REDUCTION (Mod q) OF FUSION SYSTEM AMALGAMS Geoffrey R. Robinson, Department of Mathematics, University of Aberdeen, Aberdeen AB24 3UE, Scotland, g.r.robinson@abdn.ac.uk (Revision, May 2009)

INTRODUCTION: This paper originated in the (unexpected) coalescence of parts of our recent work [6] on realising fusion systems on finite p-groups via amalgams with generalisations of some of the representation-theoretic methods of [5]. In [5], we studied a particular subgroup G_{4,8} of SO_{3}(ℝ) generated by pair of rotations

x = ( \begin{matrix}0 & 1 & 0 \\ −1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix} )

and

y = ( \begin{matrix}1 & 0 & 0 \\ 0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ 0 & \frac{−1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{matrix} )

of respective orders 4 and 8. Our original motivation for that work came from consideration of complex linear groups. A well-known theorem of Blichfeldt asserts that if G is a finite primitive complex linear group, and g ∈ G has an eigenvalue λ such that all eigenvalues of g lie within \frac{π}{3} of λ on the unit circle S^{1}, then g ∈ Z(G).

The group G_{4,8} is easily seen to be (irreducible and) primitive, but the rotation y is not central, yet has eigenvalues 1, e^{\frac{πi}{4}} , e^{\frac{−πi}{4}} . Hence G_{4,8} is infinite. On the other hand, G_{4,8} is quite close to the borderline of Blichfeldt’s result.

In fact, the group G_{4,8} had already been considered, as one of an infinite family of subgroups of SO_{3}(ℝ), each generated by a pair of rotations of finite order, by Radin and Sadun in [4], groups which they explicitly identified as amalgams. In particular, as proved in [4], the group G_{4,8} is isomorphic to the amalgam D_{16} ∗_{D_{8}} S_{4}, where D_{16} is a dihedral group of order 16. By the results of [6], such an amalgam realises (via its conjugation action on its 2-subgroups), the fusion system of any finite group X which has a dihedral Sylow 2-subgroup of order 16 and which has a normal subgroup of index 2, but no normal 2complement (for example, the group X = PGL(2, 7) has these properties).

Much of the paper [5] exploited the fact that since the entries of the given generators of G_{4,8} have entries which are (at worst) half an algebraic integer, the group G_{4,8} admits “reduction (mod p)” for all odd primes p (strictly speaking, we need to adjoin \sqrt{2} to the relevant ground ring if 2 is not a quadratic residue (mod p)). This leads to finite almost simple epimorphic image of G_{4,8} for each odd prime p (one of PSL(2, p), PGL(2, p) or PGL(2, p^{2}), depending on the congruence (mod 16) of p ). These epimorphic images always have dihedral Sylow 2-subgroups, but the order of their Sylow 2-subgroups can be arbitrarily large with a suitable choice of p.

In fact, there is a “global” explanation for these epimorphic images. In private communications to the author (2006 and 2007), J-P. Serre proved (by considering action on an appropriate tree) that (in the given 3-dimensional representation), the group G_{4,8} = SO_{3}(ℤ[\frac{1}{\sqrt{2}}]). The given representation of G_{4,8} clearly gives G_{4,8} ⊆ SO_{3}(ℤ[\frac{1}{\sqrt{2}}]). The rightmost group also admits reductions (mod p) for odd primes p (after adjoining \sqrt{2} to the field of p elements for p ≡ ±3 ( mod 8)), yielding a homomorphism into SO_{3}(p) or SO_{3}(p^{2}). One interpretation of the results of [5] is that the image of both groups under reduction (mod p) is the same for each odd prime p. Serre’s result may be interpreted as saying that there is a single global isomorphism yielding all the local isomorphisms.

Serre also proved that this global phenomenon persists in the sense that G_{4,16} = SO_{3}(ℤ[cos(\frac{π}{8} )]). However, for n > 4, Serre proved (by an argument involving Tamagawa numbers) that G_{4,2^{n}} has infinite index in SO_{3}(ℤ[cos(\frac{π}{2^{n−1}} )]), so that although, for each odd prime p, the respective images of G_{4,2^{n}} and SO_{3}(ℤ[cos(\frac{π}{2^{n−1}} )]) after reduction (mod p) are the same, the groups themselves are not isomorphic.

There are certain general principles which are illustrated by these examples. We will see in this paper that the amalgams associated to fusion systems on a finite p-group P, constructed in [6], admit certain finite dimensional complex representations with free kernels, and that these representations may be chosen to be realised over cyclotomic number fields. The representations need not be realisable over the ring of integers of the given cyclotomic fields. However, we will see that that whenever we take a rational prime q ≠ p, we may localise at a prime ideal (of the appropriate ring of integers) containing q and assume that the representation is realised over that local ring. Then we may reduce modulo the unique maximal ideal of the local ring to obtain a finite dimensional representation of our group (associated to a fusion system) over a finite field. Under suitable circumstances, the epimorphic image obtained is close to simple, and the kernel of the representation is a free normal subgroup.

An alternative strategy for reduction is to note that representations realised over a cyclotomic fields are “almost” realised over the ring of integers of that field, in the sense that it is only necessary to invert a finite number of primes, as we are dealing with finitely generated groups. Hence we obtain a single representation which admits reduction (mod π) for all but a finite number of prime ideals π (the case of G_{4,8} being a case when only a prime ideal containing 2 needs to be inverted). One advantage of this approach over the approach of working with one prime at a time is that in the latter approach, the representations which admit respective reductions (mod q) and (mod r) for different primes q and r need not be equivalent as complex representations.

In passing, we will make some remarks about permutation representations and representations with other integrality properties.

We will freely draw (sometimes without explicit reference) on results from Serre’s book [7] for basic properties of amalgams established there.

SECTION 1: REVIEW OF ALPERIN FUSION SYSTEMS

We retain the notation and terminology of [6] (and its references) for general statements and results about fusion systems.

As in [6], we refer to a fusion system F on a non-trivial finite p-group P as an Alperin fusion system if there is a finite set of finite groups L_{1}, L_{2}, …, L_{n} such that:

i) Each L_{i} has a Sylow p-subgroup P_{i} isomorphic to (and identified with) a subgroup of P and, furthermore, P_{1} = P. ii) U_{i} = F^{∗}(L_{i}) = O_{p}(L_{i}) and L_{i}/U_{i} ≅ Out_{F}(U_{i}) = Mor_{F}(U_{i}, U_{i})/Inn(U_{i}) for each i. iii) F is generated by F_{1}, F_{2}, …, F_{n}, where F_{i} is the fusion system on P_{i} induced by the conjugation action of L_{i} on its p-subgroups.

To avoid obvious redundancy, we assume from now on that there are no inclusions between any of the groups L_{i}.

Any saturated fusion system on a finite p-group is an Alperin fusion system, using the results of Broto,Castellana, Grodal, Levi and Oliver [1]. However, an Alperin fusion system on a finite p-group need not be saturated, as noted explicitly in (Leary and Stancu,[3]).

