EXOTIC NORMAL FUSION SUBSYSTEMS OF GENERAL LINEAR GROUPS

ALBERT RUIZ

Abstract. We classify the saturated fusion subsystems of index prime to p of the general linear group over 𝔽_{q} over a Sylow p-subgroup, where q is a prime power prime to an odd prime p. In this classification we get some of the exotic p-local finite groups discovered by C. Broto and J. Møller as saturated fusion subsystems of the general linear group.

1. Introduction

The concept of p-local finite group arose in the work of C. Broto, R. Levi and B. Oliver [3] as a formalization of the p-local structure of a finite group. A p-local finite group consists of a triple (S, F, L) where S is a p-group, F is a category defined in an axiomatic way which models the fusion category over S, a Sylow p-subgroup of a finite group, and L is an extension of F which contains extra information so that its p-completed nerve has many of the same properties as the p-completion of the classifying space of a finite group.

One source of examples of p-local finite groups is the ones constructed from finite groups: when G is a finite group and S is a Sylow p-subgroup in G, we can construct a triple (S, F_{S}(G), L_{S}^{c}(G)) which is a p-local finite group. We can recover the p-primary information of G using the fact that in this case |BG|_{p}^{∧} ≃ |L_{S}^{c}(G)|_{p}^{∧}.

The examples which cannot be constructed from a finite group are called exotic examples. There are known exotic examples of p-local finite groups for all primes p, and these have been constructed in two different ways:

(i) Examples by C. Broto, R. Levi and B. Oliver ([3], [4]), and A. Díaz, A. Viruel and the author ([6], [10]) are constructed in a combinatorial way: they start with the saturated fusion system of a finite group and they add morphisms to the automorphism group of some proper subgroups. Also Solomon’s example by R. Levi and B. Oliver ([7]): they show that the fusion on the Sylow 2-subgroup of Spin_{7}(q) (q an odd prime power) considered by Solomon, which is known not to occur as the fusion of any finite group, has a p-local finite group structure.

(ii) Homotopy fixed point sets of p-compact groups by C. Broto and J. Møller [5].

In this paper we use the study of saturated fusion subsystems by C. Broto, N. Castellana, J. Grodal, R. Levi and B. Oliver [2] to obtain some of the classifying spaces of exotic p-local finite groups from [3] and [5] as, up to homotopy equivalence, covering spaces of the classifying spaces of a non-exotic p-local finite groups (Remark 6.4).

Key words: 2000 Mathematics subject classification 55R35, 20D20.

Partially supported by FEDER-MEC grant MTM2004-06686.

More exactly we classify the saturated fusion subsystems of index prime to p of the general linear group GL_{n}(q) for q a prime power prime to p.

Now we will describe the contents of the paper, giving simplified versions of the main results.

In Section 3 we review the classification of the saturated fusion subsystems of index prime to p from [2]. For any (S, F, L) a p-local finite group, this classification is given in terms of the F-centric subgroups of S and O_{∗}^{p′}(F), the smallest subcategory with the same objects as F and which contains all the restrictions of all the automorphisms in F of p-power order. With this data there is a bijection between saturated fusion subsystems of index prime to p in F and subgroups of

Γ_{p′}(F) =^{def} Out_{F}(S)/ Out_{F}^{0}(S) ,

where Out_{F}^{0}(S) is defined as

Out_{F}^{0}(S) =^{def} ⟨α ∈ Out_{F}(S) | α|_{P} ∈ Mor_{O_{∗}^{p′}(F)}(P, S), some F-centric P ⟩ .

If we want to compute O_{∗}^{p′}(F) in some particular cases we would like to restrict the family of subgroups of S involved in this computation to a smaller one. So in this section we give the following general result (Theorem 3.4):

Theorem A. Let F be a saturated fusion system over S. Then for each morphism ψ ∈ Hom_{O_{∗}^{p′}(F)}(P, P′), there exists a sequence of subgroups of S

P = P_{0}, P_{1}, …, P_{k} = P′ and Q_{1}, Q_{2}, …, Q_{k},

and morphisms ψ_{i} ∈ Aut_{O_{∗}^{p′}(F)}(Q_{i}), such that

• Q_{i} is fully F-normalized, F-centric and F-radical for each i;

• P_{i−1}, P_{i} ≤ Q_{i} and ψ_{i}(P_{i−1}) = P_{i} for each i; and

• ψ(u) = ψ_{k} ∘ ψ_{k−1} ∘ ⋯ ∘ ψ_{1}(u), ∀u ∈ P.

In Sections 4 and 5 we use J.L. Alperin and P. Fong results in [1] to describe the possible Fcentric, F-radical subgroups in GL_{n}(q). This description enables us to classify the saturated fusion subsystems of index prime to p in GL_{n}(q), using the following result obtained here as Theorem 5.10:

Theorem B. Let p be a prime and q a prime power prime to p. Let e be the multiplicative order of q modulo p, and n ≥ ep. Consider (S_{n,q}, F_{n,q}, L_{n,q}) the p-local finite group induced by GL_{n}(q) over S_{n,q}, a Sylow p-subgroup. Then

Γ_{p′}(F_{n,q}) ≅ ℤ/e .

In Section 6 we identify the p-local finite groups corresponding to these saturated fusion subsystems, getting that these correspond to the p-local finite groups described by C. Broto and J. Møller as the finite Chevalley version of the generalized Grassmannians, denoted by X(e, r, m)(q′), for any positive integers e, r, m such that r divides e, e divides (p − 1) and q′ a p-adic unit. More precisely the result that we get in Proposition 6.1 and Theorem 6.3 is:

Theorem C. Fix p, q, e and (S_{n,q}, F_{n,q}, L_{n,q}) as in Theorem B, and, for each r a divisor of e, let (S_{n,q}, F_{n,q,r}, L_{n,q,r}) the unique p-local finite group such that (S_{n,q}, F_{n,q,r}) is the saturated fusion subsystem of index prime to p in (S_{n,q}, F_{n,q}) such that Out_{F_{n,q}}(S_{n,q})/ Out_{F_{n,q,r}}(S_{n,q}) ≅ ℤ/r. Then:

(a) Up to homotopy equivalence, there is a fibration

|L_{n,q,r}| → |L_{n,q}| → B(ℤ/r) .

(b) |L_{n,q,r}| ≃ BX(e, r, [n/e])(q^{e}) up to p-completion, where [n/e] is the greatest integer less or equal than n/e.

Using that BX(e, r, m)(q′) are known to be exotic p-local finite groups when m ≥ p and r > 2 [5, Proposition 11.5], this result provides us examples of extensions of p-local finite groups where one of the involved elements is exotic, and the other two correspond to finite groups. These examples answer a question proposed by B. Oliver in the Banff conference on Homotopy theory and group actions (November 2005).

Remark 1.1. These results give us a new construction of the generalized p-adic Grassmannians. For all integer j ≥ 0 we have natural maps |L_{n,q^{p^{j}},r}| → |L_{n,q^{p^{j+1}},r}| that at the level of maximal tori induce the inclusion

(ℤ/p^{l+j})^{[n/e]} ≤ (ℤ/p^{l+j+1})^{[n/e]} ,

where l = ν_{p}(q^{e} − 1). The telescope construction of these maps gives us a homotopy equivalence

BX(e, r, [n/e]) ≃ hocolim_{j} |L_{n,q^{p^{j}},r}|

up to p-completion.

Acknowledgements. The author would like to thank to Carles Broto and Antonio Viruel the many helpful discussions during our stay at the Centre Interfacultaire Bernoulli, where this project started. The author is also grateful to Bob Oliver for the helping discussion when proving last results in this paper.

The author would also like to thank to the Centre Interfacultaire Bernoulli (Lausanne, Switzerland) and the Institut Mittag-Leffler (Djursholm, Sweden) for their support and hospitality while working on this paper.

