ON DUALITY INDUCING AUTOMORPHISMS AND SOURCES OF SIMPLE MODULES IN CLASSICAL GROUPS

RADHA KESSAR

1. Introduction

Let p be a prime number, k an algebraically closed field of characteristic p, P a finite p-group and W an indecomposable kP -module. Given a finite group G and an indecomposable kG-module V, we say that (P, W) is a vertex source pair for (G, V) if there is an inclusion P ↪ G of groups, under which P is a vertex of V and W is a source of V.

Endo-permutation modules occur frequently as sources of simple modules of finite groups. For instance, every simple module of a p-soluble group has endo-permutation source; if V is a simple kG-module, where G is a finite group, lying in a nilpotent block of kG, then V has endo-permutation source, and any 2-block of a finite group whose defect groups are isomorphic to the Klein 4-group posesses a simple module with endopermutation source.

By results of T. Berger-W.Feit and independently of L. Puig, the proof of which invokes the classification of finite simple groups, if W is an endo-permutation kP module which occurs as a source of a simple kG-module, for a p-soluble group G, then the class of W is a torsion element in the Dade group of P. Further, Mazza [8] has shown that any endopermutation kP -module whose isomorphism class is a torsion element of the Dade group of P and which satisfies certain structural constraints identified by Puig does occur as source of some simple module of some p-nilpotent group. By contrast, the question of which endo-permutation modules occur as sources of simple modules in simple, quasi-simple or almost simple groups has not been extensively studied.

The smallest interesting case is the case where P is elementary abelian of rank 2, since if P is a finite cyclic group, there are no non-torsion endo-permutation kP -modules. In this paper, we study two situations wherein simple modules of groups related to the finite classical groups having vertex source pairs (P, W), where P is elementary abelian of order p^{2} and where W is endo-permutation occur and we prove that in both cases, W must be a self-dual module and hence corresponds to an element of order at most 2 in the corresponding Dade group. Notation- Let H be a finite group. For a finite dimensional kH-module M, we denote by M^{∨} the contragradient dual of M, that is M^{∨} is the kH-module given by M^{∨} := Hom_{k}(M, k) as k-vector space, and h.α(m) = α(h^{−1}m), for α ∈ M^{∨}, h ∈ H and m ∈ M .

If ϕ : G → H is an isomorphism of groups, and M is a kG-module, we will denote by ϕM the kH-module Res_{ϕ^{−1}}M.

For a finite p-group, we denote by D_{k}(P) the Dade group of P over k. For the defintion and basic properties of the Dade group, we refer to [12]. Here, we merely recall that if W is an indecomposable kP -module, with vertex P, then the (isomorphism class) of W defines a unique element [W] of D_{k}(P), the identity of D_{k}(P) is [k], and that the inverse of the element [W] in D_{k}(P) is the element [W^{∨}]. In particular, if W is self-dual, then [W] has order at most 2 in D_{k}(P).

Date: July 8, 2008.

Theorem 1.1. Let L be a finite symmetric group S_{n} or one of the finite classical groups GL_{n}(q), GU_{n}(q), O_{2n+1}(q), O_{2n}^{+}(q), O_{2n}^{−}(q), or Sp_{2n}(q), where q is a prime power. Let G be a subgroup of L containing the derived subgroup of L and let Z be a subgroup of Z(G). Let P be an elementary abelian group of order p^{2}, and let W be an indecomposable endopermutation kP -module. Suppose that there exists a simple kG/Z module with (P, W) as vertex source pair. Then W ≅ W^{∨}, and if p is 2, then W ≅ k.

Before stating our next result, we recall the following standard facts from modular representation theory. Let G be a finite group and N a normal subgroup of G such that G/N is a p-group. Suppose that U is a simple projective kN-module whose isomorphism class is stable under the conjugation action of G on N. Then the kN-module structure on U extends uniquely to a kG-module structure. The kG module U is simple, and if (P, W) is a vertex source pair of (G, V), then P is isomorphic to G/N and W is an endo-permutation kP -module.

The question of whether any W that appears in the above context is torsion in the Dade group of P has been investigated by Salminen in [10] and [11] for odd primes p. He has reduced the problem to the case where P is elementary abelian of order p^{2} and where N is a central p′-extension of a projective special linear or unitary group, or p = 3 and N is a central extension of the simple group D_{4}(q).

In this paper, we prove that groups of type A do not pose a problem.

Theorem 1.2. Let p be an odd prime. Let G be a finite group with a normal subgroup N such that [G : N] is elementary abelian of order p^{2}. Suppose that N is a quasi-simple group, with Z(N) a p′-group and such that N/Z(N) is isomorphic to P SL_{n}(q) or to P SU_{n}(q) where q is a prime power not divisible by p. Suppose that U is a simple projective kNmodule which is G-stable and let (P, W) be a vertex source pair for the kG-module U. Then W ≅ W^{∨}.