Given an Alperin fusion system F, we showed in [6] that the conjugation action on its p-subgroups of the iterated amalgam

X = X_{F} = (… ((L_{1} ∗_{P_{2}} L_{2})∗_{P_{3}}) …) ∗_{P_{n}} L_{n}

realises the fusion system F on P. We also recall from that paper that the amalgam X_{F} always has free normal subgroups of finite index. If K is any such free normal subgroup of X, then the fusion system associated to the conjugation action of X/K on its p-subgroups contains the fusion system F, since each L_{i} embeds isomorphically in X/K. We note also that since X is generated by the L_{i}, O_{p}(X) (the largest normal p-subgroup of X), is the largest subgroup of P which is normal in each L_{i}, since the hypotheses readily imply that O_{p}(X) ⊆ U_{i} for 1 ≤ i ≤ n. A subgroup Q of P with is strongly closed in P with respect to F if and only if Q^{x} ∩ P ≤ Q for each x ∈ X.

Since the notion of a simple saturated fusion system is taken, we find it convenient here to to coin the term episimple Alperin fusion system. The Alperin fusion system F is said to be episimple if P is not normal in each L_{i}, no proper non-trivial subgroup of P is strongly closed with in P with respect to F and F is generated by the fusion systems induced by the conjugation actions of O^{p′}(L_{i}) on P_{i}. Clearly, if there is episimple fusion system on a finite p-group P, then P is non-Abelian.

Then by (slight variants of) the results of [6], an episimple Alperin fusion system F on P is realised (by conjugation action on p-subgroups) within the iterated amalgam

Y = (… ((O^{p′}(L_{1}) ∗_{P_{2}} O^{p′}(L_{2}))∗_{P_{3}}) …) ∗_{P_{n}} O^{p′}(L_{n})).

Unless otherwise stated, we suppose from now on that n > 1, and that there are no inclusions among the O^{p′}(L_{i}). We also remark that each proper normal subgroup of Y is free under these hypotheses. For if P ∩ N ≠ 1 for some N ⊲ Y, then P ∩ N is strongly closed in P with respect to F, so that P ∩ N = P by hypothesis. Then O^{p′}(L_{i}) = ⟨P_{i}^{L_{i}}⟩ ≤ N for each i, so Y ≤ N. Hence a maximal free normal subgroup of Y is in fact a maximal normal subgroup. We note also that Y is perfect, for otherwise [Y, Y] is free, so that P ∩ [Y, Y] = 1, and P is Abelian, a contradiction.

LEMMA 1.1: If F is an episimple Alperin fusion system, and Y is defined as above, then every proper normal subgroup of Y is free, Y = [Y, Y] and Y/K is a finite simple group whenever K is a maximal free normal subgroup of Y of finite index.

SECTION 2: REPRESENTING ITERATED AMALGAMS

In this section, we work in somewhat more generality than is necessary for our immediate purpose, though our primary interest is still with (iterated) amalgams associated to fusion systems. Suppose that we have n (not necessarily finite) groups A_{1}, A_{2}, …, A_{n} and groups C_{2}, C_{3}, …, C_{n} such that, for 2 ≤ i ≤ n, C_{i} is identified with a subgroup of both A_{1} and A_{i} We consider the iterated amalgam

X = ((A_{1} ∗_{C_{2}} A_{2}) … A_{n−1}) ∗_{C_{n}} A_{n}.

Henceforth, we simply write

X = A_{1} ∗_{C_{2}} A_{2} ∗_{C_{3}} … ∗_{C_{n−1}} A_{n−1} ∗_{C_{n}} A_{n}.

We note that, by an inductive argument (the case n = 2 being true by virtue of the defining property of an amalgam of two groups), if T is any group, and

{σ_{i} : A_{i} → T : 1 ≤ i ≤ n}

is a set of n homomorphisms such that

Res_{C_{i}}^{A_{1}}(σ_{1}) = Res_{C_{i}}^{A_{i}}(σ_{i})

for each i > 1, then there is a unique group homomorphism σ : X → T which, for each j, restricts to σ_{j} on (the natural copy of) A_{j} in X.

This allows us to extend various kinds of representations of the groups A_{i} to representations of X. We will sometimes wish to have the freedom to define the representations σ_{i} up to equivalence only. We will still obtain existence results, but we will lose uniqueness of the extension (even up to equivalence) in general. We will see later that this lack of uniqueness can sometimes be used to our advantage.

In the case of linear representations, if we choose a ring R and a positive integer m, and only specify above that we have n homomorphisms

{τ_{i} : A_{i} → GL(m, R) : 1 ≤ i ≤ n}

such that Res_{C_{i}}^{A_{1}}(τ_{1}) is equivalent to Res_{C_{i}}^{A_{i}}(τ_{i}) for each i > 1, then for each i we may choose a representation σ_{i} from the equivalence type of τ_{i} so that Res_{C_{i}}^{A_{1}}(σ_{1}) = Res_{C_{i}}^{A_{i}}(σ_{i}) for each i > 1, and obtain a group homomorphism σ : X → GL(m, R) which restricts to σ_{i} on (the natural copy of) A_{i} in X.

It will occasionally be necessary for us to work with projective representations (in Schur’s sense). Given a ring R and an R-free R-module V of rank h, there is a homomorphism σ : X → Aut(End_{R}(V)) if and only if there are n homomorphisms

{σ_{i} : A_{i} → Aut(End_{R}(V)) : 1 ≤ i ≤ n}

such that

Res_{C_{i}}^{A_{1}}(σ_{1}) = Res_{C_{i}}^{A_{i}}(σ_{i})

for each i > 1.

Returning to genuine representations, in the particular case where the A_{i} are finite groups, and R is a field of characteristic zero, the equivalence class of the representation σ_{i} is determined by the character it affords. Hence any n characters {α_{i} : 1 ≤ i ≤ n} afforded by finite dimensional representations over R of the respective A_{i} such that Res_{C_{i}}^{A_{1}}(α_{1}) = Res_{C_{i}}^{A_{i}}(α_{i}) for each i > 1 yield an R-representation of X (and conversely, any finite dimensional representation of X over R yields n such characters). We refer to an n-tuple α = (α_{1}, …, α_{n}) of characters with this property as a compatible n-tuple of characters. The degree of α is h = α_{1}(1).

A compatible n-tuple α of complex characters is said to be irreducible if it can’t be properly decomposed as a sum of two compatible n-tuples.The ntuple α is said to be faithful if each α_{i} is faithful ( this does not, in general, guarantee that a complex representation of X which affords α is faithful in the usual sense, only that it has free kernel. For example, the free product Q_{8} ∗ S_{3} has a 2-dimensional complex representation (with image isomorphic to GL(2, 3)) which is faithful in the above sense, but its kernel is a non-trivial free group). We say that the compatible n-tuple α has finite type if it may be afforded by a complex representation of X with finite image; otherwise, we say that α is of infinite type.

In the case of permutation representations, if we choose a finite set Ω and a positive integer m, and only specify that we have n homomorphisms

{τ_{i} : A_{i} → Sym(Ω) : 1 ≤ i ≤ n}

such that Res_{C_{i}}^{A_{1}}(τ_{1}) is permutation equivalent to Res_{C_{i}}^{A_{i}}(τ_{i}) for each i > 1, then for each i we may choose a representation σ_{i} from the equivalence type of τ_{i} so that Res_{C_{i}}^{A_{1}}(σ_{1}) = Res_{C_{i}}^{A_{i}}(σ_{i}) for each i > 1, and obtain a group homomorphism σ : X → Sym(Ω) which restricts to σ_{i} on (the natural copy of) A_{i} in X.