The author also thanks the referee for the improvements suggested.

2. p-local finite groups

In this section we review the concept of a p-local finite group introduced in [3] that is based on a previous unpublished work of L. Puig, where the axioms for fusion systems are already established. See [4] for a survey on this subject.

If P and Q are subgroups of a group G we consider Hom_{G}(P, Q) the group morphisms from P to Q induced by conjugation of elements in G, and Inj(P, Q) are the injective group morphisms from P to Q.

Definition 2.1. A fusion system F over a finite p-group S is a category whose objects are the subgroups of S, and whose morphisms sets Hom_{F}(P, Q) satisfy the following two conditions:

(a) Hom_{S}(P, Q) ⊆ Hom_{F}(P, Q) ⊆ Inj(P, Q) for all P and Q subgroups of S.

(b) Every morphism in F factors as an isomorphism in F followed by an inclusion.

We say that two subgroups P,Q ≤ S are F-conjugate if there is an isomorphism between them in F. As all the morphisms are injective by condition (b), we write by Aut_{F}(P) the group Hom_{F}(P, P). Out_{F}(P) denotes the quotient group Aut_{F}(P)/ Aut_{P} (P).

The fusion systems that we consider are saturated, so we need the following definitions:

Definition 2.2. Let F be a fusion system over a p-group S.

• A subgroup P ≤ S is fully F-centralized if |C_{S}(P)| ≥ |C_{S}(P′)| for all P′ which is F-conjugate to P.

• A subgroup P ≤ S is fully F-normalized if |N_{S}(P)| ≥ |N_{S}(P′)| for all P′ which is F-conjugate to P.

• F is a saturated fusion system if the following two conditions hold:

(I) Every fully F-normalized subgroup P ≤ S is fully F-centralized and Aut_{S}(P) ∈

Syl_{p}(Aut_{F}(P)). (II) If P ≤ S and φ ∈ Hom_{F}(P, S) are such that φP is fully F-centralized, and if we

set

N_{φ} = {g ∈ N_{S}(P) | φc_{g}φ^{−1} ∈ Aut_{S}(φP)},

then there is \overline{φ} ∈ Hom_{F}(N_{φ}, S) such that \overline{φ}|_{P} = φ.

As expected, every finite group G gives rise to a saturated fusion system [3, Proposition 1.3], which provides valuable information about BG_{p}^{∧}. Some classical results for finite groups can be generalized to saturated fusion systems, as for example, Alperin’s fusion theorem for saturated fusion systems [3, Theorem A.10]:

Definition 2.3. Let F be any fusion system over a p-group S. A subgroup P ≤ S is: • F-centric if P and all its F-conjugates contain their S-centralizers.

• F-radical if Out_{F}(P) is p-reduced, that is, if Out_{F}(P) has no nontrivial normal psubgroups.

Theorem 2.4 ((Alperin’s fusion theorem for saturated fusion systems)). Let F be a saturated fusion system over S. Then for each morphism ψ ∈ Aut_{F}(P, P′), there exists a sequence of subgroups of S

P = P_{0}, P_{1}, …, P_{k} = P′ and Q_{1}, Q_{2}, …, Q_{k},

and morphisms ψ_{i} ∈ Aut_{F}(Q_{i}), such that • Q_{i} is fully F-normalized, F-centric and F-radical for each i;

• P_{i−1}, P_{i} ≤ Q_{i} and ψ_{i}(P_{i−1}) = P_{i} for each i; and

• ψ(u) = ψ_{k} ∘ ψ_{k−1} ∘ ⋯ ∘ ψ_{1}(u), ∀u ∈ P.

3. Saturated fusion subsystems of index prime to p

The saturated fusion subsystems of index prime to p in any saturated fusion system (S, F) are described in [2] in terms of:

Definition 3.1. • If G is a finite group, O^{p′}(G) is the smallest normal subgroup of index prime to p (equivalently the subgroup generated by elements of p-power order in G).

• A restriction-closed category over S is a category F such that Ob(F) is the set of subgroups of S, such that all morphisms in F are group monomorphisms between the subgroups, and with the following additional property: for each P′ ≤ P ≤ S and Q′ ≤ Q ≤ S, and each φ ∈ Hom_{F}(P, Q) such that φ(P′) ≤ Q′ then φ|_{P′} ∈ Hom_{F}(P′, Q′).

• If F is a saturated fusion system O_{∗}^{p′}(F) ⊂ F denotes the smallest restriction-closed subcategory of F whose morphism set contains O^{p′}(Aut_{F}(P)) for all subgroups P ≤ S.

• If F is a saturated fusion system over S we write:

Out_{F}^{0}(S) = ⟨α ∈ Out_{F}(S) | α|_{P} ∈ Mor_{O_{∗}^{p′}(F)}(P, S), some F-centric P ≤ S⟩ .

The result giving the classification of the saturated fusion subsystems of index prime to p is the following [2, Theorem 5.4 and Proposition 5.2]:

Theorem 3.2. For any saturated fusion system F over a p-group S, there is a bijective correspondence between subgroups

H ≤ Γ_{p′}(F) =^{def} Out_{F}(S)/ Out_{F}^{0}(S)

and saturated fusion subsystems F_{H} of F over S of index prime to p in F. The correspondence is given by associating to H the fusion system generated by θ̂^{−1}(B(H)), where B(H) is a category with one object and with morphism monoid the group H, and θ̂ is the unique functor

θ̂: F^{c} → B(Γ_{p′}(F))

with the following properties:

(a) θ̂(α) = α (modulo Out_{F}^{0}(S)) for all α ∈ Aut_{F}(S).

(b) θ̂(φ) = 1 if φ ∈ Mor(O_{∗}^{p′}(F)^{c}).

The rest of the section is dedicated to reduce the family of subgroups involved in the calculation of Out_{F}^{0}(S).

Consider ^{R}O_{∗}^{p′}(F) the smallest restriction-closed category of F whose morphism set contains O^{p′}(Aut_{F}(P)) for all fully F-normalized, F-centric, F-radical subgroups P ≤ S.

Lemma 3.3. Fix F is a saturated fusion system over S. Then:

(a) Aut_{F}(S) normalizes ^{R}O_{∗}^{p′}(F).

(b) F = ⟨^{R}O_{∗}^{p′}(F), Aut_{F}(S)⟩.

Proof. (a) Consider ψ ∈ Hom_{^{R}O_{∗}^{p′}(F)}(P, P′) and α ∈ Aut_{F}(S). We should check that αψα^{−1} ∈ Hom_{^{R}O_{∗}^{p′}(F)}(α(P), α(P′)). ψ is the restriction of composition of ϕ_{1}, ⋯, ϕ_{k}, which are p-power order elements of automorphisms of F-centric, F-radical subgroups R_{1}, …, R_{k}. But now αψα^{−1} can be written as the composition of αϕ_{j}α^{−1}, which are p-power order elements of automorphisms of α(R_{j}), which are again fully F-normalized, F-centric and F-radical because this two properties are kept under Aut_{F}(S)-conjugation.

(b) By Alperin’s theorem for saturated fusion systems (Theorem 2.4) it is enough to check that Aut_{F}(Q) ≤ ⟨^{R}O_{∗}^{p′}(F), Aut_{F}(S)⟩ for Q an F-centric, F-radical and fully F-normalized subgroup of S. We will proceed by downward induction. Aut_{F}(S) ≤ ⟨^{R}O_{∗}^{p′}(F), Aut_{F}(S)⟩, so the result holds for Q = S. Assume now that the Aut_{F}(Q′) ≤ ⟨^{R}O_{∗}^{p′}(F), Aut_{F}(S)⟩ for all Q′ of bigger order than a fixed Q, and ψ ∈ Aut_{F}(Q).