Combining the above with Salminen’s work thus proves that if p ≥ 5, then for a finite group G, and a simple kG-module U with vertex-source pair (P, W) such that Res_{N}^{G}(U) is a simple projective kN-module for some normal subgroup N of p-power order, [W] is a torsion element in D_{k}(P). We remark that this result is a special case of a long standing conjecture on the finiteness of the number of source algebra equivalences of nilpotent blocks of finite groups, a proof of which has been recently announced by Puig.

The proof of both Theorem 1.1 and Theorem 1.2 is based on the following elementary proposition.

Proposition 1.3. Suppose that G is a finite group, M is an indecomposable kG module with vertex source pair (P, W) such that P is elementary abelian of order p^{2} and W is an endo-permutation kP -module. If there exists an automorphism ϕ : G → G such that ϕM ≅ M^{∨} as kG-modules, then W is self dual and if p = 2, W = k.

Theorem 1.1 is an easy consequence of the above proposition, once one observes that the symmetric and classical groups have automorphisms which invert conjugacy classes.

However, as will be explained in more detail later on in the paper, the proof of Theorem 1.2 is somewhat more subtle and relies on the following rather curious fact, which we state below. Notation- For our next result, G will denote either the general linear group GL_{n}(q) or the general unitary group GU_{n}(q) for some prime power q.

If G = GL_{n}(q) we set ϵ = 1 and if G = GU_{n}(q), we set ϵ = −1. If G = GL_{n}(q), set N := SL_{n}(q) and if G = GU_{n}(q), set N = SU_{n}(q). If q is a prime power not divisible by p such that q = q^{′p} for some q′, then if G = GL_{n}(q), we let ϕ : G → G be the automorphism (a_{ij}) → (a_{ij}^{q′}) and if G = GU_{n}(q), we let ϕ : G → G be the automorphism (a_{ij}) → (a_{ij}^{q′})^{t^{−1}}. Note that since N is ϕ-stable and G/N is cyclic, any subgroup I of G containing N is ϕ-stable, and we have a natural inclusion of groups I ⋊ ⟨ϕ⟩ ≤ G ⋊ ⟨ϕ⟩ . This is not, in general, a normal inclusion.

Theorem 1.4. With the notation above suppose that p is an odd prime such that p|q − ϵ and p|n. Suppose further that q = q^{′p} for some prime power q′. Let c be a block of kN with central defect group and let I(c) = Stab_{G}(c). If c is stable under ϕ, then I(c) ⋊ ⟨ϕ⟩ is a normal subgroup of G ⋊ ⟨ϕ⟩.

The paper is organized into six sections. In section 2, we prove Proposition 1.3 as well as some other general results which are needed for the proofs of the main theorems. In section 3, we will prove Theorem 1.1. Section 4 contains some block theoretic results which will be needed for the proof of Theorem 1.2. In section 5, we recall some facts from the representation theory of the finite general linear and unitary groups and their commutator subgroups and prove Theorem 1.4. Theorem 1.2 is proved in the final section.

Acknowledgments. We thank Cedric Bonnafé and Markus Linckelmann for many helpful discussions and suggestions and the referee for a careful reading of the manuscript.

2. A criterion for torsion

Notation- We keep the notation of the introduction. In addition, let (K, O, k) be a p-modular system.

Definition 2.1. Let H be a finite group and let V be a kH-module. Then V is automorphically dual if there exists an automorphism ϕ : H → H such that ^{ϕ}V ≅ V^{∨} := Hom_{k}(V, k).

Clearly, any self-dual module is automorphically dual. Our results are based on the observation that the converse holds for endo-permutation modules for elementary abelian p groups of order p^{2}.

Lemma 2.2. Let P be an elementary abelian p-group of order p^{2} and let W be an indecomposable endo-permutation kP -module with vertex P. If W is automorphically dual, then W is self-dual.

Proof. Let ϕ : P → P be an automorphism of P such that ^{ϕ}W ≅ W^{∨}. The result is an easy consequence of Dade’s classification of of endo-permutation modules for abelian p-groups [2, 10.1 and 12.5]. By this classification, if p = 2, then W is isomorphic to Ω^{n}(P) for some uniquely determined n ∈ ℤ whence ^{ϕ}W is isomorphic to Ω^{n}(k) and W^{∨} is isomorphic to Ω^{−n}(P). But we are given that ^{ϕ}W ≅ W^{∨}. Hence n = 0. Now suppose that p is odd. Then

[W] = [Ω^{n}(k) ⊗_{k} V]

for some n ∈ ℤ and some self-dual kP -module V, with n and [V] uniquely determined by W whence

[^{ϕ}W] = [Ω^{n}(k) ⊗_{k} ^{ϕ}V]

and

[W^{∨}] = [Ω^{−n}(k) ⊗_{k} V^{∨}] = [Ω^{−n}(k) ⊗_{k} V].