We recall that when Ω is a finite set and C is any group, then two permutation representations σ_{1}, σ_{2} : C → Sym(Ω) are permutation equivalent if and only if kerσ_{1} = kerσ_{2} and |Ω^{Dσ_{1}}| = |Ω^{Dσ_{2}}| for every subgroup D of C containing kerσ_{1}.

A particular case is worth special mention. If each A_{i} is finite, and we have a finite set Ω such that each A_{i} acts semi-regularly on Ω, then there is an action of X on Ω in which each A_{i} acts semi-regularly. This is because, for each i > 1, the action of each C_{i} on Ω is semi-regular, so no non-identity subgroup of C_{i} fixes any point of Ω in either of the two given actions of C_{i} restricted respectively from A_{1} and A_{i} (while the identity subgroup fixes all points in either action). The kernel of this permutation representation of X is torsion free, since every element of finite order in X is conjugate to an element of some A_{i}, while each A_{i} acts semi-regularly. Since X is an iterated amalgam, the kernel of the permutation action is thus free.

Notice in this case that any set Ω on which each A_{i} acts semi-regularly has cardinality divisible by lcm{|A_{i}| : 1 ≤ i ≤ n}. On the other hand, any set Ω of cardinality lcm{|A_{i}| : 1 ≤ i ≤ n} may be endowed with a semi-regular permutation action of each A_{i}, and thus endowed with a permutation action of X with free kernel such that each A_{i} acts semi-regularly.

SECTION 3: EULER CHARACTERISTICS AND ITERATED AMALGAMS

The Euler(-Poincaré) characteristic χ(A), of a finitely generated group A is a useful invariant that (as noted by C.T.C. Wall) can be defined whenever A has a (finitely generated) free subgroup of finite index. We only need to note some formal properties (which may be found, for example, in (Serre [7])), and which we list below.

i) If A is free of rank r (the possibility r = 0 is allowed), then χ(A) = 1 − r.

ii) If A has a free subgroup B of index s, then χ(A) = \frac{χ(B)}{s}. iii) If A = B ∗_{C} D for groups B, D with a common subgroup C, then χ(A) = χ(B) + χ(D) − χ(C) whenever the three rightmost quantities are defined.

Notice that χ(A) > 0 if and only if A is finite, and χ(A) = 0 if and only if A has an infinite cyclic subgroup of finite index.

These few properties enable us to calculate the rank of certain free subgroups of X = X_{F} for an Alperin fusion system F on P, and to show that if O_{p}(X) = 1, then X/K embeds in Out(K) whenever K is a free normal subgroup of finite index of X.

We note that for our iterated amalgam X, we have

χ(X) = (∑_{i=1}^{n} \frac{1}{|L_{i}|}) − (∑_{i=2}^{n} \frac{1}{|P_{i}|}) ≤ 0

(in fact, the inequality is strict under our current assumptions. For L_{2} is not a p-group, so if P_{1} ≠ P_{2}, we have |L_{i}| ≥ 2|P_{2}| for i = 1, 2. In that case, if, furthermore, both inequalities are equalities, then p = 2, and L_{2} is a p-group, a contradiction. Hence we only need to exclude the case that P_{1} = P_{2}. If |L_{i}| = 2|P | for each i, then L_{1} = L_{2}, contrary to assumption. If L_{1} = P_{1}, then L_{1} ≤ L_{2}, contrary to assumption). Hence we have

χ(X) = \frac{d_{F}}{|P |lcm\{[L_{i} : P_{i}] : 1 ≤ i ≤ n\}}

for some negative integer d_{F}.

We remark that whenever K is a free normal subgroup of finite index of X, the index [G : K] is divisible by |P |lcm{[L_{i} : P_{i}] : 1 ≤ i ≤ n}, since each L_{i} is isomorphic to a subgroup of G/K and P = P_{1}. We set i_{K} = \frac{[X:K]}{|P |lcm\{[L_{i}:P_{i}]:1≤i≤n\}}. Then if K has rank r_{K}, we have (1 − r_{K}) = [X : K]χ(X) = i_{K}d_{F}. Hence r_{K} = 1 − i_{K}d_{F}.

THEOREM 3.1: a) Any free (not necessarily normal) subgroup of finite index of X has index divisible by |P |lcm{[L_{i} : P_{i}] : 1 ≤ i ≤ n}. Furthermore, X has a free subgroup of this minimal possible index, and any such free subgroup has rank 1 − d_{F}.

b) Suppose that O_{p}(X) = 1 (in particular, this occurs if F is episimple). Let K be a free normal subgroup of finite index of X. Then C_{X}(K) = 1 and X/K is isomorphic to a subgroup of Out(K).

PROOF: a) If F is a free subgroup of finite index of X, then each L_{i} acts semi-regularly on right cosets of F in X, for if u ∈ L and F tu = F t for some t ∈ X, then tut^{−1} is an element of finite order in the free group F, so u = 1_{X}.

On the other hand, we saw in section 2 that there is a set Ω of cardinality h = |P |lcm{[L_{j} : P_{j}] : 1 ≤ j ≤ n} which admits a permutation action of X with each L_{i} acting semi-regularly.

All point stabilizers in the permutation action must be torsion free, (and hence free), and clearly of finite index, since each element of finite order in X is conjugate to an element of some L_{i}. Thus all point stabilizers have index divisible by h. Hence the permutation action is transitive, and a point stabilizer, F say, is a free subgroup of index h. Another Euler characteristic calculation shows that F has rank 1 − d_{F}.

b) Since χ(X) < 0, we know that K is non-Abelian, so that Z(K) = 1, as K is free. Hence C_{X}(K) is isomorphic to a subgroup of X/K, and is thus finite. Since C_{X}(K)⊲X and n > 1, Theorem 2 of [6] allows us to conclude that C_{X}(K) is a finite p-group, so is contained in O_{p}(X) = 1.

SECTION 4: A FOCAL SUBGROUP THEOREM

In this section, we characterize (for a general Alperin fusion system F which is not assumed to be episimple), the focal subgroup of X = X_{F} in a manner inspired by well-known proof of D.G. Higman’s focal subgroup theorem for finite groups, a proof which makes use of Brauer’s characterization of characters.

The focal subgroup is a Sylow p-subgroup of the smallest normal subgroup of X such that the factor group is an Abelian p-group, as in the finite group case (however, the fact that such a subgroup exists needs some argument in this situation). We note that if H = X/F, where F is a free normal subgroup of finite index of X, then P is isomorphic to a p-subgroup of H, but (as we have already seen in examples) P need not be isomorphic to a Sylow p-subgroup of H in general. However, the focal subgroup theorem below shows that any Abelian p-factor group of H is an epimorphic image of P.

Let X = X_{F} be as above.

REMARK 4.1: Let N be any normal subgroup of G such that X/N is a p-group (in the usual sense that each of its elements has order a (finite) power of p). Then [X : N] is finite (in fact, of order at most |P |), and X = P N. Hence there is unique normal subgroup M on X which is minimal (that is, of maximal index) subject to X/M being a p-group.

