{5} : C_{4} ⊗ H are contained in the corresponding groups A_{4} ⊗ H, one has to consider three automorphisms in each case. But now the two possible

extensions D_{10}H.2 lead to isomorphic groups. The group D_{10} ⊃ × ⊏^{2(2)} SL_{2}(3) ⊗ A_{2} is

contained in _{∞,5}[D_{10} ⊃ × ⊏_{\sqrt{5}}^{2} SL_{2}(5)]_{4} ⊗ A_{2} and D_{10} ⊃ × ⊏^{2(6)}(SL_{2}(3) ⊗ S_{3}) is contained in

∞,3[(D_{10} ⊗ S_{3}) ⊠_{\sqrt{5}}^{2(3)} SL_{2}(5)]_{8}.

Clearly D_{10} ⊃ × ⊏^{2(3)} S̃_{3} ⊗ D_{8} is imprimitive and D_{10} ⊃ × ⊏^{2(6)} (S_{3} ⊗ D_{8}) is a subgroup of

∞,2[(D_{10} ⊗ D_{8}) ⊠_{\sqrt{5}}^{2(2)} SL_{2}(5)]_{8}.

With Corollary 7.12 one gets that D_{10} ⊃ × ⊏^{2(2)} (C_{3} ⊃ × ⊏^{2(2)} D_{8}) is contained in D_{10} ⊃ × ⊏^{2(2)} 2_{−}^{1+4}.Alt_{5}.

Since the other groups are a.i.m.f. groups, one gets the Lemma.

19. The a.i.m.f. subgroups of GL_{9}(Q)

Theorem 19.1. Let Q be a definite quaternion algebra with center ℚ and G a primitive a.i.m.f. subgroup of GL_{9}(Q). Then G is conjugate to one of the groups in the following table.

List of the primitive a.i.m.f. subgroups of GL_{9}(Q)

lattice L  |Aut(L)| r.i.m.f. supergroups
∞,2[SL_{2}(19)]_{9}  2^{3} · 3^{3} · 5 · 19 [SL_{2}(19)^{2(2)} ∘ SL_{2}(3)]_{36}
∞,2[SL_{2}(3)]_{1} ⊗ A_{9}  2^{3} · 3 · 10! F_{4} ⊗ A_{9}
∞,3[^{±}3_{+}^{1+4}.Sp_{4}(3).2]_{9}  2^{9} · 3^{9} · 5 [^{±}3_{+}^{1+4}.Sp_{4}(3).2]_{18}^{2}
∞,3[^{±}3.Alt_{6}.2^{2}]_{9}  2^{6} · 3^{3} · 5 [^{±}3.Alt_{6}.2^{2}]_{18}^{2}
∞,3[S̃_{3}]_{1} ⊗ A_{9}  2^{2} · 3 · 10! (A_{2} ⊗ A_{9})^{2}
∞,7[^{±}L_{2}(7) ⊠_{\sqrt{-7}}^{2} L_{2}(7)]_{9} 2^{9} · 3^{2} · 7^{2} [^{±}L_{2}(7) ⊠_{\sqrt{-7}}^{2} L_{2}(7)]_{18}^{2}
∞,19[^{±}L_{2}(19).2]_{9}  2^{4} · 3^{3} · 5 · 19 ((A_{18}^{(5)})^{2})
(A_{18}^{(5)})^{2}

Proof. Let Q be a definite quaternion algebra with center ℚ and G be a primitive a.i.m.f. subgroup of GL_{9}(Q). Assume that 1 ≠ N ⊴ G is a quasi-semi-simple normal subgroup of G. With Table 9.1 one finds that B := B^{∘}(N) is one of SL_{2}(5), ±L_{2}(7), ^{±}3.Alt_{6}, ^{±}L_{2}(19), SL_{2}(19), ^{±}U_{3}(3), or ^{±}S_{10}. In the last case, G is one of ∞,2[SL_{2}(3)]_{1} ⊗ A_{9}, _{∞,3}[S̃_{3}]_{1} ⊗ A_{9}, by Corollary 7.6. If B = SL_{2}(19) the group B is already absolutely irreducible and one computes and concludes that G = B is ∞,2[SL_{2}(19)]_{9}.

If B = ^{±}U_{3}(3) or B = SL_{2}(5), then Q = Q_{∞,3} resp. Q_{∞,2} and the centralizer C_{G}(B) is an absolutely irreducible subgroup of GL_{3}(ℚ). One concludes that G is imprimitive in these two cases.

If B = ^{±}L_{2}(19), the centralizer C_{G}(B) is ^{±}1 and G = B.2 = _{∞,19}[^{±}L_{2}(19).2]_{9}. If B = ^{±}3.Alt_{6}, then Q = Q_{∞,3}. If 𝔐 denotes a maximal order in Q, then Z_{𝔐}(B)

contains only one isomorphism class of lattices. For L ∈ Z_{𝔐}(B) one calculates that the Hermitian automorphism group of L is G = _{∞,3}[3.Alt_{6}.2^{2}]_{9}.

In the last case, B = ^{±}L_{2}(7). The centralizer C := C_{G}(B) is an absolutely irreducible subgroup of GL_{3}(ℚ[\sqrt{-7}]). One concludes that either G is imprimitive

or C is one of ^{±}C_{7} : C_{3} or ^{±}L_{2}(7). One finds that G = _{∞,7}[^{±}L_{2}(7) ⊠_{\sqrt{-7}}^{2} L_{2}(7)]_{9} in

this case.