Thus, ^{ϕ}W ≅ W^{∨} implies that n = 0 thanks to the uniqueness of n and [V].

□

Lemma 2.3. Let M be an indecomposable kG module and let (S, V) be a vertex-source pair for M. Then (S, V^{∨}) is a vertex-source pair for M^{∨}.

Proof. Let H be a subgroup of G and let U be a kH-module such that M is a summand of Ind_{H}^{G}(U). Then M^{∨} is a summand of (Ind_{H}^{G}(U))^{∨} ≅ Ind_{H}^{G}(U^{∨}). The result follows □

Lemma 2.4. Suppose that G is a finite group, M is an indecomposable kG module with vertex source pair (P, W). If M is automorphically dual, then so is W.

Proof. Let ϕ : G → G be such that ^{ϕ}M ≅ M^{∨}. By Lemma 2.3, (P, W^{∨}) is a vertex source pair for the kG-module M^{∨}. On the other hand, by transport of structure, (^{ϕ}P, ^{ϕ}W) is a vertex source pair for the kG-module ^{ϕ}M. Thus, by hypothesis, both (P, W^{∨}) and (^{ϕ}P, ^{ϕ}W) are vertex source pairs for M^{∨}. So, there exists g ∈ G such that P = ^{gϕ}P and W^{∨} ≅ ^{gϕ}W. The result follows by considering the automorphism (y → gϕ(y)g^{−1}) of P. □

Now we can prove Proposition 1.3:

Proof. By hypothesis, M is automorphically dual. Hence by Lemma 2.4, W is automorphically dual. The result follows by Lemma 2.2 □

3. Proof of Theorem 1.1

We keep all the notation of the previous sections.

Lemma 3.1. Let G be a finite group and τ : G → G an automorphism such that τ(g) is conjugate to g^{−1} for all g ∈ G_{p′}. If N is a normal subgroup of G which is τ-stable, then every simple kN-module is automorphically dual.

Proof. First, let M be a simple kG-module amd let ϕ_{M} be the Brauer character of M. Then by hypothesis, the Brauer character ϕ_{^{τ} M} of ^{τ} M satisfies ϕ_{^{τ} M}(g) = ϕ_{M}(g^{−1}) = ϕ_{M^{∨}}(g) for all g ∈ G_{p′}. Hence M is automorphically dual. Now, let U be a simple kN-module and let M be a simple kG-module such that U is a summand of Res_{N}M. Then U^{∨} is a summand of Res_{N}M^{∨} and ^{τ} U is a summand of Res_{N} ^{τ} M. But as shown above, ^{τ} M ≅ M^{∨}, hence τ U and U^{∨} are covered by the same simple kG-module. Thus there exists g ∈ G such that gτ U ≅ U^{∨}. The result follows by considering the automorphism y → gτ(y)g^{−1} of N.

□ Lemma 3.2. Let L be a finite symmetric group S_{n} or one of the finite classical groups GL_{n}(q), GU_{n}(q), O_{2n+1}(q), O_{2n}^{+}(q), O_{2n}^{−}(q), or Sp_{2n}(q), where q is a prime power. Let G be a subgroup of L such that [L, L] ≤ G and let Z be a central subgroup of G. Every simple module kG/Z-module is automorphically dual.

Proof. First, note that since Z(G) is cyclic, it suffices to show that every simple kG-module is automorphically dual. In each case above, there is an automorphism τ : L → L such that τ(x) is conjugate to x^{−1} for all x ∈ L: If L is a symmetric group, then take τ to be the identity map. If L is an orthogonal group, or L is a symplectic group Sp_{2n}(q), where q is a power of 2, then by results of Gow (Lemma 2.1 and Lemma 2.2 of [6]) and Wonenburger [13], every element of L is conjugate to its inverse, hence again τ may be taken to be the identity map. If L = Sp_{2n}(q) where q is odd, then by Lemma 2.1(b) of [6], every element of L is the product of two skew symplectic involutions. Hence, we may take τ to be conjugation by any fixed skew-symplectic involution. If L = GL_{n}(q) we take τ to be the transpose inverse map. Finally, if L = GU_{n}(q) we let τ be the map which raises every entry of every matrix to its q-th power (considering GU_{n}(q) as the fixed point subgroup of GL_{n}(𝔽̄_{q}) under the map which sends a matrix (a_{ij}) to the matrix ((a_{ij}^{q}))^{−t}). It is easy to check that in each case G is τ-stable. Now the result follows from Lemma 3.1. □

Now we can prove Theorem 1.1:

Proof. Immediate from Lemma 3.2 and Proposition 1.3. □

4. Some results from block theory

Notation We keep the notation of the previous sections. Let G be a finite group. By a block of kG (or of OG), we will mean a primitive idempotent of the center Z(kG) of the group algebra kG (or of the center Z(OG) of the group algebra OG).