Consider the subgroup K =^{def} ψ Aut_{S}(Q)ψ^{−1}, which is a p-subgroup of Aut_{F}(Q), so K ≤ Aut_{^{R}O_{∗}^{p′}(F)}(Q). Since Q is fully F-normalized, Aut_{S}(Q) is a Sylow p-subgroup of Aut_{F}(Q). As Aut_{^{R}O_{∗}^{p′}(F)}(Q) ≤ Aut_{F}(Q), we get that both Aut_{S}(Q) and K are Sylow p-subgroups in Aut_{^{R}O_{∗}^{p′}(F)}(Q), so they are conjugated by an element ϕ ∈ Aut_{^{R}O_{∗}^{p′}(F)}(Q):

ϕKϕ^{−1} ≤ Aut_{S}(Q) .  (1)

So, as F is saturated, by condition (II) in Definition 2.2 ϕψ must extend to a morphism \overline{ϕψ} defined over N_{ϕψ} which by Equation (1) is equal to N_{S}(Q), so it is always bigger than Q. So by induction hypothesis \overline{ϕψ}: N_{S}(Q) → S is an element in ⟨^{R}O_{∗}^{p′}(F), Aut_{F}(S)⟩, so its restriction ϕψ is again in ⟨^{R}O_{∗}^{p′}(F), Aut_{F}(S)⟩. Recall now that ϕ ∈ Aut_{^{R}O_{∗}^{p′}(F)}(Q) so ψ = ϕ^{−1}(ϕψ) is an element in ⟨^{R}O_{∗}^{p′}(F), Aut_{F}(S)⟩. □

Now we are ready to prove the main result of this section, which is analogous to the Alperin’s Theorem (Theorem 2.4) for the category O_{∗}^{p′}(F):

Theorem 3.4. Let F be a saturated fusion system over S. Then for each morphism ψ ∈ Hom_{O_{∗}^{p′}(F)}(P, P′), there exists a sequence of subgroups of S

P = P_{0}, P_{1}, …, P_{k} = P′ and Q_{1}, Q_{2}, …, Q_{k},

and morphisms ψ_{i} ∈ Aut_{O_{∗}^{p′}(F)}(Q_{i}), such that • Q_{i} is fully F-normalized, F-centric and F-radical for each i;

• P_{i−1}, P_{i} ≤ Q_{i} and ψ_{i}(P_{i−1}) = P_{i} for each i; and

• ψ(u) = ψ_{k} ∘ ψ_{k−1} ∘ ⋯ ∘ ψ_{1}(u), ∀u ∈ P.

Proof. We have just to check that O_{∗}^{p′}(F) ≤ ^{R}O_{∗}^{p′}(F).

Consider α: P → P′ an element in O_{∗}^{p′}(F). Then α is the composition of restrictions of automorphisms of order a power of p in Aut_{F}(Q_{i}), for some subgroups Q_{1}, …, Q_{r}. So it is enough to check that these elements are in ^{R}O_{∗}^{p′}(F).

Consider then α: Q → Q an element of order a power of p, for Q any subgroup in S. We will check that it is a morphism in ^{R}O_{∗}^{p′}(F) by downward induction. The result is true for Q = S (both morphisms sets are empty sets). Assume now that for a fixed Q, all the elements of order a power of p in Aut_{F}(Q′) are in ^{R}O_{∗}^{p′}(F) for all Q′ of order bigger than the order of Q.

If Q is not fully F-normalized, consider g: Q → Q′ an isomorphism in Hom_{F}(Q, Q′) such that Q′ is fully F-normalized. Using Lemma 3.3 and [2, Lemma 3.4 (c)] we get that there exists β ∈ Aut_{F}(S) and φ ∈ Hom_{^{R}O_{∗}^{p′}(F)}(β(Q), Q′) such that g = φ ∘ β|_{Q}.

If g ∘ α ∘ g^{−1} ∈ Hom_{^{R}O_{∗}^{p′}(F)}(Q′, Q′) then β|_{Q} ∘ α ∘ β^{−1}|_{β(Q)} = φ^{−1} ∘ g ∘ α ∘ g^{−1} ∘ φ is also in Hom_{^{R}O_{∗}^{p′}(F)}(Q′, Q′). Use now that α is Aut_{F}(S)-conjugated to β|_{Q} ∘α ∘β^{−1}|_{β(Q)} and Aut_{F}(S) normalizes ^{R}O_{∗}^{p′}(F) and we obtain that α is a morphism in ^{R}O_{∗}^{p′}(F). So we can assume that Q is fully F-normalized.

If Q is not F-centric, then by condition (II) in [3, Definition 1.2] we get that α extends to a morphism \overline{φ} ∈ Hom_{F}(Q · C_{S}(Q), S). The image of \overline{φ} must be again in Q · C_{S}(Q), so \overline{φ} ∈ Aut_{F}(Q · C_{S}(Q)) and \overline{φ}|_{Q} = φ. If the order of \overline{φ} is a power of p, we can apply the induction hypothesis and the result follows. If not, we can consider an integer r such that \overline{φ}^{r}|_{Q} = φ and the order of \overline{φ}^{r}|_{Q} is a power of p, and apply the induction hypothesis.

Assume now that Q is F-centric and fully F-normalized, and α ∈ Aut_{F}(Q) an element of p-power order. If Q is F-radical, we have finished, so suppose that Q is not F-radical. That means that O_{p}(Out_{F}(Q)) ≠ ⟨1⟩, and so K =^{def} O_{p}(Aut_{F}(Q)) is strictly bigger than Aut_{Q}(Q). As Q is fully F-normalized, we have that Aut_{S}(Q) is a Sylow p-subgroup for Aut_{F}(Q), and so K ≤ Aut_{S}(Q). Consider

N_{S}^{K}(Q) =^{def} {x ∈ N_{S}(Q) | c_{x} ∈ K} ,

which is strictly bigger than Q. If x ∈ N_{S}^{K}(Q), αc_{x}α^{−1} is an element in K (K is a normal subgroup in Aut_{F}(Q)), so, in this case,

N_{S}^{K}(Q) ≤ N_{α} =^{def} {x ∈ N_{S}(Q) | αc_{x}α^{−1} ∈ Aut_{S}(Q)}

and by condition (II) in [3, Definition 1.2] α extends to a morphism \overline{α}: N_{S}^{K}(Q) → S. Moreover the image of \overline{α} is contained in N_{S}^{K}(Q): the extension to elements of the normalizer in S of Q must give elements in the normalizer of the image, which is again Q; also c_{\overline{α}(x)} = α ∘ c_{x} ∘ α^{−1}, so if c_{x} is in K, c_{\overline{α}(x)} is again in K. So we have an extension of α, which can be taken of ppower order (take \overline{α}^{r} as before if necessary) and which is in ^{R}O_{∗}^{p′}(F) by induction hypothesis. As we are in a restriction closed category, α is also a morphism in ^{R}O_{∗}^{p′}(F). □

4. A representation of the extraspecial group p_{+}^{1+2γ} in GL_{n}(q)

In this section we introduce some subgroups needed to understand the fusion system of GL_{n}(q) over a prime p such that p ∤ q. We divide this study in two subsections, the first one concerning to the particular case q ≡ 1 mod p and the second one, where we extend the results to the general case.

Notation 4.1. Consider the extraspecial group of order p^{1+2γ} and exponent p, noted as p_{+}^{1+2γ}, as the group generated by A_{0}, …, A_{γ−1}, B_{0}, …, B_{γ−1} and C with the relations [A_{i}, A_{j}] = 1, [B_{i}, B_{j}] = 1 and [A_{i}, B_{j}] = C^{δ_{ij}} for all i, j ∈ {0, 1, …, γ − 1}.