Now assume that G does not contain a quasi-semi-simple normal subgroup. Then the Fitting subgroup of G is a self-centralizing normal subgroup. By Table 8.7 one has the following possibilities for F it(G): ^{±}C_{19}, ^{±}C_{7}, ^{±}3_{+}^{1+2}YC_{9}, or ±3_{+}^{1+4}, because 3 does not divide the order of Out(F it(G))/B^{∘}(F it(G)) and 9 does not divide the degree of the corresponding irreducible character of F it(G) in the other cases. In the first case, G is a proper subgroup of _{∞,19}[^{±}L_{2}(19).2]_{9} by Lemma 11.2. The second and third case lead to reducible groups and in the last case, G contains B^{∘}(F it(G)) = ^{±}3_{+}^{1+4}.Sp_{4}(3) of index 2. One concludes that G = _{∞,3}[^{±}3_{+}^{1+4}.Sp_{4}(3).2]_{9}.

20. The a.i.m.f. subgroups of GL_{10}(Q)

Theorem 20.1. Let Q be a definite quaternion algebra with center ℚ and G a primitive a.i.m.f. subgroup of GL_{10}(Q). Then G is one of the groups listed in the following table:

FINITE QUATERNIONIC MATRIX GROUPS 213

List of the primitive a.i.m.f. subgroups of GL_{10}(Q).

lattice L  |Aut(L)|  r.i.m.f. supergroups
∞,2[2.U_{4}(2)]_{10}  2^{7} · 3^{4} · 5  [2.U_{4}(2)^{2(2)} ∘ SL_{2}(3)]_{40}
∞,2[SL_{2}(11)]_{5} ⊗ A_{2}  2^{4} · 3^{2} · 5 · 11 A_{2} ⊗ [SL_{2}(11)^{2(2)} ∘ SL_{2}(3)]_{20}
∞,2[SL_{2}(11) ⊗^{2(2)} D_{8}]_{10}  2^{6} · 3 · 5 · 11 [SL_{2}(11) ⊗^{2(2)} 2_{−}^{1+4}.Alt_{5}]_{40}
∞,2[^{±}U_{5}(2)]_{5} ⊗ A_{2}  2^{12} · 3^{6} · 5 · 11 A_{2} ⊗ [^{±}U_{5}(2)^{2(2)} ∘ SL_{2}(3)]_{20}
∞,2[U_{5}(2) ⊗^{2(2)} D_{8}]_{10}  2^{14} · 3^{5} · 5 · 11 [^{±}U_{5}(2) ⊗^{2(2)} 2_{−}^{1+4}.Alt_{5}]_{40}
∞,2[S_{6} ⊗^{2(2)} SL_{2}(3)]_{10}  2^{8} · 3^{3} · 5  F_{4} ⊗^{2(2)} [^{±}S_{6}]_{10}
∞,2[SL_{2}(3)]_{1} ⊗ [^{±}U_{4}(2) □^{2} C_{3}]_{10}  2^{10} · 3^{6} · 5  F_{4} ⊗ [^{±}U_{4}(2) □^{2} C_{3}]_{10}
∞,2[SL_{2}(3)]_{1} ⊗ A_{10}  2^{3} · 3 · 11!  A_{10} ⊗ F_{4}
∞,2[SL_{2}(3)]_{1} ⊗ A_{10}^{(2)}  2^{6} · 3^{2} · 5 · 11 A_{10}^{(2)} ⊗ F_{4}
∞,2[SL_{2}(3)]_{1} ⊗ A_{10}^{(3)}  2^{6} · 3^{2} · 5 · 11 A_{10}^{(3)} ⊗ F_{4}
∞,2[2_{−}^{1+4}.Alt_{5}]_{2} ⊗ A_{5}  2^{7} · 3 · 5 · 6! A_{5} ⊗ E_{8}
∞,3[C_{3} ⊃ × ⊏^{2(2)}SL_{2}(11)]_{10}  2^{4} · 3^{2} · 5 · 11 [SL_{2}(11)^{2(2)} ∘ SL_{2}(3)]_{20}^{2}
∞,3[(C_{3} ∘ U_{4}(2)) ⊃ × ⊏_{\sqrt{-3}}^{2(2)}SL_{2}(3)]_{10} 2^{10} · 3^{6} · 5 [^{±}U_{5}(2)^{2(2)} ∘ SL_{2}(3)]_{20}^{2}
∞,3[C_{3} ⊃ × ⊏^{2(2)}^{±}U_{5}(2)]_{10}  2^{12} · 3^{6} · 5 · 11 [^{±}U_{5}(2)^{2(2)} ∘ SL_{2}(3)]_{20}^{2}
∞,3[^{±}U_{4}(2) □^{2} C_{3}]_{10} 2^{8} · 3^{5} · 5 [^{±}U_{4}(2) □^{2} C_{3}]_{20}^{2}
∞,3[2.U_{4}(3).4]_{10}  2^{10} · 3^{6} · 5 · 7 [2.U_{4}(3).4^{2(3)} ∘ S̃_{3}]_{40}
∞,3[SL_{2}(19)]_{10}  2^{3} · 3^{2} · 5 · 19 [SL_{2}(19)^{2(3)} ∘ S̃_{3}]_{40}
∞,3[2.Alt_{7}]_{10a}  2^{4} · 3^{2} · 5 · 7 [2.Alt_{7}^{2(3)} ∘ S̃_{3}]_{40}
∞,3[2.Alt_{7}]_{10b}  2^{4} · 3^{2} · 5 · 7 [2.Alt_{7} ∘ S̃_{3}]_{40}
∞,3[L_{2}(11) ⊗^{2(3)} S̃_{3}]_{10}  2^{5} · 3^{2} · 5 · 11 [L_{2}(11) ⊗^{2(3)} D_{12}]_{20}^{2}
∞,3[L_{2}(11) ⊃ × ⊏^{2(3)} S̃_{3}]_{10}  2^{5} · 3^{2} · 5 · 11 [L_{2}(11) ⊃ × ⊏^{2(3)}D_{12}]_{20}^{2}
∞,3[S̃_{3}]_{1} ⊗ A_{10} 2^{2} · 3 · 11! (A_{10} ⊗ A_{2})^{2}
∞,3[S̃_{3}]_{1} ⊗ A_{10}^{(2)}  2^{5} · 3^{2} · 5 · 11 (A_{10}^{(2)} ⊗ A_{2})^{2}
∞,3[S̃_{3}]_{1} ⊗ A_{10}^{(3)}  2^{5} · 3^{2} · 5 · 11 (A_{10}^{(3)} ⊗ A_{2})^{2}
∞,3[SL_{2}(9)]_{2} ⊗ A_{5}  2^{4} · 3^{2} · 5 · 6! A_{5} ⊗ E_{8}
∞,5[^{±}U_{3}(5) : 3]_{10}  2^{5} · 3^{3} · 5^{3} · 7 [^{±}U_{3}(5) : 3 □^{2} C_{3}]_{40}
∞,5[^{±}5_{+}^{1+2} : SL_{2}(5).4]_{10} 2^{6} · 3 · 5^{4} [^{±}5_{+}^{1+2} : SL_{2}(5).2 □^{2} SL_{2}(5)]_{40}
∞,5[SL_{2}(5).2]_{2} ⊗ A_{5}  2^{4} · 3 · 5 · 6! A_{5} ⊗ E_{8}
∞,5[SL_{2}(5) : 2]_{2} ⊗ A_{5}  2^{4} · 3 · 5 · 6! A_{5} ⊗ [(SL_{2}(5) □^{2} SL_{2}(5)) : 2]_{8}
∞,7[2.S_{7}]_{10}  2^{5} · 3^{2} · 5 · 7 ((Λ^{3}A_{6})^{2})
∞,7[2.L_{3}(4).2^{2}]_{10}  2^{9} · 3^{2} · 5 · 7 [2.L_{3}(4).2^{2}]_{20}^{2}
∞,11[L_{2}(11) ⊃ × ⊏_{\sqrt{-11}}^{2(2)}SL_{2}(3)]_{10}  2^{6} · 3^{2} · 5 · 11 [U_{5}(2)^{2(2)} ∘ SL_{2}(3)]_{20}^{2}