Lemma 4.1. Let H be a finite group, and let N, J be normal subgroups of H such that N ≤ J. Suppose that H/J is a p-group and that J/N is a cyclic p′-group. Suppose further that b is an H-stable block of kN of defect 0. Then there is a block f of kJ which is H stable and such that bf = f. Further, if J/N is central in H/N, then any block f of kJ such that bf = f is H-stable.

Proof. Let W be the unique simple kNb-module. Then W is H-stable and hence W is J-stable. Now J/N being a cyclic p′-group means that W extends in [J : N] ways to a kJ-module and these are all the simple kJ-modules covering W. Let S be the set of isomorphism classes of these extensions. Since W is H-stable, H acts on S by conjugation. Since the normal subgroup J of H is clearly in the kernel of this action, H/J acts on S. But H/J is a p-group and |S| = [J : N] is prime to p, so this action must have a fixed point. In other words, there exists a kJ-module V such that Res_{N}^{J}(V) ≅ W and such that xV ≅ V for all x ∈ H. Let f be the block of kJ containing V. Then f has the required properties.

Now, suppose that J/N is central in H/N, let V be as above be an H-stable extension of W to J, and let U be any extension of W to J. Then, there is a 1-dimensional kJ/Nmodule Z such that U ≅ V ⊗_{k} Inf_{J/N}^{J}(Z), where Inf_{J/N}Z is the inflation to J, via the canonical map J → J/N, and where V ⊗_{k} Inf_{J/N}^{J}(Z) has the kJ-module structure given by x.(v⊗z) = xv⊗xz, for x ∈ J , v ∈ V and z ∈ Inf_{J/N}^{J}(Z). Since J/N is central in H/N, gJZ ≅ ^{gJ}Z, and hence ^{g}U ≅ U for any g ∈ H. The result now follows as above. □

In the sequel, we will use the following fact about block idempotents without comment: If N is a normal subgroup of a finite group G, such that G/N is a p-group, then any block of kG is a central idempotent of kN, and consequently, if b is a block of kN which is stable under the conjugation action of G on N, then b is a block of kG.

Lemma 4.2. Let L be a finite group, and let G be a subgroup of L and N, J be normal subgroups of L. Suppose that L = JG, J ∩ G = N, that G/N is a p-group and that J/N is a cyclic p′-group. Suppose further that b is an L-stable block of kN of defect 0 and that f is an L-stable block of kJ such that bf = f. Then,

(i) The blocks kGb and kLf are nilpotent.

Furthermore, letting U be the unique (upto isomorphism) kGb-module and V the unique (upto isomorphism) simple kLf-module, and letting P be a defect group of kGb, we have

(ii) Res_{G}^{L}(V) ≅ U.

(iii) P is a defect group of kLf.

(iv) The map

kGb → kLf, a → af

is an interior P -algebra isomorphism, and induces an isomorphism of interior P -algebras between a source algebra of kGb and that of kLf.

(v) There is a pair (P, W) which is a vertex-source pair of the kG-module U and which is also a vertex-source pair for the kL-module V.

Proof. Since G/N is a p-group, and b is a G-stable block of kN, it is clear that kGb is a nilpotent block, and that P ≅ G/N. Since J/N is a p′ group, and f covers b, kJf is a block of defect 0. Also, L/J = GJ/J ≅ G/N, and G/N is a p-group, so it follows that kLf is nilpotent with defect groups isomorphic to G/N ≅ P. Now, b = Tr_{P}^{G}(x), for some x ∈ (kG)^{P}. Since f ∈ Z(kL) this yields

f = bf = Tr_{P}^{G}(xf).

The fact that [L : G] is prime to p, then gives

f = Tr_{G}^{L}(\frac{1}{[L : G]}f) = Tr_{P}^{L}(\frac{1}{[L : G]}xf),

from which it follows that P is contained in a defect group of kLf. But P is isomorphic to the defect groups of kLf, proving (iii).