To see these facts, notice that (since P_{i} is a Sylow p-subgroup of L_{i}), L_{i} = P_{i}(L_{i} ∩ N) for each i, so that L_{i} ≤ P N for each i. Thus X = P N, as X is generated by the L_{i}. In particular, [X : N] ≤ |P |. If A and B are two minimal choices of normal subgroup with X = P A = P B, then X/(A∩B) is also a finite p-group, so that X = P (A ∩ B) as above. Hence (A ∩ B) = A = B by the maximality of [X : A] and [X : B].

We note also that when X = P N and N ⊲ X, then P ∩ N is a Sylow psubgroup of N. For let Q be a any maximal p-subgroup of N. Then Q^{g} ≤ P for some g ∈ X, as P is (up to conjugacy) the unique maximal p-subgroup of X. Hence we may suppose that Q ≤ P, in which case Q = P ∩N by maximality of Q. Then Q⊲P. If S is any other maximal p-subgroup of N, then S^{x} ≤ P ∩N(= Q) for some x ∈ X, so S^{x} = Q by maximality of S and Q. Now x^{−1} = un for some u ∈ P, some n ∈ N, so that S = Q^{n} is an N-conjugate of Q.

THEOREM 4.2: Let P_{0} = ⟨x^{−1}x^{g} : x, x^{g} ∈ P, g ∈ X⟩. Then:

a) X has a normal subgroup H with X = P H and P ∩ H = P_{0} (so that X/H ≅ P/P_{0}). Furthermore, P ∩H is a Sylow p-subgroup of H and P ∩H = P ∩[X, X]. b) P ∩ [X, X] = ⟨P_{i} ∩ [L_{i}, L_{i}] : 1 ≤ i ≤ n⟩.

PROOF: Let P_{∞} = ⟨P_{i} ∩ [L_{i}, L_{i}] : 1 ≤ i ≤ n⟩. By the focal subgroup theorem for finite groups, P_{0} contains P_{i} ∩ [L_{i}, L_{i}] for each i, so that P_{∞} ≤ P_{0} ≤ P ∩ [X, X]. Let λ be any (complex) linear character of P/P_{∞}, inflated to a linear character of P. Then, for each i, Res_{P_{i}}^{P}(λ) extends to a linear character, say μ_{i}, of L_{i}. As X is an iterated amalgam of the L_{i}, there is a common extension of the μ_{i} to a linear character, μ say, of X. Furthermore, X/kerμ is a finite Abelian p-group.

Performing this construction for each linear character λ of P/P_{∞}, and letting H be the intersection of the kernels of all the corresponding linear characters μ of X, we find that X = HP, and that H ∩P = P_{∞} (the former equality following from Remark 4.1, and the latter because the intersection of the kernels of all such linear characters λ of P is P_{∞}). Now X/H is Abelian, so that P ∩[X, X] ≤ P ∩ H = P_{∞}. We now have

P_{∞} = P ∩ H ≤ P_{0} ≤ P ∩ [X, X] ≤ P_{∞},

so that P_{0} = P ∩[X, X] = P_{∞}. By the discussion preceding the Theorem, P ∩H is a Sylow p-subgroup of H.

EXAMPLE 4.3: As remarked in the introduction, the amalgam X = D_{16}∗_{D_{8}} S_{4} is isomorphic to SO_{3}(ℤ[\frac{1}{\sqrt{2}}]). The above focal subgroup theorem shows directly that X has a normal subgroup of index 2, since X = X_{F} where F is the Alperin fusion system on the dihedral group P of order 16 corresponding to n = 2, L_{1} = P, L_{2} = S_{4}. Then

P ∩ [X, X] = ⟨[P, P], P_{2} ∩ [L_{2}, L_{2}]⟩ ≠ P,

as [P, P] ≤ Φ(P) and P_{2} ≠ P.

SECTION 5: REDUCTION (Mod q) FOR PRIMES q ≠ p; LIFTING REPRESENTATIONS

We discuss further some themes begun in Section 2, and consider questions of integrality of representations. As in that section, we work in somewhat more generality than necessary for the fusion system amalgams. We consider n finite groups A_{1}, A_{2}, … A_{n} and n − 1 groups C_{2}, C_{3}, …, C_{n} such that C_{i} is identified with a subgroup of both A_{1} and A_{i} for each i. We again set X = A_{1} ∗_{C_{2}} A_{2} ∗_{C_{3}} A_{3} … ∗_{C_{n}} A_{n}. We set d = lcm{|A_{i}| : 1 ≤ i ≤ n}, and we let ω be a complex primitive d-th root of unity. Let q be a prime which does not divide lcm{|C_{i}| : 2 ≤ i ≤ n}. Let π be a prime ideal of ℤ[ω] which contains q, and let R = ℤ[ω]_{(π)} (the incomplete localization) - we retain the notation π for the unique maximal ideal of R. We let α = {(α_{1}, α_{2}, …, α_{n})} be a compatible n-tuple of characters of the respective A_{i}, and we let h be its degree.

THEOREM 5.1: The compatible n-tuple α may be realized by a ℚ[ω]Xmodule. If the n-tuple is faithful, then the subgroup of elements of X acting trivially on this module is free.

PROOF: The first claim follows from the discussion of Section 2, using the fact (first proved by R.Brauer) that each representation of each A_{i} is realizable over ℚ[ω]. Furthermore, the kernel of the action of X on this module is torsion free, since every element of finite order of X is conjugate to an element of some L_{i}, but the action of each L_{i} is faithful.

We now begin to address issues of integrality.

THEOREM 5.2 : i) The compatible n-tuple α may be realized by an RXmodule M. Furthermore, if α_{i} is faithful and O_{q}(A_{i}) = 1 for each i, then the kernel of the action of X on M/πM is a free normal subgroup of finite index. If the n-tuple is faithful and irreducible, and q does not divide d, then the action of X on M/πM is absolutely irreducible.

ii) Suppose that q does not divide d, and let M be an absolutely irreducible kX-module, for a finite field k of characteristic q. Then M lifts to an R-free RX-module.

PROOF: i) Since R is a principal ideal domain whose field of fractions is a splitting field for each A_{i}, each of the irreducible characters α_{i} may be afforded by an RA_{i}-module. Since q does not divide |C_{i}| for any i > 1, two R-free RC_{i}- modules (for any i) are isomorphic if and only if they afford the same character. Hence, by the discussions of Section 2, we obtain an R-free RX-module M which affords α.

We claim that if α_{i} is faithful and O_{q}(A_{i}) = 1 for each i, then K, the kernel of the action of X on M/πM, is torsion free, hence free.

Any element of finite order in K is conjugate to an element of some A_{i}, and K ∩ A_{i} ≤ O_{q}(A_{i}) = 1, so the second claim follows (that the index is finite is clear because GL(h, R/πR) is finite).

Now suppose that the n-tuple is faithful and irreducible. If q |̸d, then each A_{i} acts completely reducibly on M/πM. If the representation of X, on M/πM is not absolutely irreducible, then, since each A_{i} has order prime to q, we see that the n-tuple α is not irreducible, on lifting the (restriction to each A_{i} of the) Brauer character from characteristic q. For this properly decomposes α as a sum of compatible n-tuples, contrary to irreducibility. Part ii) follows by a similar argument.