[(L_{2}(11)⊗ \sqrt{-11} SL_{2}(3) ⊗ S_{3}).2]_{40}

∞,11[L_{2}(11) ⊃ × ⊏^{2(3)}S_{3}]_{10}  2^{5} · 3^{2} · 5 · 11 [L_{2}(11) ⊃ × ⊏^{2(3)}D_{12}]_{20}^{2}

(A_{10}^{(3)} ⊗ A_{2})^{2}

∞,11[^{±}L_{2}(11).2]_{5} ⊗ A_{2}  2^{5} · 3^{2} · 5 · 11 (A_{10}^{(3)} ⊗ A_{2})^{2}

[L_{2}(11) ⊃ × ⊏^{2(3)}D_{12}]_{20}^{2}

∞,19[SL_{2}(19).2]_{10}  2^{4} · 3^{2} · 5 · 19 (B_{40})

(B_{40})

The proof is split into lemmata. For the rest of this chapter let Q be a definite quaternion algebra with center ℚ and G be a primitive a.i.m.f. subgroup of GL_{10}(Q).

By Table 9.1 and Lemma 7.2 the possibilities for quasi-semi-simple normal subgroups N of G are SL_{2}(5), Alt_{6} (2 matrix groups), SL_{2}(9) (2 matrix groups), L_{2}(11) (3 matrix groups), SL_{2}(11) (2 matrix groups), Alt_{7}, 2.Alt_{7} (2 matrix groups), SL_{2}(19) (2 matrix groups), M_{11}, 2.L_{3}(4), U_{4}(2) (2 matrix groups), 2.U_{4}(2), 2.M_{12}, U_{3}(5), 2.M_{22}, 2.U_{4}(3), U_{5}(2), and Alt_{11}. By Corollary 7.7, N is not conjugate to one of M_{11}, 2.M_{12}, or 2.M_{22}.

Lemma 20.2. G has no normal subgroup SL_{2}(11) with character χ_{10a} + χ_{10b}.

Proof. Assume that G has a normal subgroup N conjugate to SL_{2}(11), where the restriction of the natural character of G to N is χ_{10a} + χ_{10b}. Then G contains the normal subgroup NC_{G}(N) of index ≤ 2. Since the outer automorphism of N does not interchange the two Galois conjugate characters χ_{10a} and χ_{10b}, the character field of the natural character of G is ℚ[\sqrt{3}]. Therefore G is not absolutely irreducible.

Lemma 20.3. If G contains a normal subgroup N ≅ Alt_{6} with character χ_{5}, then G is one of _{∞,2}[2_{−}^{1+4}.Alt_{5}]_{2} ⊗ A_{5}, _{∞,3}[SL_{2}(9)]_{2} ⊗ A_{5}, _{∞,5}[SL_{2}(5).2]_{2} ⊗ A_{5}, or _{∞,5}[SL_{2}(5) : 2]_{2} ⊗ A_{5}.

Proof. By Corollary 7.6 G is of the form A_{5} ⊗ H, where H ≤ GL_{2}(Q) is a primitive a.i.m.f. group. Hence by Theorem 12.1 H is one of _{∞,2}[(D_{8} ⊗ Q_{8}).Alt_{5}]_{2},

∞,2[SL_{2}(3)]_{1} ⊗ A_{2}, _{∞,3}[SL_{2}(9)]_{2}, _{∞,3}[SL_{2}(3) □^{2} C_{3}]_{2}, _{∞,5}[SL_{2}(5).2]_{2}, or

∞,5[SL_{2}(5) : 2]_{2}. If H = _{∞,2}[SL_{2}(3)]_{1} ⊗ A_{2}, then G = _{∞,2}[SL_{2}(3)]_{1} ⊗ (A_{2} ⊗ A_{5}) is

contained in _{∞,2}[SL_{2}(3)]_{1} ⊗ [^{±}U_{4}(2) □^{2} C_{3}]_{10}.

If H = _{∞,3}[SL_{2}(3) □^{2} C_{3}]_{2}, one computes that G is a proper subgroup of

∞,3[(C_{3} ∘ U_{4}(2)) ⊃ × ⊏_{\sqrt{-3}}^{2(2)}SL_{2}(3)]_{10}.