4.1. Case q ≡ 1 mod p. Fix first q a prime power such that p | (q − 1) and ζ a p-root of the unity in 𝔽_{q}.

We will construct a representation of p_{+}^{1+2γ} in GL_{p^{γ}}(q), so we assume that n ≥ p^{γ}. In the case n > p^{γ} we consider the inclusion of p_{+}^{1+2γ} in GL_{n}(q), as the composition p_{+}^{1+2γ} ≤ GL_{p^{γ}} ≤ GL_{n}(q).

A faithful representation can be constructed as follows: consider V a 𝔽_{q}-vector space of dimension p^{γ}, with basis {v_{0}, …, v_{p^{γ}−1}}; write each integer k such that 0 ≤ k ≤ p^{γ} − 1 in base p, getting γ unique numbers a_{i} ∈ {0, …, p − 1}:

k = a_{0} + a_{1}p + a_{2}p^{2} + ⋯ + a_{γ−1}p^{γ−1} .

Then, using this basis consider the following morphisms:

• A_{i} is the linear transformation which sends

v_{a_{0}+···+a_{i}p^{i}+···} ↦ ζ^{a_{i}}v_{a_{0}+···+a_{i}p^{i}+···} ,

• B_{i} is the permutation in the basis which sends

v_{a_{0}+···+a_{i}p^{i}+···} ↦ v_{a_{0}+···+(a_{i}+1)p^{i}+···} ,

where the coefficient (a_{i} + 1) is reduced mod p.

Lemma 4.2. The elements A_{i} and B_{i} defined above give a faithful representation of p_{+}^{1+2γ} in GL_{p^{γ}}(q).

Proof. A direct computation shows us that these elements carry out the relations in Notation 4.1, taking C as ζ Id. □

We will also be interested in the following transformation: for i ∈ {0, 1, …, n−1} we consider σ_{i}: V → V as the linear map which in the basis v_{0}, …, v_{p^{γ}−1} acts as:

σ_{i}(v_{a_{0}+···+a_{i}p^{i}+···}) =^{def} v_{a_{0}+···+(ξa_{i})p^{i}+···} ,  (2)

where ξ is a (p − 1)-root of unity in 𝔽_{p} and again the coefficient (ξa_{i−1}) is reduced mod p. We can check with a direct computation that:

σ_{i}A_{j}σ_{i}^{−1} = \{ \begin{matrix}A_{j} & if i ≠ j \\ A_{j}^{ξ^{−1}} & if i = j \end{matrix}  (3)

σ_{i}B_{j}σ_{i}^{−1} = \{ \begin{matrix}B_{j} & if i ≠ j \\ B_{j}^{ξ} & if i = j \end{matrix}  (4)

where A_{j}^{ξ^{−1}} (respectively B_{j}^{ξ}) means the matrix A_{j} to the power ξ^{−1} (respectively to the power ξ).

4.2. General case. Now we consider e the order of q mod p. We assume that e > 1, and observe that e must divide (p − 1).

Consider 𝔽_{q^{e}} a Galois extension of 𝔽_{q}, with Galois group ℤ/eℤ. 𝔽_{q^{e}} can be regarded as the quotient 𝔽_{q}[x]/r(x), where r(x) is an irreducible polynomial of degree e with coefficients in 𝔽_{q}.

Now consider 𝔽_{q^{e}} as 𝔽_{q}-vector space with basis {1, x, …, x^{e−1}}. This gives an inclusion of GL_{1}(q^{e}) ⊂ GL_{e}(q), and more generally of GL_{n}(q^{e}) ⊂ GL_{en}(q) (convert every coefficient of a matrix in GL_{n}(q^{e}) to an e × e-matrix with coefficients in 𝔽_{q}).

Using this inclusion we get that we can apply the previous subsection to the study of GL_{n}(q^{e}), getting the corresponding elements A_{j}, B_{j} and σ_{j} in GL_{en}(q), and also the corresponding inclusion of p_{+}^{1+2γ} ≤ GL_{ep^{γ}}(q).

Consider the Frobenius automorphism of 𝔽_{q^{e}}, which is the 𝔽_{q}-linear map which sends every y =^{def} a_{0} + ⋯ + a_{e−1}x^{e−1} to y^{q}, and is a generator of the Galois group of 𝔽_{q^{e}} over 𝔽_{q}. This can be thought as an e × e matrix with coefficients in 𝔽_{q}.

Consider the following fixed isomorphism 𝔽_{q^{e}}^{n} ≅ 𝔽_{q}^{en} as 𝔽_{q}-vector space: if {v_{1}, …, v_{n}} is a basis of 𝔽_{q^{e}}^{n} as 𝔽_{q^{e}}-vector space, then

{v_{1}, xv_{1}, …, x^{e−1}v_{1}, v_{2}, xv_{2}, …, x^{e−1}v_{2}, …, v_{n}, xv_{n}, …, x^{e−1}v_{n}}

is a basis of 𝔽_{q}^{en} as 𝔽_{q}-vector space.

Now define φ_{n} the 𝔽_{q}-linear automorphism in 𝔽_{q^{e}}^{n} which sends a vector

φ_{n}(r_{1}(x)v_{1} + ⋯ + r_{n}(x)v_{n}) =^{def} r_{1}(x)^{q}v_{1} + ⋯ + r_{n}(x)^{q}v_{n} ,  (5)

and regard it as an automorphism of 𝔽_{q}^{en} in the previous basis.

With these definitions, we can compute the following conjugations:

• φ_{n}A_{j}φ_{n}^{−1} = A_{j}^{q} and

• φ_{n}B_{j}φ_{n}^{−1} = B_{j}. Note that, as the order of the Frobenius automorphism is q^{e}, the inverse φ_{n}^{−1} is given by:

φ_{n}^{−1}(r_{1}(x)v_{1} + ⋯ + r_{n}(x)v_{n}) = r_{1}(x)^{q^{e−1}}v_{1} + ⋯ + r_{n}(x)^{q^{e−1}}v_{n} ,

So φ_{n}^{−1} sends r_{i}(x)v_{i} to r_{i}(x)^{q^{e−1}}v_{i}. If we apply A_{j}, each r_{i}(x)^{q^{e−1}}v_{i} is multiplied by a coefficient a_{ij} which only depends on the expression of i in base p and j. In we apply now φ_{n} to a_{ij}r_{i}(x)^{q^{e−1}}v_{i} we get a_{ij}^{q}v_{i}. Now as A_{j} are diagonal matrices, we get A_{j}^{q}.

B_{j} permutes the basis v_{i}, so it commutes with the action of φ_{n}.

5. The fusion systems of GL_{n}(q)

Notation 5.1. In all this section fix p and odd prime and q a prime power prime to p. Let e be the multiplicative order of q modulo p, and l = ν_{p}(q^{e} − 1). Consider also V an n-dimensional 𝔽_{q}-vector space, in such a way that we write GL(V) instead of GL_{n}(q) if we have to deal with subgroups.

We are interested in the F-centric, F-radical subgroups of the general linear groups. Fix S a Sylow p-subgroup of a finite group G, and P a subgroup of S.

Note that when G is a finite group and S a Sylow p-subgroup, then a subgroup P ≤ S is p-radical in G when N_{G}(P)/P does not contain any non trivial normal p-subgroup. Also a subgroup P ≤ S is p-centric in G if C_{G}(P) = Z(P) × C′_{G}(P), where C′_{G}(P) has order prime to p.

For P, being p-radical in G and being F_{S}(G)-radical (F_{S}(G) is the saturated fusion system given by G over S) are independent definitions. If we require P to be p-centric in G, then we have that when P is F_{S}(G)-radical, P is also p-radical in G. So the list of subgroups of G that we are interested in is contained in the list of p-radical subgroups of G.