The following three Lemmas may be well-known, but we include a proof for convenience. They are motivated by a desire to understand the interplay between integrality of representations and representability by unitary matrices.

LEMMA 5.3: Let σ be a representations of the finite group G over the ring of algebraic integers R of a number field K. Suppose that σ affords an irreducible complex character χ (of degree h, say) of G. Let M = [m_{ij}] be a matrix in GL(h, K) which normalizes Gσ. Then:

a) For each (i, j) the matrix

m_{ij} \frac{[G : Z(G)]}{χ(1)}M^{−1}

has all its entries in R.

b) If M has finite order, then

\frac{[G : Z(G)]}{χ(1)}M

has all its entries in R.

PROOF: We claim that for each (i, j) the matrix m_{ij} \frac{[G:Z(G)]}{χ(1)}M^{−1} has all entries algebraic integers, from which a) of the Lemma follows. The integrality (or otherwise) of these quantities is unaffected by replacing M by a non-zero scalar multiple (even if the scalar lies outside K).

Conjugation by M induces a permutation of the elements of Gσ. Hence some finite power of M centralizes Gσ. Thus some finite power of M is a scalar matrix. Hence we may suppose (possibly after extending K and R) that M has finite order.

Now choose a prime ideal π of R. Since M has finite order, there is some (i, j) such that the valuation ν_{π}(m_{ij}) ≤ 0. Set gτ = MgσM^{−1} for each g ∈ G (these matrices are all in GL(n, R)). Let E_{ji} be the n × n matrix whose (j, i)-entry is 1 and whose other entries are all 0.

Now we have

∑_{g∈G/Z(G)}(gσ)^{−1}E_{ji}gτ

= ∑_{g∈G/Z(G)}(gσ)^{−1}E_{ji}MgσM^{−1}

(by consideration of trace)

= \frac{[G : Z(G)]trace(E_{ji}M)}{χ(1)}M^{−1}.

Since this matrix has entries in R, since π was an arbitrary prime ideal of R, and since trace(E_{ji}M) = m_{ij}, our claim of integrality (and, in fact, the integrality of

\frac{[G : Z(G)]}{χ(1)}M^{−1}

in the case that M has finite order) is established. Part b) follows on replacing M by M^{−1}.

LEMMA 5.4: Let H be an irreducible monomial finite subgroup of GL(s, ℤ[ω]) for some root of unity ω and suppose that H ⊲T for some larger finite subgroup, T, of GL(s, ℂ). Then T consists of unitary matrices, and,

\frac{[H : Z(H)]}{s}T ⊆ M_{s}(ℤ[η]),

for some complex root of unity η.

PROOF: The given representation of T is equivalent to some unitary representation, and the equivalence may be achieved via a unitary conjugating matrix, since H is a unitary irreducible subgroup of T. Thus T itself consists of unitary matrices. It follows from Lemma 5 that T consists of matrices which have integral entries after multiplication by \frac{[H:Z(H)]}{s}. Since we know as in earlier discussions that

t\frac{[H : C_{H}(t)]trace(t^{−1})}{s} = ∑_{y∈H/C_{H}(t)}[t, y],

for each t ∈ T, we see that the entries of t lie in a cyclotomic field for each t ∈ T (notice that this issue of realizability only depends on the coset tH, and that since ∑_{h∈H} |trace(th)|^{2} = |H|, we may choose a coset representative for tH of non-zero trace).

REMARK 5.5: Let α be a faithful compatible n-tuple of infinite type of X. Let σ be an associated matrix representation of X affording α via a ℚ[ω]Xmodule, as in Theorem 4. Then, as X is finitely generated, this may be viewed as homomorphism σ : X → GL(m, ℤ[ω, \frac{1}{t}]) for some positive integer t. Notice that this single representation allows us to perform reduction (mod π) for all but a finite number of prime ideals π of ℤ[ω].

The next Lemma is obvious, but we consider it worth recording explicitly.

LEMMA 5.6: For each prime ideal π of ℤ[ω] not containing the integer td, let σ_{π} : X → GL(h, ℤ[ω]/π) denote the composition of σ with reduction (mod π), and let K_{π} denote its kernel. Then each K_{π} is a free normal subgroup of finite index. Furthermore, there are infinitely many distinct normal subgroups K_{π}, and infinitely many non-isomorphic possibilities for X/K_{π}.

PROOF: Choose a prime ideal π. Since σ_{π} has finite image, the assertion that K_{π} has finite index is clear. The fact that K_{π} is free follows from Lemma 5.2. Let q be the unique rational prime contained in π. If q does not divide [X : K_{π}], then the given characteristic q representation lifts to a complex representation of the finite group X/K_{π}, contrary to the fact that α has infinite type. Thus [X : K_{π}] is divisible by q.

As π was arbitrary, the claims of the Lemma follow.

REMARK 5.7: In the case that X = X_{F} is the amalgam of Section 1, reduction (mod q) may be performed for any prime q ≠ p. We remark that in the cases where K_{π} is a free normal subgroup, it need not be maximal.

The next result illustrates that if we can realise a faithful representation of an infinite group via unitary matrices with entries in a cyclotomic number field, then the representation can’t be realized over the ring of algebraic integers of that field.

LEMMA 5.8: Let 𝕂 be a subfield of a finite cyclotomic extension of ℚ, and let R be the ring of algebraic integers in 𝕂. Let G be an infinite subgroup of U(h, 𝕂) for some positive integer h. Then G is not conjugate within GL(h, ℂ) to any subgroup of GL(h, R).

PROOF: Let s = [𝕂 : ℚ] and set G = Gal(𝕂/ℚ). As usual, let ∥A∥ = max{∥vA∥ : ∥v∥ = 1} for each A ∈ M_{h}(ℂ), where ∥v∥ denotes the usual Euclidean norm of the vector v ∈ ℂ^{h}, and ℂ^{h} is viewed as the space of h-long complex row vectors. Then all unitary h × h matrices are in the unit ball of M_{h}(ℂ). Let B denote the direct product of s copies of M_{h}(ℂ), endowed with the (componentwise) max norm ∥(A_{1}, A_{2}, … A_{s})∥′ = max{∥A_{i}∥}. Then B is a finite-dimensional complex Banach algebra, and its closed unit ball is compact.

We note that G acts as a group of automorphisms of U(h, 𝕂). Let us label the elements of G as {τ_{1}, … τ_{s}}, according to some chosen fixed ordering. Then ϕ : M_{h}(𝕂) → B defined by Aϕ = (A^{τ_{1}}, …, A^{τ_{s}}) is an injection which maps U(h, 𝕂) into the unit ball of B. We note also that ∥Aϕ∥′ ≥ 1 whenever A = [a_{ij}] is a non-zero matrix in M_{h}(R), since for all τ ∈ G, we have ∥A^{τ} ∥ ≥ max{|a_{ij}^{τ}| : 1 ≤ i, j ≤ h}, and ∏_{τ∈G} |a_{kℓ}^{τ}| ≥ 1 whenever a_{kℓ} ≠ 0.