Similarly one gets the next two lemmata:

Lemma 20.4. If G contains a normal subgroup N ≅ Alt_{11} with character χ_{10}, then G is one of _{∞,2}[SL_{2}(3)]_{1} ⊗ A_{10} or _{∞,3}[S̃_{3}]_{1} ⊗ A_{10}.

Lemma 20.5. If G contains a normal subgroup N ≅ L_{2}(11) with character χ_{10a}, then G is one of _{∞,2}[SL_{2}(3)]_{1} ⊗ A_{10}^{(2)} or _{∞,3}[S̃_{3}]_{1} ⊗ A_{10}^{(2)}.

Lemma 20.6. If G contains a normal subgroup N ≅ SL_{2}(9) with character χ_{4}, then G is conjugate to _{∞,3}[SL_{2}(9)]_{2} ⊗ A_{5}.

Proof. By 2.18 one has Q = Q_{∞,3} and C := C_{G}(N) embeds into GL_{5}(ℚ). Since G contains NC of index ≤ 2, C is an absolutely irreducible subgroup of GL_{5}(ℚ). Therefore C^{(∞)} is one of Alt_{5} or Alt_{6}. Since in the first case C ⋬ B^{∘}(C), one has C^{(∞)} = Alt_{6} and the lemma follows from 20.3.

Similarly one gets

Lemma 20.7. If G contains a normal subgroup N ≅ SL_{2}(5) with character χ_{2a} + χ_{2b}, then G is conjugate to one of _{∞,5}[SL_{2}(5) : 2]_{2} ⊗ A_{5} or _{∞,5}[SL_{2}(5).2]_{2} ⊗ A_{5}.

Proof. Now C := C_{G}(N) is a centrally irreducible subgroup of GL_{5}(ℚ[\sqrt{5}]). Again C^{(∞)} is one of Alt_{5} or Alt_{6} and the Lemma follows from 20.3.

The next lemma deals with the absolutely irreducible candidates for normal subgroups N.

Lemma 20.8. If G contains a normal subgroup N isomorphic to 2.Alt_{7} with character χ_{20a}, 2.Alt_{7} with character χ_{20b}, SL_{2}(19) with character χ_{20}, 2.U_{4}(2) with character χ_{20}, U_{3}(5) with character χ_{20}, resp. 2.U_{4}(3) with character χ_{20}, then G is conjugate to one of _{∞,3}[2.Alt_{7}]_{10a}, _{∞,3}[2.Alt_{7}]_{10b}, _{∞,3}[SL_{2}(19)]_{10}, _{∞,2}[2.U_{4}(2)]_{10}, ∞,5[^{±}U_{3}(5) : 3]_{10}, resp. _{∞,3}[2.U_{4}(3).4]_{10}.

Proof. In all cases N is already absolutely irreducible. One finds G = B^{∘}(N).

Next we treat those candidates for normal subgroups N in G, such that C_{G}(N) has to be contained in B^{∘}(N).

Lemma 20.9. G has no normal subgroup N isomorphic to SL_{2}(9) with character χ_{10a} + χ_{10b}. If G contains a normal subgroup N isomorphic to Alt_{7} with character χ_{10a} + χ_{10b}, SL_{2}(19) with character χ_{10a} + χ_{10b}, 2.L_{3}(4) with character χ_{10a} + χ_{10b}, resp. U_{4}(2) with character χ_{10a} + χ_{10b}, then G = B^{∘}(N).2 is conjugate to

∞,7[2.S_{7}]_{10}, _{∞,19}[SL_{2}(19).2]_{10}, _{∞,7}[2.L_{3}(4).2^{2}]_{10}, resp. _{∞,3}[^{±}U_{4}(2) □^{2} C_{3}]_{10}.

Proof. In all cases the centralizer C_{G}(N) embeds into the enveloping algebra of N and hence is contained in B^{∘}(N). Assume first, that N is isomorphic to SL_{2}(9). Since the character field of the extension of the character χ_{10a} to 2.P GL_{2}(9) is of degree 4 over ℚ (cf. [CCNPW 85]), G is isomorphic to 2.S_{6}. (Note that the outer automorphism of S_{6} interchanges the two isoclinism classes of groups 2.S_{6}, so there is only one group to be considered.) But then G is not maximal finite, since it is contained in the a.i.m.f. group _{∞,2}[2.U_{4}(2)]_{10}.

In all the other cases Glide(N) = 1 and there is an automorphism of N inducing the Galois automorphism of the character field ℚ[χ]. Since ℚ[χ] is an imaginary quadratic number field, Remark (I.13) of [Neb 96] implies that there is a unique extension G = B^{∘}(N).2 with real Schur index 2. Computing the automorphism groups of the G-invariant lattices, one finds that in all cases G is a maximal finite subgroup of GL_{10}(Q).

Lemma 20.10. If G has a normal subgroup N ≅ Alt_{6} with character χ_{10}, then G

is conjugate to _{∞,2}[S_{6} ⊗^{2(2)} SL_{2}(3)]_{10}.

Proof. By Table 9.1 the group N is nearly tensor decomposing over ℚ with parameter 2. Let B := B^{∘}(N) = ^{±}S_{6} and C := C_{G}(N) = C_{G}(B). By 10.1 G is either G = BC where C is an a.i.m.f. subgroup of GL_{1}(Q) such that (C, 2, Q) is not a

maximal triple or of the form B ⊗^{2(2)} C or B ⊃ × ⊏^{2(2)} C where (C, 2, Q) is a maximal

triple. Table 10.2 G is one of _{∞,3}[^{±}S_{6} ⊗ S̃_{3}]_{10} or _{∞,2}[^{±}S_{6} ⊗^{2(2)} SL_{2}(3)]_{10}. The first

group is not maximal finite but contained in _{∞,3}[^{±}U_{4}(2) □^{2} C_{3}]_{10}.