The possible p-radical subgroups of GL_{n}(q) over the prime p are described in [1, Section 4] and depend on some parameters α, γ, m, and 𝔠 = (c_{1}, …, c_{t}). These are constructed as follows:

Fixed α ≥ 0 and γ ≥ 0 consider (ℤ/p^{l+α})p_{+}^{1+2γ} the central product of the cyclic group of order p^{l+α} and an extraspecial group of order p^{1+2γ} and exponent p over the center of the extraspecial group. Consider R_{α,γ} the image of (ℤ/p^{l+α})p^{1+2γ} by the composition

(ℤ/p^{l+α})p_{+}^{1+2γ} ≤ GL_{p^{γ}}(q^{ep^{α}}) ≤ GL_{ep^{α+γ}}(q) ,  (6)

where the first inclusion works as follows:

• The subgroup ℤ/p^{l+α} corresponds to the matrices of the form λ Id, where λ is a p^{l+α}-root of unity in 𝔽_{q^{ep^{α}}}.

• The inclusion p_{+}^{1+2γ} ≤ GL_{p^{γ}}(q^{ep^{α}}) is as in Section 4. Fix now m ≥ 1 and R_{m,α,γ} the image of R_{α,γ} in GL_{mep^{α+γ}}(q) which sends g to the m-fold

diagonal map.

Finally consider 𝔠 = (c_{1}, …, c_{t}) a sequence of positive integers and define F_{𝔠} =^{def} (ℤ/p)^{c_{1}} ≀

(ℤ/p)^{c_{2}} ≀ ⋯ ≀ (ℤ/p)^{c_{t}}. Let d =^{def} mep^{α+γ+c_{1}+···+c_{t}} and denote by R_{m,α,γ,𝔠} the image of R_{m,α,γ} ≀ F_{𝔠}

in GL_{d}(q). Call a subgroup of this type as a basic subgroup of GL_{d}(q).

Theorem 5.2 ([1]). Fix V an n-dimensional 𝔽_{q}-vector space. Let G = GL(V), and R be a p-radical subgroup of G. Then there exist decompositions

\begin{matrix}W = V_{0} ⊕ V_{1} ⊕ ⋯ ⊕ V_{s}, \\ R = R_{0} × R_{1} × ⋯ × R_{s}, \end{matrix}

such that R_{0} is the trivial subgroup of GL(V_{0}), and R_{i} are basic subgroups of GL(V_{i}) for i ≥ 1.

Now we are able to give the Sylow p-subgroup in terms of e, (the order of q modulo p), l (the p-adic valuation of q^{e} − 1) and the coefficients a_{0}, …, a_{k}, all of them 0 ≤ a_{i} ≤ (p − 1) such that:

[n/e] = a_{0} + a_{1}p + ⋯ + a_{k}p^{k} .

Lemma 5.3. S_{n,q}, a Sylow p-subgroup of GL_{n}(q), is given by the following construction:

\begin{aligned}S_{n,q} =^{def} & (ℤ/p^{l} ≀ ℤ/p≀ ⋯^{(k)} ≀ℤ/p)^{a_{k}} × (ℤ/p^{l} ≀ ℤ/p≀ ⋯^{(k−1)} ≀ℤ/p)^{a_{k−1}} × ⋯ \\ & ⋯ × (ℤ/p^{l} ≀ ℤ/p)^{a_{1}} × (ℤ/p^{l})^{a_{0}} . \end{aligned}

where e, l and a_{1}, …, a_{k} are defined above.

Moreover, if (S_{n,q}, F_{n,q}) is the saturated fusion system induced by GL_{n}(q) over S_{n,q}, then Out_{F_{n,q}}(S_{n,q}) is isomorphic to

((ℤ/e × ℤ/(p − 1)^{k}) ≀ Σ_{a_{k}}) × ((ℤ/e × ℤ/(p − 1)^{k−1}) ≀ Σ_{a_{k−1}}) × ⋯ × ((ℤ/e) ≀ Σ_{a_{0}}) .

Proof. This combination of direct products and wreath products is included in GL_{e[n/e]}(q) ≤ GL_{n}(q). Checking the orders of S_{n,q} and the Sylow p-subgroup of GL_{n}(q) we have finished.

Finally the outer automorphisms group is computed in [1, Section 4]. □

For fixed p and q we can compute e, and the Sylow p-subgroup and the fusion does not change between GL_{n}(q) and GL_{e[n/e]}(q). Sometimes we will consider that we are working in rank multiples of e and we will write in GL_{em}(q) instead of GL_{n}(q).

Notation 5.4. Consider S_{n,q} a Sylow p-subgroup of GL_{n}(q) as computed in Lemma 5.3, and (S_{n,q}, F_{n,q}, L_{n,q}) the corresponding p-local finite group.

Let us begin now with the first non-trivial case:

Lemma 5.5. Let p be an odd prime and q a prime power such that p|(q − 1). Consider (S_{p,q}, F_{p,q}) as in Notation 5.4. Then Out_{F_{p,q}}^{0}(S) = Out_{F_{p,q}}(S).

Proof. To simplify the notation, consider in this proof (S, F) =^{def} (S_{p,q}, F_{p,q}). In this case, by Lemma 5.3 we have Out_{F}(S) ≅ ℤ/(p − 1). We can see that σ defined in Equation (2) is a generator, so we must check that σ ∈ Out_{F}^{0}(S).

Consider R_{0,1} as in Equation (6). We get that R_{0,1} is isomorphic to a central extension of p_{+}^{1+2}, generated by A and B as defined in Equations (3) and (4), by ℤ/p^{l}, the p-primary part in the center of GL_{p}(q).

A direct computation tells us that R_{0,1} is F-centric.

Equation (2) gives us the action of A and B:

σAσ^{−1} = A^{ζ^{−1}} and σBσ^{−1} = B^{ζ} ,

where ζ is a (p − 1)-root in 𝔽_{p}. So σ restricts to an automorphism of R_{0,1} which, as element in Out_{F}(R_{0,1}), must be considered as the matrix ( \begin{matrix}ζ^{−1} & 0 \\ 0 & ζ \end{matrix} ).

Use now that in this case, [1, Section 4] tells us that Out_{F}(R_{0,1}) ≅ SL_{2}(p), and now apply the fact that SL_{2}(p) is generated by its elements of order p. □

Proposition 5.6. Consider (S_{n,q}, F_{n,q}) as in Notation 5.4, with n ≥ ep. There is a subgroup H in Out_{F_{n,q}}(S_{n,q}) such that Out_{F_{n,q}}(S_{n,q})/H ≅ ℤ/e and Out_{F_{n,q}}^{0}(S) ≤ H.

Proof. Let us consider the form of Out_{F_{n,q}}(S_{n,q}) given in Lemma 5.3.

Observe that all the elements in Out_{F_{n,q}}(S_{n,q}) restrict to automorphisms of the maximal torus 𝕋_{p^{l}}^{[n/e]}, and that different elements in Out_{F_{n,q}}(S_{n,q}) give different restrictions. This implies that if F_{0} is the minimal saturated fusion subsystem of (S_{n,q}, F_{n,q}) of index prime to p over S, then

Out_{F_{n,q}}(S_{n,q})/ Out_{F_{n,q}}^{0}(S_{n,q}) ≅ Out_{F_{n,q}}(S_{n,q})/ Out_{F_{0}}(S) ≅

≅ Out_{F_{n,q}}(𝕋_{p^{l}}^{[n/e]})/ Out_{F_{0}}(𝕋_{p^{l}}^{[n/e]}) . (7)

We have that Out_{F_{n,q}}(𝕋_{p^{l}}^{[n/e]}) ≅ ℤ/e ≀ Σ_{[n/e]}. Now consider O_{p′}(Out_{F_{n,q}}(𝕋_{p^{l}}^{[n/e]})), the subgroup generated by all the elements of order a power of p. As p ≤ [n/e], there are elements of order p in Σ_{[n/e]}, and as O_{p′}(Out_{F_{n,q}}(𝕋_{p^{l}}^{[n/e]})) is a normal subgroup, it musts contain all the alternating group A_{[n/e]} ≤ Σ_{[n/e]} and its normal closure in (ℤ/e)^{[n/e]} ⋊ A_{[n/e]}, which is (ℤ/e)^{[n/e]−1} ⋊ A_{[n/e]} ((ℤ/e)^{[n/e]−1} ≤ (ℤ/e)^{[n/e]} is the kernel of the map (a_{1}, …, a_{[n/e]}) ↦ ∑ a_{i}).