Now suppose that T^{−1}GT ⊆ GL(h, R) for some T ∈ GL(h, ℂ). Then, since G is unitary, hence completely reducible, T may be assumed to lie in GL(h, 𝕂), by the usual argument using elementary linear algebra.

For g ≠ h ∈ G, we have

1 ≤ ∥(T^{−1}(g − h)T)ϕ∥′ ≤ ∥(g − h)ϕ∥′∥T ϕ∥′∥(T^{−1})ϕ∥′.

Since Gϕ is infinite, and the unit ball of B is compact, this is a contradiction.

SECTION 6: MINIMAL PROJECTIVE REPRESENTATIONS

The next Lemma, which will be useful in the remainder of this section, is related to some results of (Feit-Tits [2]), but the presence of a large prime divisor allows a slightly cleaner conclusion.

LEMMA 6.1: Let G be a finite subgroup of GL(h, K) for an algebraically closed field K, and let S be a simple non-Abelian section of G. Then S is isomorphic to a subgroup of PGL(h, F) for some field F. If, furthermore, |S| is divisible by a prime q > h + 1, then we may take F = K.

PROOF: By induction, we may suppose G is irreducible, and that S ≅ G/M for some maximal normal subgroup M of G which has no section isomorphic to S. Also, by repeated Frattini arguments and use of the Schur-Zassenhaus theorem, we may suppose that M = Φ(G) (so is nilpotent), and that all prime divisors of |M| are divisors of |S|. We may further suppose that G = G′, and that M ≰ Z(G). We are done in case some covering group of S is a component (that is, quasi-simple subnormal subgroup) of G. From now on, then, we also suppose that F^{∗}(G) = Φ(G). Hence O_{r}(G) ≰ Z(G) for some prime r, and S is isomorphic to a subgroup of Out(O_{r}(G)) for any such prime r. If the given representation is (up to equivalence) induced from a representation of a proper subgroup H containing M, then S is isomorphic to a subgroup of the symmetric group S_{[G:H]} and [G : H] ≤ h, so the result is clear. Thus we may suppose that the representation is primitive, and hence that underlying KG-module is homogeneous as KM-module.

Now h is divisible by r^{w} for some integer w such that S is isomorphic to subgroup of Sp(2w, r) , and the first claim is established, since 2w ≤ r^{w}. For the second claim, notice that when |S| is divisible by a prime q > h + 1, we have q > h ≥ r^{w} ≥ r. Thus q divides r^{j} ± 1 for some j ≤ w, since |Sp(2w, r)| = r^{w^{2}} ∏_{i=1}^{w}(r^{2i} − 1). In particular, q ≤ h + 1, a contradiction.

REMARKS 6.2: For the rest of this section, we confine our discussion to the case of an episimple Alperin fusion system F on a p-group P, and we define the amalgam Y as before. Given a finite-dimensional ℂ-vector space V, and n injective homomorphisms σ_{i} : L_{i} → Aut(End_{ℂ}(V)) such that σ_{1} and σ_{i} agree on P_{i} for 2 ≤ i ≤ n, there is a unique homomomorphism σ : Y → Aut(End_{ℂ}(V)) which extends each σ_{i}, and the kernel of σ, say K, is a free normal subgroup.

In that case, V has the structure of a projective (in Schur’s sense) ℂY - module. If, in addition, V is chosen to have minimal dimension, h say, subject to this, we say that σ affords a minimal faithful projective complex representation of Y. We note that there is always such a projective representation of Y, since Y has a free subgroup of finite index, which gives rise to a complex permutation representation (of finite degree) of Y with free kernel.

If σ may be chosen so that [Y : K] is finite, we say that F has a finite complex linear minimal projective realization, and Lemma 6.1 (and/or [2]) may be applied in that situation. We suppose in what follows that Y has no finite complex linear minimal projective realization.

Slight modifications of earlier arguments show that we may choose a perfect central extension Ỹ of Y and a complex root of unity ω such that V is equivalent to an absolutely irreducible ℚ[ω]Ỹ -module (and such that, for each i, L̃_{i} has finite centre). We let K̃ ⊲ Ỹ consist of all those elements of Ỹ which act as scalars on V.

The minimality of the dimension of V shows that V is primitive when viewed as ℂỸ -module (for if Ỹ has a proper subgroup H of finite index (less than or equal to h), then Ỹ ≠ HZ(Ỹ), so we may, and do, suppose that H contains Z(Ỹ). The image of the associated permutation representation of Y acts faithfully on some complement to a (one-dimensional) trivial submodule. By minimality of the choice of dimension, the kernel of the permutation representation is not free, so contains P by episimplicity, and then is all of Y, a contradiction).

As in earlier discussions, there is an integer t such that V is afforded by a ℤ[ω, \frac{1}{t}]Ỹ -module, and we may reduce the representation (mod q) for each prime q which does not divide pt. We now consider such primes q are also greater than h+1, for convenience. For each such choice of q, this yields a normal subgroup of finite index, say K̃(q), of Ỹ . Namely, K̃(q) consists of all the elements which act trivially in the representation of Ỹ afforded by the direct sum of all composition factors of (a) reduction (mod q) of V. By unimodularity, (recall that Ỹ is perfect), the image of K̃(q) in its action on V contains only finitely many scalar matrices, each of which has order dividing h. But since q does not divide h, we see that this image of K̃(q) contains no non-identity scalar matrix, as elements of finite order in the kernel of a reduction (mod q) have order a power of q, and any element of order prime to q which acts trivially on all the composition factors of the reduced module act trivially on the reduced module itself. Now K̃(q) is isomorphic to a (normal) subgroup, K(q) say, of Y. This (proper) subgroup contains no element of order p, since F is episimple. Hence K̃(q) is free (of finite index). We note also that O_{q}(Ỹ /K̃(q)) = 1, since the leftmost group acts trivially on the direct sum of all composition factors of the (chosen) reduction (mod q) of V.

The minimality of the dimension of V, and standard Clifford theory, show that if K̃(q) does not act trivially on V, then (as it is not represented by scalars) it must act irreducibly. For otherwise, we would obtain a non-trivial projective representation of Ỹ /K̃(q) of dimension smaller than h = dim(V)). This would lead to a projective representation of Y of dimension less than h, with free kernel. On the other hand, if K̃(q) does act trivially on V, then Ỹ is finite, contrary to current assumptions.

Since Ỹ is infinite, we now know that K̃(q) acts irreducibly on V for each choice of prime q > h + 1 not dividing pt. Let M(q) be a maximal free normal subgroup of Y containing K(q), and let M̃(q) be the full pre-image of M(q) in Ỹ . We know that Y/M(q) is a non-Abelian finite simple group. Suppose that this group has order prime to q. Then (as in Lemma 5.5), we may choose a supplement W (q)/K(q) to M(q)/K(q) in Y/K(q) such that (W (q)∩M(q))/K(q) is a nilpotent q′-group (in fact, contained in the Frattini subgroup of Y/K(q)). Now Y = W (q)M(q) and Y/M(q) ≅ W (q)/(W (q) ∩ M(q)). Hence W (q)/K(q) is a q′-group. Let W̃ (q) be the full pre-image of W (q) in Ỹ .