Now, Res_{N}^{G}(U) is the unique (upto isomorphism) simple kNb-module, so the fact that b is L-stable, means that Res_{N}^{G}(U) is J-stable. Since J/N is a cyclic p′-group, Res_{N}^{G}(U) extends in precisely [J : N] ways to a kJ-module, each of which has the same dimension as U, and furthermore, these [J : N] extensions are the only simple kJ-modules whose restriction to N contain direct summands isomorphic to Res_{N}^{G}(U). On the other hand, Res_{J}^{L}(V) is a simple kNf-module, so the fact that fb = f means that Res_{J}^{L}(V) is one of these [J : N] extensions. Thus,

Res_{N}^{G}(Res_{G}^{L}(V)) = Res_{N}^{J}(Res_{J}^{L}(V)) = Res_{N}^{G}(U).

But U is the unique extension of Res_{N}^{G}(U) to a kG-module, hence Res_{G}^{L}(V) ≅ U, proving (ii).

We now prove (iv). Note that the second assertion of (iv) is an immediate consequence of the first assertion of (iv) and (iii). Thus, it suffices to prove that the map a → af is an isomorphism of interior P -algebras between kGb and kLf. Since f is a central idempotent of kL, the map is clearly a P -algebra homomorphism. Furthermore, since by (i), (ii) and (iii) both kLf and kGb are nilpotent, with defect groups of the same order and with simple modules of the same k-dimension, it follows that kLf and kGb have the same k-dimension, thus it suffices to prove that the map a → af is injective. So, let a ∈ kGb be such that af = 0. Let P be the projective cover of the simple kLf-module V. Then af = 0 implies that aP = afP = 0. But it is easy to see that Res_{G}^{L}(P) is a projective cover of U. Thus, a annihilates the unique projective kGb-module, which means akGb = 0, that is a = 0. Thus, the map a → af is injective, finishing the proof of (iv) Statement (v) is a consequence of (iv). □

Lemma 4.3. Let G be a finite group with O_{p}(G) = 1. Let N be a normal subgroup of G. If G/N is a p-group, then C_{G}(N) = Z(N) is a p′-group. Consequently, the canonical homomorphism of G/N into Out(N) is injective.

Proof. Let P be a Sylow p-subgroup of G. Then G = NP from which it follows that C_{P} (N) is a normal subgroup of G (as C_{P} (N) is normalised both by N and by P) . By hypothesis, C_{P} (N) = 1. Thus, C_{G}(N) is a p′-group from which it also follows that C_{G}(N) ≤ Z(N). □

Lemma 4.4. Let N, G, G′ be finite groups such that both G and G′ contain N as a normal subgroup. Let γ_{G} : G → Aut(N) and γ_{G′} : G′ → Aut(N) be the canonical homomorphisms and let π : Aut(N) → Out(N) be the canonical surjection. Suppose that G/N and G′/N are both p-groups, that O_{p}(G) = 1 = O_{p}(G′), and that kN has a G-stable block of defect

0. If π ∘ γ_{G}(G) and π ∘ γ_{G′}(G′) are conjugate subgroups of Out(N), then G and G′ are isomorphic through an isomorphism which stabilises N. Furthermore, if π ∘ γ_{G}(G) = π ∘ γ_{G′}(G′), then the above isomorphism may be chosen to be the identity on N.

Proof. Since G is a p extension of N and b is G-stable, b is a block of kG. Let P be a defect group of kGb. Then G = NP, a semi-direct product. Set P_{0} := γ_{G}(P) and let

ϕ : G → N ⋊ P_{0}

be defined by

ϕ(nx) = nγ_{G}(x), n ∈ N, x ∈ P.

By Lemma 4.3, it follows that γ_{G} induces an isomorphism between P and P_{0}, that ϕ is an isomorphism and that γ_{G}(G) = Inn(N)P_{0}, a semidirect product.

Now G and G′ both contain N. Hence, by hypothesis, there exists α ∈ Aut(N) such that

γ_{G′}(G′) = ^{α}γ_{G}(G) = Inn(N)^{α}P_{0}.

If π ∘ γ_{G}(G) = π ∘ γ_{G′}(G′), then choose α to be the identity.

Let U be the full inverse image of ^{α}P_{0} in G′ through γ_{G′} and let S be a Sylow psubgroup of U. By Lemma 4.3, applied to G′, γ_{G′} induces an isomorphism between S and αP_{0}. Now,

γ_{G′}(NS) = Inn(N)^{α}P_{0} = γ_{G′}(G′)

and by Lemma 4.3 Ker(γ_{G′}) = Z(N) ≤ N, hence G′ = NS. On the other hand,

|N||S| = |N||^{α}P_{0}| = |Z(N)||Inn(N)||^{α}P_{0}| = |Ker(γ_{G′})||γ_{G′}(G′)| = |G′|.

Thus G = NS, a semidirect product and the map

φ : G′ → N ⋊ ^{α}P_{0}

defined by

φ(nx) = nγ_{G′}(x), n ∈ N, x ∈ S,

is an isomorphism.