To finish the proof, take the restriction of σ, which by Lemma 5.5 is in Out_{F_{n,q}}^{0}(S), and which gives an odd permutation of Σ_{[n/e]}, getting that

H_{𝕋} =^{def} (ℤ/e)^{[n/e]−1} ⋊ Σ_{[n/e]} ≤ Out_{F_{0}}(𝕋_{p^{l}}^{[n/e]}) .

But, by Equation (7), Out_{F_{n,q}}^{0}(S) must be contained in H defined as the kernel of the morphism:

Out_{F_{n,q}}(S) → Out_{F_{n,q}}(𝕋_{p^{l}}^{[n/e]})/H_{𝕋} ≅ ℤ/e ,

and the result follows. □

From the previous proposition we get that Out_{F_{n,q}}(S)/ Out_{F_{n,q}}^{0}(S) has at most e elements. The rest of the section is dedicated to prove that there is an element of order e in Out_{F_{n,q}}(S)/ Out_{F_{n,q}}^{0}(S).

As we are interested just in the F_{n,q}-centric, F_{n,q}-radical subgroups, we can remove some subgroups in the list:

Lemma 5.7. If m > 1 then R_{m,α,γ,𝔠} is not p-centric in S.

Proof. Assume m > 1. We are considering the inclusion of R_{α,γ} in GL_{mep^{α+γ}}(q) diagonally:

g ↦ ( \begin{matrix}g & 0 & ⋯ & 0 \\ 0 & g & ⋯ & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & ⋯ & g \end{matrix} ) .

Fix g a non-trivial element in the center or R_{α,γ}. We have that

h =^{def} ( \begin{matrix}g & 0 & ⋯ & 0 \\ 0 & 1 & ⋯ & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & ⋯ & 1 \end{matrix} )

is an element which centralizes R_{m,α,γ} which is not in R_{m,α,γ}.

Recall now that in a semidirect product G ⋊ H, if we have an element g in the center of G invariant under the action of H, then (g, 1) is in the centre of G ⋊ H. Apply this argument to the element (h, h, …, h) ∈ (R_{m,α,γ})^{p^{c_{1}+···+cr}}, which is invariant by the action of Σ_{n}. □

So, from now on, we just consider the case m = 1.

Lemma 5.8. Consider λ a p^{l}-root of unity in 𝔽_{q^{e}}. The maximal torus 𝕋_{p^{l}}^{m} can be seen as the image of (ℤ/p^{l})^{m} under the composition

(a_{1}, …, a_{m}) ↦ diag(λ^{a_{1}}, …, λ^{a_{m}}) ∈ GL_{m}(q^{e}) ≤ GL_{em}(q) .

Fix R an F_{em,q}-centric, F_{em,q}-radical subgroup in S_{em,q} (Notation 5.4) and consider χ the image in GL_{em}(q) of the subset (not a subgroup) of elements in (ℤ/p^{l})^{m} such that the product a_{1} ⋯ a_{m} = 1. If ψ ∈ O^{p′}(Aut_{F_{em,q}}(R)), then ψ(χ ∩ R) = χ ∩ R.

Proof. To simplify the notation in this proof consider (S, F) =^{def} (S_{em,q}, F_{em,q}) Consider R an F-centric, F-radical subgroup of GL(V), with V an n-dimensional 𝔽_{p}-vector space. By the description in Theorem 5.2 we get that there is a decomposition which can be given, after reordering and grouping by isomorphism type, as:

V = V_{0} ⊕ V_{1,1} ⊕ ⋯ ⊕ V_{1,m_{1}} ⊕ V_{2,1} ⊕ ⋯ ⊕ V_{k,m_{k}}

such that R can be written as:

R = R_{0} × (R_{1})^{m_{1}} × ⋯ × (R_{k})^{m_{k}}

with R_{i} basic subgroups in GL(V_{i,j}), and such that m_{1}d_{1} + ⋯ + m_{k}d_{k} = m, where d_{i} = dim(V_{i,1}). For each i ∈ {1, …, k} and j ∈ {1, …, m_{i}}, consider H_{i,j} the image of GL(V_{i,j}) in GL(V). Consider \overline{S}_{i,j} = S ∩ H_{i,j}, and \overline{F}_{i,j} the saturated fusion system F_{\overline{S}_{i,j}}(GL(V_{i,j})). Observe that \overline{S}_{i,j} ≅ \overline{S}_{i,j′} and \overline{F}_{i,j} ≅ \overline{F}_{i,j′} for j, j′ ∈ \{1, …, m_{i}\}.

With all this notation, we have the isomorphism:

Aut_{F}(R)/R ≅ ∏_{i=1}^{k}(Aut_{\overline{F}_{i,1}}(R_{i})/R_{i}) ≀ Σ_{m_{i}}

The action of R on itself by conjugation is included in the action of S over R, and this preserves χ, as these are permutations.

Consider now an element in Aut_{F}(R) such that its class belongs to O^{p′}(Out_{F}(R)). We have to deal with two possibilities:

• As R_{i} are \overline{F}_{i,j}-centric basic subgroups, using Lemma 5.7 and [1, Section 4] all the elements of order a power of p in Out_{\overline{F}_{i,1}}(R_{i}) are permutations, so they preserve χ.

• If there exists i such that m_{i} ≥ p then the elements of the form τσ(τ^{−1}) are also in O^{p′}(Out_{F}(R)), for all σ in A_{m_{i}} ≤ Σ_{m_{i}} and all τ ∈ Aut_{F′_{i,1}}(R_{i}) (A_{m_{i}} the alternating subgroup in Σ_{i}). To simplify notation consider the elements in χ∩S_{i,j} as (a_{1}, …, a_{d_{i}}) ∈ (ℤ/p^{l})^{d_{i}}. But τ sends (a_{1}, …, a_{d_{i}}) to (a_{τ(1)}q_{1}, …, a_{τ(d_{i})}q_{d_{i}}), and so σ(τ^{−1}) permutes the components and multiply each one by q_{i}^{−1} respectively. In any case, the product of the elements is

a_{1} ⋯ a_{d_{i}}q_{1} ⋯ q_{d_{i}}q_{1}^{−1} ⋯ q_{d_{i}}^{−1} = a_{1} ⋯ a_{d_{i}}.

So the product a_{1} ⋯ a_{d_{i}} doesn’t change. This implies that the elements of ψ(χ ∩ R) ≤ χ, but as ψ is an automorphism of R the result follows. □

To get an element of order e in Aut_{F_{n,q}}^{0}(S_{n,q}) we use the automorphism φ_{1} defined in Equation (5) as follows: as we consider the case n ≥ e, consider ϕ the 𝔽_{q}-linear map defined as

ϕ =^{def} φ_{1} ⊕ Id_{[n/e]−1} .  (8)

Observe that ϕ is an element of order e.

Proposition 5.9. Consider (S_{n,q}, F_{n,q}) as in Notation 5.4, with n ≥ e. For every F_{n,q}- centric subgroup P, and j such that 1 ≤ j ≤ e − 1, the restriction of ϕ^{j}, where ϕ is defined in Equation (8), to P is not in Hom_{O_{∗}^{p′}(F_{n,q})}(P, S_{n,q}).