Now \overline{W} (q) = W̃ (q)/K̃(q) is also a q′-group, and “lifts” to a finite complex linear group which has Y/M(q) as an epimorphic image. In fact, each composition factor of the reduction (mod q) of V lifts to a ℂỸ -module which yields a homomorphism from Ỹ to a finite complex linear group. At least one of these finite images has a composition factor isomorphic to Y/M(q). By the minimality of h, the reduction (mod q) of V is thus absolutely irreducible. But now F has a finite complex linear minimal projective realization, contrary to assumption. Thus the simple group Y/M(q) does have order divisible by q. Since q > h + 1, we may conclude from Lemma 6.1 that some finite central extension of Y/M(q) has an irreducible representation of degree at most h over an algebraically closed field of characteristic q.

THEOREM 6.3: Let F be an episimple Alperin fusion theorem on a p-group P. Let Y be the iterated amalgam of the O^{p′}(L_{i}) as before, and let h be the dimension of a minimal faithful complex projective representation of Y . Then one of the following occurs: a) F has no finite complex linear minimal projective realization, and for all but a finite number of primes q, Y has a finite simple epimorphic image Y (q) of order divisible by q which has a finite central extension having an absolutely irreducible representation σ of degree at most h in characteristic q.

b) Y has a finite complex linear minimal projective realization.

PROOF: We choose any prime q > h + 1 which does not divide p.t. If Y/M(q) has order divisible by q, then it has a faithful projective representation of degree at most h over an algebraically closed field of characteristic q by Lemma 5.5, and the discussion preceding the theorem.

If Y/M(q) has order prime to q, then F has a finite complex linear minimal projective realization by the discussion preceding the theorem.

REMARK 6.4: This leads us to seek cases where the minimal degree faithful complex projective representation of the fusion system F can’t be realised by a finite group. Since this representation is primitive, one method for ensuring that this is the case is to find a primitive complex representation such that some non-central elements have eigenvalues sufficiently close together on the unit circle S^{1}. In general, other ad hoc methods may be available in a particular case. We illustrate some of the earlier discussions with an example.

EXAMPLE 7.1: TWISTED AND UNTWISTED IMAGES OF A SINGLE AMALGAM

Let p = 2 and P_{1} be a semi-dihedral 2-group of order 16. Let F denote the Alperin fusion system on P_{1} with L_{1} = GL(2, 3) and L_{2} = S_{4}, where P_{2} is the dihedral subgroup of P_{1} of order 8, and is identified with a Sylow 2-subgroup of L_{2}, while P_{1} is identified with a Sylow 2-subgroup of L_{1}. Since neither S_{4} nor GL(2, 3) has a non-trivial epimorphic image of odd order, F is episimple with U_{1} quaternion of order 8 and U_{2} a Klein 4-group. We may realise F via the amalgam Y = GL(2, 3) ∗_{D_{8}} S_{4}. By Lemma 1.1, we know that Y = [Y, Y], and than any maximal free normal subgroup of finite index is a maximal normal subgroup. We will exhibit a 3-dimensional complex representation of Y which is minimal faithful projective complex representation of Y. Such a representation has kernel of infinite index, for otherwise the finite image of Y would be a perfect finite primitive complex linear group containing a subgroup H isomorphic to GL(2, 3). An element of order 6 in H would be non-central, yet would have eigenvalues 1, e^{\frac{πi}{3}} and e^{\frac{−πi}{3}} , contary to Blichfeldt’s result.

We begin by writing GL(2, 3) as a group of unitary 2 × 2 matrices with entries in ℤ[\frac{1}{\sqrt{−2}}]. Starting from the dihedral group D of order eight

⟨( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) , ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} )⟩.

We wish to adjoin a unitary matrix of order 8 which normalizes this group. Using the procedure of Lemma 5.4, for example, one such matrix is

( \begin{matrix}\frac{−1}{\sqrt{−2}} & \frac{−1}{\sqrt{−2}} \\ \frac{1}{\sqrt{−2}} & \frac{−1}{\sqrt{−2}} \end{matrix} ) .

Then the group

P = ⟨( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) , ( \begin{matrix}\frac{−1}{\sqrt{−2}} & \frac{−1}{\sqrt{−2}} \\ \frac{1}{\sqrt{−2}} & \frac{−1}{\sqrt{−2}} \end{matrix} )⟩

is semi-dihedral of order 16 (we note that the first matrix conjugates the second to its cube). This group has a unique quaternion subgroup of order 8, which is

Q = ⟨( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) , ( \begin{matrix}\frac{−1}{\sqrt{−2}} & \frac{−1}{\sqrt{−2}} \\ \frac{−1}{\sqrt{−2}} & \frac{1}{\sqrt{−2}} \end{matrix} )⟩.

We now seek a unitary matrix of order 3 which normalizes Q. Such a matrix is

( \begin{matrix}\frac{−1+\sqrt{−2}}{2} & \frac{1}{2} \\ \frac{−1}{2} & \frac{−1−\sqrt{−2}}{2} \end{matrix} ) .

Then Q is normalized by

⟨( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) , ( \begin{matrix}\frac{−1}{\sqrt{−2}} & \frac{−1}{\sqrt{−2}} \\ \frac{1}{\sqrt{−2}} & \frac{−1}{\sqrt{−2}} \end{matrix} ) , ( \begin{matrix}\frac{−1+\sqrt{−2}}{2} & \frac{1}{2} \\ \frac{−1}{2} & \frac{−1−\sqrt{−2}}{2} \end{matrix} )⟩,

which is a group of order at least 48. This gives the desired unitary representation of GL(2, 3). Another way to verify this is to notice that the product

( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) ( \begin{matrix}\frac{−1+\sqrt{−2}}{2} & \frac{1}{2} \\ \frac{−1}{2} & \frac{−1−\sqrt{−2}}{2} \end{matrix} )

is a unitary matrix of determinant −1 and trace \sqrt{−2}, so is a matrix of order

eight. Taking images modulo ⟨( \begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix} )⟩, we have an element of order 2

and an element of order 3 whose product has order 4, so we obtain a group isomorphic to S_{4}. Thus

⟨( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) , ( \begin{matrix}\frac{−1+\sqrt{−2}}{2} & \frac{1}{2} \\ \frac{−1}{2} & \frac{−1−\sqrt{−2}}{2} \end{matrix} )⟩

is isomorphic to GL(2, 3) (it can’t be the binary octahedral group, as it contains a non-central element of order 2).

Hence there is (faithful in the earlier sense that its kernel is free) representation of Y as a (primitive, absolutely irreducible) subgroup of SU(3, ℤ[\frac{1}{\sqrt{−2}}]) via

G = ⟨( \begin{matrix}1 & 0 & 0 \\ 0 & −1 & 0 \\ 0 & 0 & −1 \end{matrix} ) , ( \begin{matrix}\frac{−1+\sqrt{−2}}{2} & \frac{1}{2} & 0 \\ \frac{−1}{2} & \frac{−1−\sqrt{−2}}{2} & 0 \\ 0 & 0 & 1 \end{matrix} ) , ( \begin{matrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{matrix} )⟩.