Let

ψ : N ⋊ P_{0} → N ⋊ ^{α}P_{0}

be defined by

ψ(nx) = α(n)^{α}x, n ∈ N, x ∈ P_{0}.

Then, φ^{−1} ∘ ψ ∘ ϕ : G → G′ is an isomorphism with the required properties. □

Remarks. (i) The condition that kN has a G-stable block of defect 0 in the above may be replaced by the weaker condition that N has a complement in G.

(ii) The above proposition may be also understood more structurally as a consequence of the fact that, as Z(N) is an abelian p′-group, and as G/N (respectively G′/N) is a p-group, restriction from G/N (respectively G′/N) induces an injective map from H^{2}(G/Z(N), Z(N)) (respectively G′/N) to H^{2}(N/Z(N), Z(N)) and of the fact that G/Z(N) and G′/Z(N) are isomorphic groups.

5. On characters and blocks of finite general linear and unitary groups and their commutator subgroups

The aim of this section is to prove Theorem 1.4. For this, we recall some facts from the Lusztig parametrization of characters of finite groups of Lie type in our special situation. We will follow [1] and [3]. Notation. We keep the notation of the previous sections and that introduced for the statement of Theorem 1.4. In particular, G will denote either a finite general linear group or a finite unitary group. For s a semi-simple element of G, we let [s] denote the Gconjugacy class of s.

5.A. On the ordinary characters of G. For s a semi-simple element of G, we let E(G, [s]) denote the rational Lusztig series corresponding to [s]. The subsets E(G, [s]) as [s] runs over the semi-simple classes of G partition the set of ordinary irreducible characters of G; if an irreducible character χ belongs to the subset E(G, [s]), we will say that χ has semisimple label [s]. Now, let ϕ be the automorphism of G which appears in the statement of Theorem 1.4 (so ϕ is only defined when q is a p-th power).

The above labelling of characters is compatible with the action of ϕ in the following sense.

Lemma 5.1. Let χ be an irreducible character of G. Let [s] be the semi-simple label of χ and [t] be the semi-simple label of ^{ϕ}χ. Then [s] = [ϕ(t)].

Proof. This is immediate from Corollary 2.4 of [9]. □

5.B. On the blocks of kG. For the rest of this section, we will assume that p|q −ϵ. Then by the Fong-Srinivasan classification of the blocks of the finite general linear and unitary groups [4], two ordinary irreducible characters χ and χ′ of G lie in the same p-block of G if and only if the the p′-parts of s and of s′ are conjugate in G where [s] and [s′] are the semi-simple labels of χ and χ′ respectively. Thus the p-blocks of G are partitioned by G conjugacy classes of p′-semi-simple elements of G. If b is a block of G, the p′-part of the semi-simple label of whose irreducible characters is the conjugacy class [t], we say that b has semi-simple label [t]. If b has label [t], then any Sylow p-subgroup of C_{G}(t) is a defect group of b.

5.C. On characters of N. Let ψ be an ordinary irreducible character of N, let χ be an irreducible character of G covering ψ and let [s] be the semi-simple label of χ. Let I(ψ) be the stabiliser of ψ in G. If χ′ is another character of G covering ψ, the semisimple label of χ′ is [sz] for some z ∈ Z(G) and conversely, for any z ∈ Z(G), there exists some irreducible character χ′ of G with label [sz] covering ψ(see for instance [1, Proposition 11.7]). If we set d_{[s]} to be the number of distinct conjugacy classes of G of the form [sz], where z ∈ Z(G), we get that [I(ψ) : N] is divisible by d_{[s]} (see [1, Corollary 11.13]).

Now let s be as above. Multiplication yields a transitive action of Z(G) on the set of conjugacy classes of G of the form [sz], z ∈ Z(G). Denote by Z(s) the stabiliser of [s] under this action, so that d_{[s]} = \frac{q−ϵ}{|Z(s)|} and the number of G-conjugates of ψ divides |Z(s)|.