Proof. Consider the restriction of ϕ^{j} an F_{n,q}-centric subgroup P. As P is F_{n,q}-centric, it contains the center of S_{n,q}, which can be seen as the image of the matrices λ Id in the inclusion GL_{m}(q^{e}) ≤ GL_{em}(q). Assume now that ϕ^{j}|_{P} is in Hom_{O_{∗}^{p′}(F_{n,q})}(P, S_{n,q}).

By Theorem 3.4, ϕ^{j} can be written as a composition of restriction of automorphisms ϕ_{i} ∈ O^{p′}(Aut_{F_{n,q}}(R_{i})) (i ∈ {1, …, k}), for R_{i} F_{n,q}-centric, F_{n,q}-radical subgroups. Consider now ρ Id, ρ a primitive p-root of unity in 𝔽_{q^{e}}, which is an element in χ∩P (χ as defined in Lemma 5.8). This implies ϕ^{j}(ρ Id) = ϕ_{k} ∘ ϕ_{k−1} ∘ ⋯ ∘ ϕ_{1}(ρ Id) ∈ χ.

A direct computation show us that ϕ(ρ Id) = ρ^{q^{−1}} Id_{1} ⊕ρ Id_{[n/e]−1}, and then ϕ^{j}(ρ Id) = ρ^{q^{−j}} Id_{1} ⊕ρ Id_{[n/e]−1}. So this element is not in χ, as q^{−j} ≠ 1 mod p.

So we get a contradiction which comes from assuming that

ϕ^{j}|_{P} ∈ Hom_{O_{∗}^{p′}(F_{n,q})}(P, S_{n,q}) .

□

Theorem 5.10. Let p be an odd prime and q prime power such that p ∤ q. Consider e the multiplicative order of q modulo p. Fix S_{n,q} a Sylow p-subgroup of GL_{n}(q) and (S_{n,q}, F_{n,q}) the induced saturated fusion system. Then,

Out_{F}(S_{n,q})/ Out_{F}^{0}(S_{n,q}) ≅ \begin{cases}(ℤ/e) ≀ Σ_{[n/e]} & if e ≤ n < ep, \\ ℤ/e & if ep ≤ n. \end{cases}

Proof. We have to consider n ≥ e to have a nontrivial Sylow p-subgroup in GL_{n}(q).

For n < ep we have that S_{n,q} ≅ (ℤ/p^{l})^{[n/e]}, which is abelian. So the only F-centric subgroup is the total and Out_{F_{n,q}}^{0}(S_{n,q}) is trivial.

When n ≥ ep, use Proposition 5.6 to see that the quotient is at most ℤ/e and Proposition 5.9 implies the result. □

6. Exotic fusion subsystems in the general linear group

Fix as before p a prime, q a prime power prime to p. Fix e the multiplicative order of q modulo p. Consider (S_{n,q}, F_{n,q}, L_{n,q}) the saturated fusion system of GL_{n}(q) at the prime p, with n ≥ e.

Proposition 6.1. (a) For each r dividing e there is a p-local finite group (S_{n,q}, F_{n,q,r}, L_{n,q,r}), such that F_{n,q,r} is of index prime to p in F_{n,q} over S_{n,q} and Out_{F_{n,q,r}}(S_{n,q}) is a subgroup of index r in Out_{F_{n,q}}(S_{n,q}).

(b) If n ≥ ep, there is just one (S_{n,q}, F_{n,q,r}, L_{n,q,r}) satisfying (a).

(c) Up to homotopy equivalence, there is a fibration

|L_{n,q,r}| ⟶ |L_{n,q}| ⟶ B(ℤ/r) .

Proof. For e ≤ n < ep consider the saturated fusion subsystem of index prime to p corresponding to the kernel of the group epimorphism from

Out_{F}(S_{n,q})/ Out_{F}^{0}(S_{n,q}) ≅ (ℤ/e) ≀ Σ_{[n/e]}

to ℤ/r defined as

\begin{matrix}(ℤ/e)^{[n/e]} ⋊ Σ_{[n/e]} & ⟶ & ℤ/r \\ (b_{1}, …, b_{[n/e]}; σ) & ↦ & \frac{e}{r}(b_{1} + ⋯ + b_{[n/e]}) . \end{matrix}

getting (a) and (c) for this case, applying [2, Theorem 5.5].

For n ≥ ep we have Out_{F}(S_{n,q})/ Out_{F}^{0}(S_{n,q}) ≅ (ℤ/e), so (a), (b) and (c) follows directly again from [2, Theorem 5.5]. □

We proceed now to identify these saturated fusion subsystems with the ones studied in [5, Section 11].

Let p be an odd prime and r ≥ 1, e ≥ 1 natural numbers such that r|e|(p − 1). Consider G(e, r, m) the subgroup of GL_{m}(ℤ_{p}^{∧}) as the subgroup generated by:

A(e, r, m) = {diag(a_{1}, …, a_{m}) | a_{i}^{e} = 1 and (a_{1} ⋯ a_{m})^{e/r} = 1}

and the matrices corresponding to permutations in the coordinates. As a group, we have that G(e, r, m) = A(e, r, m) ⋊ Σ_{n}.

Consider now m ≥ 1 and let X(e, r, m) be the p-compact group which realizes the pseudoreflection group G(e, r, m) ≤ GL_{m}(ℤ_{p}^{∧}) (see [8] and [9] for further information). If m ≥ 2 these are called the Generalized p-adic Grassmannians and we are interested in their finite Chevalley version. That is the space BX(e, r, m)(q) defined as the pullback diagram:

\xymatrix{BX(e, r, m)(q) \ar[r] \ar[d] & BX(e, r, m) \ar[d]^{Δ} \\ BX(e, r, m) \ar[r]^{1×φ^{q}} & BX(e, r, m) × BX(e, r, m)}

where Δ is the diagonal map and φ^{q} is the unstable Adams operation of exponent q, where q is a p-adic unit.

So if we consider q a power of a prime not divisible by p, it is a p-adic unit and the previous construction makes sense. Consider e the multiplicative order of q modulo p. In this case we have the following equivalences up to p-completion [5, Remark 11.1]:

• BX(1, 1, n)(q) ≃ BGL_{n}(q).

• BX(1, 1, n)(q) ≃ BX(1, 1, e[n/e])(q).

• If q′ is a another prime power such that p|(q′ − 1) and ν_{p}(q′ − 1) = ν_{p}(q^{e} − 1) then

BX(1, 1, n)(q) ≃ BX(e, 1, [n/e])(q′). Observe that we can take q′ = q^{e}.

Where [n/e] is the greater integer less or equal than n/e.

Before the main statement of the section we need to compute some centralizers.

Lemma 6.2. Consider (S_{n,q}, F_{n,q,r}, L_{n,q,r}) as in Proposition 6.1, with n ≥ e, and V a nontrivial elementary abelian p-subgroup contained in S_{n,q}. Then

(a) V is F_{n,q,r}-conjugate to a subgroup of the maximal torus 𝕋_{p^{l}}^{[n/e]}.

(b) For V contained in 𝕋_{p^{l}}^{[n/e]}, consider the centralizer p-local finite group

(C_{S_{n,q}}(V), C_{F_{n,q,r}}(V), C_{L_{n,q,r}}(V))

defined in [3, Proposition 2.5]. Then there are integers m_{0}, m_{1}, …, m_{s} such that

|C_{L_{n,q,r}}(V)| ≃ |L_{em_{0},q,r}| × BGL_{m_{1}}(q^{e}) × ⋯ × BGL_{m_{s}}(q^{e})

up to p-completion. Moreover em_{0} < n and [n/e] = m_{0} + m_{1} + ⋯ + m_{s}.

Proof. Consider V a nontrivial elementary abelian p-group contained in S. Consider the saturated fusion system (S_{n,q}, F_{S_{n,q}}(GL_{[n/e]}(q^{e}))) which is contained in (S_{n,q}, F_{n,q,r}) for all r|e. Then we can use the fact that V is F_{S_{n,q}}(GL_{[n/e]}(q^{e}))-conjugate to a toral subgroup, getting (a).