Similarly one finds the next lemma, because _{∞,2}[L_{2}(11) ⊗ SL_{2}(3)]_{10} is contained in A_{10} ⊗_{∞,2}[SL_{2}(3)]_{1}.

Lemma 20.11. If G has a normal subgroup N ≅ L_{2}(11) with character χ_{10}, then

G is conjugate to _{∞,3}[L_{2}(11) ⊗^{2(3)} S̃_{3}]_{10}.

The next two lemmata deal with similar situations, where now the enveloping algebra of N is a matrix ring over a definite quaternion algebra over ℚ.

Lemma 20.12. If G has a normal subgroup N ≅ SL_{2}(11) with character 2χ_{10},

then G is conjugate to one of _{∞,2}[SL_{2}(11)]_{5} ⊗ A_{2}, _{∞,2}[SL_{2}(11) ⊗^{2(2)} D_{8}]_{10}, or

∞,3[C_{3} ⊃ × ⊏^{2(2)}SL_{2}(11)]_{10}.

Proof. By Table 9.1 the group N is nearly tensor decomposing over ℚ with parameter 2. By 10.1 G is either G = NC where C := C_{G}(N) is an a.i.m.f. subgroup of

GL_{1}(D) such that (C, 2, D) is not a maximal triple or of the form N ⊗^{2(2)}C or N ⊃ × ⊏^{2(2)}C where (C, 2, D) is a maximal triple and D is an indefinite quaternion algebra over ℚ. By Table 10.2 G is one of the three groups in the lemma.

Completely analogously one gets

Lemma 20.13. If G has a normal subgroup N ≅ U_{5}(2) with character 2χ_{10}, then

G is conjugate to one of _{∞,2}[^{±}U_{5}(2)]_{5} ⊗ A_{2}, _{∞,2}[^{±}U_{5}(2) ⊗^{2(2)} D_{8}]_{10}, or

∞,3[C_{3} ⊃ × ⊏^{2(2)}^{±}U_{5}(2)]_{10}.

In the last two cases, the center of the enveloping algebra of N is an imaginary quadratic field. Hence here the situation is not so tight.

Lemma 20.14. If G has a normal subgroup N ≅ L_{2}(11) with character χ_{5a} + χ_{5b},

then G is conjugate to one of _{∞,2}[SL_{2}(3)]_{1} ⊗ A_{10}^{(3)}, _{∞,3}[L_{2}(11) ⊃ × ⊏^{2(3)} S̃_{3}]_{10}, _{∞,3}[S̃_{3}]_{1} ⊗

A_{10}^{(3)}, _{∞,11}[L_{2}(11) ⊃ × ⊏_{\sqrt{-11}}^{2(2)}SL_{2}(3)]_{10}, _{∞,11}[L_{2}(11) ⊃ × ⊏^{2(3)}S_{3}]_{10}, or _{∞,11}[^{±}L_{2}(11).2]_{5} ⊗ A_{2}.

Proof. The centralizer C := C_{G}(N) is a centrally irreducible subgroup of GL_{1}(D) where D is a quaternion algebra over ℚ[\sqrt{-11}] and G contains NC of index 2. Using the classification of finite subgroups of GL_{2}(ℂ) in [Bli 17], one finds that C is one of D_{8}, ^{±}S_{3}, SL_{2}(3), or S̃_{3}. Distinguish 2 cases:

a) C_{G}(C) > ^{±}N. Then G = (^{±}N.2)C is one of _{∞,11}[^{±}L_{2}(11).2]_{5} ⊗ [D_{8}]_{2}, ∞,11[^{±}L_{2}(11).2]_{5} ⊗ A_{2}, _{∞,2}[SL_{2}(3)]_{1} ⊗ A_{10}^{(3)}, or _{∞,3}[S̃_{3}]_{1} ⊗ A_{10}^{(3)}. In the first case, G is imprimitive and contained in _{∞,11}[^{±}L_{2}(11).2]_{5}^{2}.

b) C_{G}(C) = ^{±}N. The groups C are nearly tensor decomposing over ℚ with parameter 2, 3, 2, resp. 3. Since C = C_{G}(N) Lemma 10.1 implies that G is

one of _{∞,2}[L_{2}(11) ⊃ × ⊏^{2(2)} D_{8}]_{10}, _{∞,11}[L_{2}(11) ⊃ × ⊏_{\sqrt{-11}}^{2(2)} SL_{2}(3)]_{10}, _{∞,11}[L_{2}(11) ⊃ × ⊏^{2(3)} S_{3}]_{10}, or

∞,3[L_{2}(11) ⊃ × ⊏^{2(3)} S̃_{3}]_{10}. Note that 3 is decomposed and 2 is inert in ℚ[\sqrt{-11}]. The first

group is not maximal finite but contained in _{∞,2}[^{±}U_{5}(2) ⊗^{2(2)} D_{8}]_{10}.

Lemma 20.15. If G has a normal subgroup N ≅ U_{4}(2) with character χ_{5a} + χ_{5b},

then G is conjugate to _{∞,2}[SL_{2}(3)]_{1} ⊗ [^{±}U_{4}(2) □^{2} C_{3}]_{10} or _{∞,3}[(C_{3} ∘ U_{4}(2)) ⊃ × ⊏_{\sqrt{-3}}^{2(2)}

SL_{2}(3)]_{10}.

Proof. Let B := B^{∘}(N) ≅ ^{±}C_{3} ∘ N. The centralizer C := C_{G}(N) is a centrally irreducible subgroup of GL_{1}(D) where D is a quaternion algebra over ℚ[\sqrt{-3}] and G contains BC of index 2. As in the last lemma C is one of D_{8}, ^{±}S_{3}, SL_{2}(3), or S̃_{3}.