Suppose z ∈ Z(s). If G = GL_{n}(q), then s being semi-simple there is a diagonal matrix diag(α_{1}, ⋯, α_{n}), α_{i} ∈ 𝔽̄_{q}^{×}, such that s is conjugate to diag(α_{1}, ⋯, α_{n}) in GL_{n}(𝔽̄_{q}). Thus sz is conjugate to diag(ζα_{1}, ζα_{2}, ⋯, ζα_{n}), where z = diag(ζ, ⋯, ζ), ζ ∈ 𝔽_{q}^{×}. Hence diag(α_{1}, α_{2}, ⋯, α_{n}) is conjugate to diag(ζα_{1}, ⋯, ζα_{n}). Taking determinants, one sees that ζ^{n} = 1. Furthermore, letting o(s) denote the order of s, we have that 1 = (diag(α_{1}, α_{2}, ⋯, α_{n}))^{o(s)} is conjugate to z^{o(s)}, hence z^{o(s)} = 1. Of course z^{q−ϵ} = 1. Thus Z(s) is a cyclic group of order dividing gcd(n, q − 1, o(s)). Arguing similarly when G = GU_{n}(q) and noting that Z(GU_{n}(q)) consists of scalar matrices of order dividing q +1, we obtain that Z(s) is a cyclic group of order dividing gcd(n, q + 1, o(s)).

Summarising the above discussion we get :

Proposition 5.2. Let χ be an ordinary irreducible character G and let [s] be the semisimple label of χ. Let ψ be an irreducible character of N covered by χ.

(i) The number of G-conjugates of ψ is a divisor of |Z(s)|, and I(ψ)/N has order divisible by \frac{q−ϵ}{|Z(s)|}.

(ii) |Z(s)| is a divisor of gcd(n, q − ϵ, o(s)).

We also get an analogous result for blocks:

Proposition 5.3. Let b be a block of G and let [t] be the semi-simple label of b. Let c be a block of N covered by b. If b′ is a block of G covering c, then the semi-simple label of b′ is of the form [tz] for some z ∈ Z(G); the number of elements in the G-orbit of c is a divisor of |Z(t)| and in particular, this number is prime to p.

Proof. Since b and b′ cover the same blocks of N, there is a sequence b =: b_{1}, b_{2}, ⋯, b_{r} := b′ of blocks of G and ordinary characters χ_{i} in b_{i}, 1 ≤ i ≤ r such that for each i, 1 ≤ i ≤ r−1, χ_{i} and χ_{i+1} cover a common character of N. Let [s_{i}] be the semi-simple label of χ_{i}, 1 ≤ i ≤ r. Then from the above discussion, it follows that for each i, 1 ≤ i ≤ r − 1 there is a z_{i} ∈ Z(G) such that [s_{i+1}] = [s_{1}z_{i}]. Setting t_{i} to be the p′-part of s_{i} and v_{i} to be the p′-part of z_{i}, it follows that [t_{i+1}] = [t_{i}v_{i}] for all i, 1 ≤ i ≤ r − 1. This proves the first assertion.

The block b contains an irreducible character, say χ with semi-simple label [t] and c contains a character, say ψ covered by χ. Since I(b) contains I(ψ), the second statement follows from Proposition 5.2(i). Since t is a p′-element, the last assertion is immediate from Proposition 5.2(ii). □

In what follows we will identify GL_{n}(q) with the group of invertible linear transformations of an n-dimensional vector space over a field of q-elements, and identify GU_{n}(q) with a subgroup of GL_{n}(q^{2}) in the natural way. As before, we assume that p divides q − ϵ . In addition, we assume from now on that q = q^{′p}, and ϕ is the automorphism of G appearing in the statement of Theorem 1.4.

Lemma 5.4. Let t be a semi-simple element of GL_{n}(q) and suppose that the minimal polynomial of t over 𝔽_{q} has an irreducible factor whose degree is distinct from the degrees of all other irreducible factors of the minimal polynomial of t. Suppose that [ϕ(t)] = [tz] for some z = diag(η, η, ⋯, η) ∈ Z(GL_{n}(q)). Then, |Z(t)||q′ − 1.

Proof. Let p(x) be an irreducible factor of the minimal polynomial of t with degree distinct from that of every other irreducible factor. Let λ be a root of p(x) in 𝔽̄_{q}. and let r be the order of λ. We claim that there exists a positive integer v such that v is relatively prime to p and such that the order r is a factor of (q^{′v} − 1)(q − 1). Indeed, by hypothesis, λ^{q′} is an eigen value of tz, i.e. λ^{q′} = λ′η for some eigen value λ′ of t. The minimal polynomials of λ and λ^{q′} over 𝔽_{q} have the same degree and the minimal polynomials of λ′η and λ′ over 𝔽_{q} also have the same degree. Hence, it follows that λ′ is a root of p(x), that is λ^{q′} = λ^{q^{u}}η for some u. This gives λ^{q′(q^{′pu−1}−1)} = η ∈ 𝔽_{q}, whence λ^{(q^{′pu−1}−1)} ∈ 𝔽_{q}. The claim follows by setting v = pu − 1.