The distinguished copy of GL(2, 3) is that generated by the first two matrices. The distinguished copy of S_{4} is

⟨( \begin{matrix}1 & 0 & 0 \\ 0 & −1 & 0 \\ 0 & 0 & −1 \end{matrix} ) , ( \begin{matrix}0 & −1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix} ) , ( \begin{matrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{matrix} )⟩

and our original dihedral group D is the intersection of these distinguished subgroups.

Now G is an infinite group by Blichfeldt’s result, since it is primitive, and the distinguished GL(2, 3) subgroup contains a non-central element of order 6 with eigenvalues 1, e^{\frac{πi}{3}} , e^{\frac{−πi}{3}} . We notice that entrywise complex conjugation of its matrices induces an automorphism of order 2 of G.

This unitary complex representation of Y may be reduced (mod q) (strictly, modulo any prime ideal of ℤ[\sqrt{−2}] containing q) for each odd prime q. The kernel (in Y) of any one of these reductions (mod q) is a free normal subgroup of Y. If q ≡ 1 ( mod 8), or q ≡ 3 (mod 8), then −2 is a quadratic residue (mod q), and the reduction is realised over GF(q). Otherwise −2 is a quadratic non-residue (mod q), the reduction as given is realised over GF(q^{2}), but (on consideration of the trace) is not realisable (even up to equivalence) over GF(q). Let Y (q) denote (the image of) this reduction (mod q). Then Y (q) is always a primitive absolutely irreducible group. We note that when −2 is not a quadratic residue (mod q), then the automorphism of Y (q) induced by complex conjugation on G is that induced by the Frobenius automorphism of GF(q^{2}). Hence, in that case, Y (q) is isomorphic to a subgroup of SU(3, q).

Since Y (q) is not equivalent to a monomial group, we see that its Fitting subgroup is either central of order dividing 3, or else is an extra-special 3-group of order 27 and exponent 3. The latter case may be excluded, since otherwise, Y (q)/F (Y (q)) is isomorphic to a subgroup of SL(2, 3), while Y (q)/F (Y (q)) has a subgroup isomorphic to GL(2, 3). Hence Y (q) has central Fitting subgroup. Since the generalized Fitting subgroup of Y (q) contains its centralizer, we see that Y (q) has a component (that is to say, quasi-simple subnormal subgroup) L(q), which acts absolutely irreducibly by Clifford’s theorem, so is unique, as distinct components centralize each other.

If L(q) is a q′-group and q > 3, then the Hall-Higman theorem forces Y (q) to be a q′-group. Otherwise, an element of order q centralizes L(q) and hence is represented by a scalar matrix of determinant 1 (using the fact that Y (q) is perfect). In that case, Y (q) is liftable to a finite subgroup of GL(3, ℂ), a possibility we excluded in the discussion at the beginning of this section.

For q = 3, a similar argument forces 3 to divide |Y (3)| in the case that L(3) is a 3′-group. The Hall-Higman theorem then forces an element x of order 3 to act with cubic minimum polynomial, since [L(3), x] is not a 2-group, so that x must act non-trivially on a Sylow r-subgroup of L(3) for some odd prime r. In that case, C_{L(3)}(x) consists of scalar matrices (which must then be trivial by unimodularity). But then L(3) is nilpotent, a contradiction.

Hence in all cases, the component L(q) has order divisible by q. Thus O_{q′}(Y (q)) is central (again of order dividing 3 on consideration of determinants). The component L(q) is not isomorphic to SU(3, 4), since SU(3, 4) has no 3-dimensional representation over any field of odd characteristic.

We note also that (by unimodularity), Y (q) contains no elementary Abelian subgroup of order 8. Also, since a Sylow 2-subgroup of Y (q) has a 3-dimensional representation over a field of odd order (which becomes monomial (up to equivalence) over some extension field), a Sylow 2-subgroup, say S(q), of Y (q) has an Abelian (normal) subgroup of index 2. Since Y (q) has no factor group of order 2 and S(q) does contain a semi-dihedral subgroup of order 16, S(q) is either semi-dihedral or wreathed. Let S_{0}(q) = S(q) ∩ L. Then either S_{0}(q) is generalized quaternion, or else S_{0}(q) contains a Klein 4-group. In either case, consideration of the representation of S_{0}(q), together with unimodularity, shows that C_{Y (q)}(S_{0}(q)) is Abelian. It follows by a Frattini argument that Y (q)/L is solvable, as Out(S_{0}(q)) is solvable in all cases. Since Y (q) is perfect, we conclude that Y (q) = L(q) is quasi-simple.

Since we have exhibited a 3-dimensional representation of Y (q) over a field of q or q^{2} elements, further analysis (such as use of characterization theorems, (either of low-dimensional primitive linear groups, or of finite simple groups with semi-dihedral or wreathed Sylow 2-subgroups) leads to the conclusion that if q ≡ 5 (mod 8) or q ≡ 7 (mod 8), then Y (q) is isomorphic to SU(3, q), while if q ≡ 1 (mod 8), or q ≡ 3 (mod 8), then Y (q) is isomorphic to SL(3, q). It is interesting to note that Y (q) has wreathed Sylow 2-subgroups if and only if q ≡ ±1 (mod 8), whereas the Sylow 2-subgroup of Y is semi-dihedral of order 16.

We note that the case Y (q) = M_{11} can’t occur because M_{11} contains a Frobenius group of order 55 and a Frobenius group of order 20, so has no faithful 3-dimensional representation over any field.

REMARK 7.2: The example above illustrates a general methodology, implicit in the results of this paper. Using the fact that SL(3, 3) is an epimorphic image of the amalgam Y, one way to view the example is that we have regarded the natural 3-dimensional representation of SL(3, 3) as a GF(3)Y -module, and lifted it to a ℂY -module. This yields a representation of Y which is realisable over ℤ[1/\sqrt{−2}], allowing reduction (mod q) for all other odd primes, yielding infinitely many non-isomorphic finite homomorphic images of Y.

The result is slightly atypical, in that the liftability of the representation of Y from characteristic 3 to characteristic 0 implicitly uses the liftabilty for S_{4} and GL(2, 3), which, despite the presence of the prime 3, follows using the Fong -Swan theorem.

The general method is to take the iterated amalgam X associated to an Alperin fusion system F, and a finite epimorphic image, say H, with a faithful (without loss of generality) representation over a field of characteristic q not dividing any |L_{i}|. The Brauer characters of the L_{i} afforded by the given representation then lift to complex characters. Furthermore, the complex characters so obtained are automatically compatible, since we started with a genuine representation of P in characteristic q ≠ p. Hence we obtain a ℂX-module, and the associated representation may be realised over a cyclotomic field. This representation may then be reduced (mod r) for all but finitely many primes r.

The condition that q does not divide |L_{i}| for any i can be relaxed somewhat, as the example and discussion above illustrated. The method can be applied as long as q ≠ p and L_{i} is q-solvable whenever q divides |L_{i}|.

REMARK 7.3: In a recent private communication to the author, J-P. Serre has now proved (using the above representation) that the group G above is isomorphic to both SU(3, ℤ[\frac{1}{\sqrt{−2}}]) and to Y = GL(2, 3) ∗_{D_{8}} S_{4}.

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