But now O_{3}(B) ≅ C_{3} and therefore all three groups _{∞,3}[S̃_{3}]_{1} ⊗ [^{±}U_{4}(2) □^{2} C_{3}]_{10},

A_{2} ⊗_{∞,3}[^{±}U_{4}(2) □^{2} C_{3}]_{5}, and D_{8} ⊗_{∞,3}[^{±}U_{4}(2) □^{2} C_{3}]_{5} are imprimitive and one only finds one a.i.m.f. group in the case C_{G}(C) > B. In the case C_{G}(C) = B one again uses 10.1 to deduce that G is one of

∞,2[^{±}U_{4}(2) ∘ C_{3} ⊃ × ⊏^{2(2)}D_{8}]_{10}, _{∞,3}[^{±}U_{4}(2) ∘ C_{3} ⊃ × ⊏_{\sqrt{-3}}^{2(2)}SL_{2}(3)]_{10},

∞,3[^{±}U_{4}(2) ∘ C_{3} ⊃ × ⊏^{2(3)}S_{3}]_{10} or _{∞,3}[^{±}U_{4}(2) ∘ C_{3} ⊃ × ⊏_{\sqrt{-3}}^{2(3)} S̃_{3}]_{10}.

The first group is not maximal finite but contained in _{∞,2}[^{±}U_{5}(2) ⊗^{2(2)} D_{8}]_{10} and the last two groups are imprimitive.

For the rest of the proof of 20.1 we assume that G contains no quasi-semi-simple normal subgroup. Then the Fitting subgroup F it(G) := ∏_{p||G|} O_{p}(G) is a selfcentralizing normal subgroup of G, and hence an irreducible subgroup of GL_{10}(Q) by Lemma 8.11. From Table 8.7 one gets that F it(G) is one of ^{±}C_{25}, ^{±}5_{+}^{1+2}, of ±C_{11}.

The first possibility immediately leads to a contradiction.

Lemma 20.16. If O_{5}(G) = 5_{+}^{1+2}, then G = _{∞,5}[^{±}5_{+}^{1+2} : SL_{2}(5).4]_{10}.

Proof. Then G contains the group B := B^{∘}(O_{5}(G)) = ^{±}5_{+}^{1+2} : SL_{2}(5) with G/B ≅ C_{4}(= Gal(ℚ[ζ_{5}]/ℚ)). Since the split extension B : C_{4} has real Schur index 1 one concludes G = _{∞,5}[^{±}5_{+}^{1+2} : SL_{2}(5).4]_{10}.

Lemma 20.17. O_{11}(G) ≠ C_{11}.

Proof. The centralizer C := C_{G}(O_{11}(G)) is a centrally irreducible subgroup of GL(D) where D is a quaternion algebra over ℚ[ζ_{11}]. As in 20.14 C is one of the groups D_{8}, ^{±}S_{3}, SL_{2}(3), or S̃_{3}. Since these groups have no automorphism of order 5, G contains the group ^{±}C_{11} : C_{5}C of index 2. In all cases, G is a proper subgroup of one of the groups of 20.14.

21. Appendix

Some invariants of the occurring primitive r.i.m.f. subgroups of GL_{32}(ℚ), that are not tensor products:

lattice L  det(L) min(L) |L_{min}|  |Aut(L)|

lattice sparse

[2_{+}^{1+10}.O_{10}^{+}(2)]_{32}  1  4  146880 2^{31} · 3^{5} · 5^{2} · 7 · 17 · 31 p ≠ 2
\begin{matrix}[((SL_{2}(5) ∘ SL_{2}(5)) ⊠_{\sqrt{5}}^{2} \\ (SL_{2}(5) ∘ SL_{2}(5))) : S_{4}]_{32,1} \end{matrix}  5^{16}  8  21600  2^{13} · 3^{5} · 5^{4}  +

\begin{matrix}[((SL_{2}(5) ∘ SL_{2}(5)) ⊠_{\sqrt{5}}^{2} \\ (SL_{2}(5) ∘ SL_{2}(5))) : S_{4}]_{32,2} \end{matrix}  1  4  {43200+ \atop 103680}  2^{13} · 3^{5} · 5^{4}  p ≠ 5
[4.L_{3}(4).2^{2}]_{32,1}  5^{16}  8  11520+  2^{10} · 3^{2} · 5 · 7  ℚ[\sqrt{-5}], +
10080  +
[4.L_{3}(4).2^{2}]_{32,2} 1 4 8064+ 2^{10} · 3^{2} · 5 · 7 ℚ[\sqrt{-5}]
20160+  +