To compute the centralizer fusion system, consider the point-wise stabilizer of V in

Aut_{F_{n,q,r}}(𝕋_{p^{l}}^{[n/e]}) = G(e, r, [n/e]),

which is isomorphic to G(e, r, m_{0}) × Σ_{m_{1}} × ⋯ × Σ_{m_{s}}. Note that as V is a nontrivial group, m_{0} < [n/e].

We can see that C_{F_{n,q,r}}(V) is a saturated fusion system over C_{S_{n,q}}(V), which can be written as C_{S_{n,q}}(V) ≅ S_{em_{0},q} ×S_{m_{1},q^{e}} ×⋯×S_{m_{s},q^{e}}, where we are using that Lemma 5.3 identifies S_{m,q^{e}} and S_{em,q}. Also C_{F_{n,q,r}}(V) contains F_{em_{0},q,r} × F_{S_{m_{1},q^{e}}}(GL_{m_{1}}(q^{e})) × ⋯ × F_{S_{ms,q^{e}}}(GL_{m_{s}}(q^{e})), as all the morphisms in this last saturated fusion system centralize V and are contained in F_{n,q,r}.

Considering now that F_{n,q,r} is a fusion subsystem of F_{n,q}, the morphisms in the centralizer of V in F_{n,q,r} are again morphisms in the centralizer of V in F_{n,q}.

The centralizer p-local finite group of V in F_{n,q} is computed in [5, Proposition 11.2], and its classifying space is mod p equivalent to BGL_{em_{0}}(q) × BGL_{m_{1}}(q^{e}) × ⋯ × BGL_{m_{s}}(q^{e}), and also has the same Sylow p-subgroup as C_{F_{n,q,r}}(V).

With all this data we get that C_{F_{n,q,r}}(V) is a saturated fusion subsystem of index prime to p in C_{F_{n,q}}(V) such that in the maximal torus Aut_{C_{Fn,q,r} (V)}(𝕋_{p^{l}}^{[n/e]}) is an index r subgroup of Aut_{C_{Fn,q} (V)}(𝕋_{p^{l}}^{[n/e]}), and so

C_{F_{n,q,r}}(V) ≃ F_{em_{0},q,r} × F_{S_{m_{1},q^{e}}}(GL_{m_{1}}(q^{e})) × ⋯ × F_{S_{ms,q^{e}}}(GL_{m_{s}}(q^{e})) .

So (b) follows considering the centric linking systems associated to these saturated fusion systems. □

Theorem 6.3. For each r dividing e and n ≥ e consider (S_{n}, F_{n,q,r}, L_{n,q,r}), the p-local finite group from Proposition 6.1. Then

|L_{n,q,r}| ≃ BX(e, r, [n/e])(q^{e})

up to p-completion.

Proof. We proceed by induction on n, beginning by the smallest considered case, that is n = e.

For GL_{e}(q) we have that the p-local finite group (S_{e,q}, F_{e,q,r}, L_{e,q,r}) is characterized by S_{e,q} = ℤ/p^{l}, and Aut_{F_{e,q,r}}(S_{e,q}) = ℤ/h, where h =^{def} e/r. So it corresponds to the fusion system of ℤ/p^{l} ⋊ ℤ/h, where h =^{def} e/r. The Sylow p-subgroup of BX(e, h, 1)(q) is also isomorphic to ℤ/p^{l}, with outer automorphism group G(e, h, 1) = ℤ/h, so both are the same p-local finite group.

Now assume that the result is true for any n′ ≤ n.

To proceed with the induction argument we use the centralizer decomposition from [3, Theorem 2.6].

Fix V ≤ S an elementary abelian fully F_{n,q,r}-centralized subgroup. If necessary we can conjugate it to get V contained in the torus, and still being fully F_{n,q,r}-centralized.

Consider (C_{S_{n,q}}(V), C_{F_{n,q,r}}(V), C_{L_{n,q,r}}(V)) the centralizer p-local finite group, whose classifying space is, by Lemma 6.2, homotopy equivalent up to p-completion to

|L_{em_{0},q,r}| × BGL_{m_{1}}(q^{e}) × ⋯ × BGL_{m_{s}}(q^{e}) ,

with em_{0} < n. By induction hypothesis we have, up to p-completion, |L_{em_{0},q,r}| ≃ BX(e, r, m_{0})(q^{e}) and so

|L_{em_{0},q,r}| × BGL_{m_{1}}(q^{e}) × ⋯ × BGL_{m_{s}}(q^{e}) ≃

BX(e, r, m_{0})(q^{e}) × BGL_{m_{1}}(q^{e}) × ⋯ × BGL_{m_{s}}(q^{e}) . (9)

Now consider the centralizer decomposition from [3, Theorem 2.6], obtaining

hocolim_{\overrightarrow{V ∈(F^{e})^{op}}} |C_{L_{n,q,r}}(V)| ≃ |L_{n,q,r}|

while using Equation (9) and [5, Proposition 11.3] we also get

hocolim_{\overrightarrow{V ∈(F^{e})^{op}}} |C_{L_{n,q,r}}(V)| ≃ BX(e, r, [n/e])(q^{e})

getting the result. □

We finally get one of the results which motivated this work.

Remark 6.4. Consider p a prime, q a prime power such that e, the order of q modulo p, is bigger than 2. Consider r a divisor of e such that r > 2. Then, by Proposition 6.1 we have the fibration (up to homotopy equivalence):

|L_{n,q,r}| ⟶ |L_{n,q}| ⟶ B(ℤ/r) .

By Theorem 6.3, we have

|L_{n,q,r}| ≃ BX(e, r, [n/e])(q^{e})

up to p-completion, and by [5, Proposition 11.5], when r > 2 and n ≥ ep, these are classifying spaces of exotic p-local finite groups. So we get examples of extensions of p-local finite groups where two of them correspond to finite groups and the third one is an exotic p-local finite group.

References

[1] J.L. Alperin and P. Fong, ‘Weights for Symmetric and General Linear Groups’, J. Algebra 131 (1990) n.1 2–22.

[2] C. Broto, N. Castellana, J. Grodal, R. Levi and R. Oliver, ‘Extensions of p-local finite groups’, Prepublicacions UAB 04/2005 (2005).

[3] C. Broto, R. Levi and R. Oliver, ‘The homotopy theory of fusion systems’, J. Amer. Math. Soc. vol. 16 (2003), no. 4, 779–856.

[4] C. Broto, R. Levi and R. Oliver, ‘The theory of p-local groups: a survey’, Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic K-theory, 51–84, Contemp. Math., 346, Amer. Math. Soc., Providence, RI, 2004.

[5] C. Broto and J.M. Møller, ‘Finite Chevalley versions of p-compact groups’, Prepublicacions UAB 15/2004 (2004).

[6] A. Díaz, A. Ruiz and A. Viruel, ‘All p-local finite groups of rank two for odd prime p’, Trans. Amer. Math. Soc. (to appear).

[7] R. Levi and R. Oliver, ‘Construction of 2-local finite groups of a type studied by Solomon and Benson’, Geom. Topol. 6 (2002), 917–990.

[8] D. Notbohm, ‘Topological realization of a family of pseudoreflection groups’, Fund. Math. 155 (1998), no. 1, 1–31.

[9] D. Quillen, ‘On the cohomology and K-theory of the general linear groups over a finite field’, Ann. of Math. (2) 96 (1972), 552–586.

[10] A. Ruiz and A. Viruel, ‘The classification of p-local finite groups over the extraspecial group of order p^{3} and exponent p. Math. Z. 248 (2004), no. 1, 45–65.

Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Cerdanyola del Vallès, Spain.

E-mail address: Albert.Ruiz@uab.cat