2 · 23040+ 32256+ 40320

[SL_{2}(17) ⊃ ⊏^{2(3)} S̃_{3}]_{32,1} 1 4 3 · 4896+ 2^{7} · 3^{3} · 17 p ≠ 3, 17

4 · 14688

[SL_{2}(17) ⊃ ⊏^{2(3)} S̃_{3}]_{32,2}  17^{16}  12  1632  2^{7} · 3^{3} · 17  p ≠ 3
[(2.Alt_{7} ⊠_{\sqrt{-7}}^{2} 2.Alt_{7}) : 2]_{32}  7^{16}  8  5040  2^{9} · 3^{4} · 5^{2} · 7^{2}  +
[2.Alt_{7} ⊠_{\sqrt{-7}}^{2(3)} (SL_{2}(3) □^{2} C_{3})]_{32} 2^{16} · 7^{16} 12  6720  2^{8} · 3^{4} · 5 · 7  +
[(Sp_{4}(3)⊗ \sqrt{-3} Sp_{4}(3)) : 2 □^{2} C_{3}]_{32} 3^{16}  6  9600  2^{15} · 3^{9} · 5^{2}  +
[SL_{2}(5) ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+6}.O_{6}^{−}(2)]_{32}  5^{16}  8  21600  2^{16} · 3^{5} · 5^{2}  p ≠ 2
[SL_{2}(9).2 ⊃ × ⊏_{∞,2}^{2(2)}2_{−}^{1+4}.Alt_{5}]_{32} 2^{16} · 3^{16} 8  7200  2^{12} · 3^{3} · 5^{2}  +
[SL_{2}(5) ⊃ × ⊏_{∞,3}^{2(3)}(Sp_{4}(3) □^{2} C_{3})]_{32} 3^{16} · 5^{16} 12  4800  2^{11} · 3^{6} · 5^{2}  +
[SL_{2}(17)^{2(3)} ∘ S̃_{3}]_{32}  17^{4}  6  233376^{∗}  2^{7} · 3^{3} · 17  p ≠ 3
[SL_{2}(7) ⊠_{\sqrt{-7}}^{2} 2.Alt_{7}]_{32}  2^{8} · 7^{16} 12  47040  2^{8} · 3^{3} · 5 · 7^{2}  +
[SL_{2}(9) ⊗ D_{10} □^{2} SL_{2}(5)]_{32} 3^{16} · 5^{8} 8  3600  2^{8} · 3^{3} · 5^{3}  p ≠ 5
[SL_{2}(7) ⊃ × ⊏_{\sqrt{-7}}^{2(3)}(SL_{2}(3) □^{2} C_{3})]_{32} 2^{12} · 7^{16} 10  1344  2^{8} · 3^{3} · 7  p ≠ 2
[SL_{2}(7) ⊗_{∞,3}^{2(3)} (SL_{2}(3) □^{2} C_{3})]_{32} 2^{16} · 7^{8} 6  1344  2^{8} · 3^{3} · 7  p ≠ 3

∗ not decomposed into orbits under Aut(L)

lattice L det(L) min(L) |L_{min}| |Aut(L)|

lattice sparse

[(SL_{2}(3) ∘ C_{4}).2 ⊃ × ⊏_{\sqrt{-1}}^{2(3)}SL_{2}(7)]_{32}  2^{16} · 3^{16} · 7^{8} 12  672+ 2^{9} · 3^{2} · 7  +

1344

[SL_{2}(7) ⊗_{∞,3}^{2(3)} SL_{2}(9)]_{32}  7^{8}  6  6720+ 2^{8} · 3^{3} · 5 · 7  p ≠ 3

20160

[SL_{2}(9) □^{2} SL_{2}(5)]_{32}  2^{8} · 5^{8}  6  4800 2^{7} · 3^{3} · 5^{2}  p ≠ 5
[((SL_{2}(5) ∘ SL_{2}(5)) : 2 ⊃ × ⊏_{\sqrt{5}}^{2(6)}(C_{3} ⊃ × ⊏^{2(2)}D_{8})]_{32,1} 2^{16} · 3^{16}  8  1440+ 2^{10} · 3^{3} · 5^{2}  p ≠ 5

2880

[((SL_{2}(5) ∘ SL_{2}(5)) : 2 ⊃ × ⊏_{\sqrt{5}}^{2(6)}(C_{3} ⊃ × ⊏^{2(2)}D_{8})]_{32,2} 2^{16} · 3^{16} · 5^{16} 16  1440 2^{10} · 3^{3} · 5^{2}  +
[C_{15} : C_{4} ⊃ × ⊏^{2(2)}F_{4}]_{32}  3^{16} · 5^{8}  8  1440+ 2^{10} · 3^{3} · 5  p ≠ 2, 5

2160

[(2_{−}^{1+4}.Alt_{5}⊗_{∞,2} SL_{2}(5)) ⊠_{\sqrt{5}}^{2(2)} D_{10}]_{32}  2^{16} · 5^{8}  8  9600+ 2^{11} · 3^{2} · 5^{3}  p ≠ 5

12000+ 14400

[(SL_{2}(5)⊗ \sqrt{5} D_{10}) ⊃ × ⊏_{∞,3}^{2(3)}(SL_{2}(3) □^{2} C_{3})]_{32} 2^{16} · 3^{16} · 5^{8} 12 4800+ 2^{8} · 3^{3} · 5^{2}  p ≠ 5

5760

[SL_{2}(5) ∘ (C_{5} ⊃ × ⊏^{2}D_{24})]_{32} 11^{8} 6 3 · 1440 2^{6} · 3^{2} · 5^{2} ℚ[\sqrt{3}, \sqrt{5}]
+2400  p ≠ 5
[SL_{2}(3)^{2(2)} ∘ (C_{5} ⊃ × ⊏^{2}D_{24})]_{32} 11^{8} 6 5 · 720 2^{7} · 3^{2} · 5 ℚ[\sqrt{3}, \sqrt{5}]
+960  p ≠ 5
[SL_{2}(5) ∘ (C_{5} ⊃ × ⊏_{\sqrt{5}′}^{2} Q_{24})]_{32}  5^{8} · 11^{8}  8  1440 2^{6} · 3^{2} · 5^{2} ℚ[\sqrt{3}, \sqrt{5}], +
[SL_{2}(3)^{2(2)} ∘ (C_{5} ⊃ × ⊏_{\sqrt{5}′}^{2} Q_{24})]_{32}  5^{8} · 11^{8}  8  720 2^{7} · 3^{2} · 5 ℚ[\sqrt{3}, \sqrt{5}], +

The tables are organized as follows:

First a name of the rational irreducible maximal finite (r.i.m.f.) subgroup G of GL_{32}(ℚ) resp. of a invariant lattice L of minimal determinant is given (cf. Section 5). The next columns indicate the abelian invariants of the discriminant group L^{#}/L of L, the minimum of the square lengths of the nonzero vectors in L and the number of these minimal vectors decomposed into orbit lengths under G. The fourth column gives the order of G and the last column allows one to deduce some information on the lattice of G-invariant lattices. A + in this column indicates that G is lattice sparse, that is that all invariant lattices are obtained from L by multiplying with invertible elements in the commuting algebra of G (which is ℚ except for the groups 4 and 5 and the last four groups), taking duals with respect to positive definite invariant quadratic forms (which are unique up to scalar multiples except for the last four groups), and taking intersections and sums. If G is not lattice sparse the primes p are indicated such that all G-sublattices of L of p-power index can be obtained by a combination of the four operations above.

The next two tables are built up similarly.

Theorem 21.1. The groups in this table are maximal finite subgroups of GL_{32}(ℚ).

Proof. For all groups except the last four groups G the theorem follows easily by showing that G is the automorphism group of all its invariant lattices. Only the last four groups are not uniform. Let H be a r.i.m.f. supergroup of one of these groups G. As for the other groups one easily shows that the space of invariant quadratic forms of H is a proper subspace of F(G). Therefore C_{ℚ^{32×32}}(H) is isomorphic to one of the proper subfields of ℚ[\sqrt{3}, \sqrt{5}] and either H is uniform or dim(F(H)) = 2 and H satisfies the conditions of [NeP 95, Theorem (II.4)]. Thus there is (F, L) ∈ F_{>0}(H) × Z(H) such that F is integral on L and the prime divisors of the determinant det(F, L) of a Gram matrix of F on L divide the group order |H|. By the formula in [Schu 05] the largest prime which may divide the order of H is 31. Since the determinants of the integral positive definite lattices (F, L) ∈ F_{>0}(G) × Z(G) which involve only prime divisors ≤ 31 are divisible by 11, one concludes that 11 divides |H|. Moreover H is primitive, because G is primitive. Since the possible normal p-subgroups of H do not admit an automorphism of order 11 it follows that H has a quasi-semi-simple normal subgroup. Let (F, L) ∈ F_{>0}(H) × Z(H) be an H-invariant integral lattice of minimal determinant. The 11-modular representation δ : G → GL_{8}(11) obtained from the action of G on L^{#}/L is faithful because 11 does not divide the order of G and extends to a representation of H. So H has an image H̄ with G ≤ H̄ ≤ GL_{8}(11). Then the determination of the minimal degrees of a projective representation of a finite Chevalley group in nondefining characteristic in [LaS 74] resp. [SeZ 93] show that the simple composition factors of H are contained in [CCNPW 85]. One now gets the result from the classification of the nonabelian finite simple groups and [CCNPW 85].

Only one primitive r.i.m.f. group of GL_{36}(ℚ) whose lattices are not tensor products turns up:

lattice L det(L) min(L) |L_{min}| |Aut(L)|

lattice sparse

[SL_{2}(19)^{2(2)} ∘ SL_{2}(3)]_{36} 2^{18} · 19^{8}  10  4104 2^{6} · 3^{3} · 5 · 19  +

Some invariants of the occurring primitive r.i.m.f. groups of dimension 40, that are not tensor products:

lattice L  det(L) min(L) |L_{min}|  |Aut(L)|

lattice sparse

[^{±}U_{3}(5) : 3 □^{2} C_{3}]_{40}  5^{20}  8  10500 2^{6} · 3^{4} · 5^{3} · 7 p ≠ 3
[^{±}5_{+}^{1+2} : SL_{2}(5).2 □^{2} SL_{2}(5)]_{40} 5^{4}  4  3600  2^{8} · 3^{2} · 5^{5} p ≠ 5
[SL_{2}(11) ⊗_{∞,2}^{2(2)} 2_{−}^{1+4}.Alt_{5}]_{40}  11^{8}  6  13200 2^{10} · 3^{2} · 5^{2} · 11 p ≠ 2
[U_{5}(2) ⊗_{∞,2}^{2(2)} 2_{−}^{1+4}.Alt_{5}]_{40}  1  4  39600 2^{18} · 3^{6} · 5^{2} · 11 p ≠ 2

[2.U_{4}(2)^{2(2)} ∘ SL_{2}(3)]_{40}  2^{8} · 3^{8} 6  960+ 11520+ 12960+ 17280+ 25920  2^{10} · 3^{5} · 5 p ≠ 2

[SL_{2}(11) ⊃ ⊏^{2(3)}C_{12}.C_{2}]_{40} 11^{8} · 2^{20} 8  1320  2^{6} · 3^{2} · 5 · 11 p ≠ 11

[SL_{2}(11) ⊃ ⊏^{2(2)}SL_{2}(3)]_{40}  11^{8}  6  \begin{matrix}2 · 1320 \\ +3960 \\ +5280 \\ +7920 \end{matrix} 2^{6} · 3^{2} · 5 · 11 p ≠ 2, 11

[SL_{2}(19)^{2(3)} ∘ S̃_{3}]_{40}  3^{20} · 19^{8} 10  4104  2^{5} · 3^{3} · 5 · 19 +
[2.Alt_{7}^{2(3)} ∘ S̃_{3}]_{40}  3^{8} · 7^{8} 6  {5040 \atop +3360}  2^{6} · 3^{3} · 5 · 7 p ≠ 3
[2.Alt_{7} ∘ S̃_{3}]_{40}  2^{12} · 3^{8} 6  \begin{matrix}3 · 1680 \\ +10080 \end{matrix} 2^{5} · 3^{3} · 5 · 7 p ≠ 2, 3
[2.U_{4}(3).4^{2(3)} ∘ S̃_{3}]_{40}  3^{20}  6  3360  2^{12} · 3^{7} · 5 · 7 +
F_{4} ⊗^{2(2)} [^{±}S_{6}]_{10}  2^{16} · 3^{16} 6  {960 \atop +1440}  2^{12} · 3^{4} · 5 p ≠ 2

[(L_{2}(11)⊗ \sqrt{-11} SL_{2}(3) ⊗ S_{3}).2]_{40} 2^{20}  6 \begin{matrix}2 · 2640 \\ +15840 \\ +2 · 31680 \end{matrix} 2^{7} · 3^{3} · 5 · 11 p ≠ 3, 11

[2.M_{12}.2⊗ \sqrt{-2} GL_{2}(3)]_{40}  2^{20}  6  {21120 \atop +63360} 2^{11} · 3^{4} · 5 · 11 ℚ[\sqrt{-2}]

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