Now, let y = diag(ζ, ζ, ⋯, ζ) ∈ Z(t), and let λ be a root of p(x). We claim that ζ = λ^{q^{m}−1}, for some integer m. Indeed, the eigen values of yt are of the form ζα, where α is an eigen value of t. Hence [t] = [yt] implies λζ is also an eigen value of t. Again, since ζ ∈ 𝔽_{q}, the minimal polynomial of λζ over 𝔽_{q} has the same degree as the minimal polynomial of λ over 𝔽_{q} which means that λζ is a root of p(x), so λζ = λ^{q^{m}} for some m, proving the claim.

Since ζ^{q−1} = 1, it follows from the claim that r is a divisor of (q^{m} − 1)(q − 1) = (q^{′pm} − 1)(q − 1). Combining this with what we showed previously, it follows that r is a factor of gcd((q^{′pm} − 1)(q − 1), (q^{′v} − 1)(q − 1)) = (q^{′gcd(m,v)} − 1)(q − 1), the last equality holding because v is relatively prime to p. Since q^{m} − 1 is divisible by q^{′gcd(m,v)} − 1, and by q − 1 and since gcd(q^{′gcd(m,v)} − 1, q − 1) = q′ − 1, this gives that r is a factor of (q^{m} − 1)(q′ − 1). Thus,

ζ^{(q′−1)} = λ^{(q^{m}−1)(q′−1)} = 1.

Thus Z(t) has order dividing q′ − 1. □

We need an analogous result for the unitary groups.

Lemma 5.5. Let t be a semi-simple element of GU_{n}(q), and suppose that t has an eigen value λ satisfying the following:

For any eigen value λ′ and any element diag(ζ, ζ, ⋯, ζ) ∈ Z(GU_{n}(q)) such that either λ^{−q′} or λ equals ζλ′, we have λ′ = λ^{(−q)^{m}} for some non-negative integer m.

If [ϕ(t)] = [tz] for some z = (η, η, ⋯, η) ∈ Z(GU_{n}(q)), then |Z(t)||q′ + 1.

Proof. Let r be the order of λ. We claim that there exists a positive integer v such that v is relatively prime to p and such that the order r is a factor of ((−q′)^{v} − 1)(q + 1). Indeed, by hypothesis, λ^{−q′} = λ′η for some eigen value λ′ of t, and some η ∈ 𝔽̄ such that η^{q+1} = 1. Thus, λ^{−q′} = λ^{(−q)^{u}}η for some u. This gives λ^{q′((−q)^{′pu−1}−1)} = η, and η^{q+1} = 1. The claim follows by setting v = pu − 1.

Now, let y = diag(ζ, ζ, ⋯, ζ) ∈ Z(t). So [t] = [yt] implies λζ is also an eigen value of t. Again, the nature of λ gives that λζ = λ^{(−q)^{m}} for some m. Since ζ^{q+1} = 1, it follows from the above that r is a divisor of ((−q)^{m} − 1)(q + 1) = ((−q′)^{pm} − 1)(q + 1). Combining this with what we showed previously, it follows that r is a factor of gcd((−q′)^{pm} − 1)(q + 1), ((−q′)^{v} − 1)(q + 1)).

We claim that d := gcd(q + 1, (−q′)^{v} − 1) = q′ + 1. Indeed suppose first that v is even, say v = 2w. Then (−q′)^{v} = q^{′v} and d is a factor of gcd(q^{′2p} − 1, q^{′2w} − 1) = q^{′2} − 1, the last equality holding because v is relatively prime to p. Thus d|q^{′2} − 1 and d|q^{′p} + 1. Now gcd(q′ − 1, q^{′p} + 1) = 1 if q′ is even and is 2 otherwise. On the other hand, since p is odd, \frac{q^{′p}+1}{q′+1} is odd if q is odd. The claim follows. Suppose next that v is odd. In this case, (−q′)^{v} = −q′^{v}, and d = gcd(q + 1, q^{′v} + 1). Again, the fact that v is relatively prime to p, and that p is odd, will imply the claim.

Since q + 1 is a factor of (−q′)^{pm} − 1, it follows from the claim that gcd(((−q′)^{pm} − 1)(q + 1), ((−q′)^{v} − 1)(q + 1)) is a factor of (−q′)^{pm} − 1)(q′ + 1), from which we get that

ζ^{(q′+1)} = λ^{((−q′)^{pm}−1)(q′+1)} = 1.

The above shows that Z(t) has order dividing q′ + 1. □

Notation - For any positive integer m, let m_{+} denote the p-part of m and let m_{−} denote the p′ part of m.

Lemma 5.6. Suppose that p|(q − 1) and p|n. Suppose also that SL_{n}(q) has a block c with central defect group. Let b be a block of GL_{n}(q) covering SL_{n}(q), let [t] be the semi-simple label of b, and let f(x) be the characteristic polynomial of t. One of the following holds:

(i) f(x) is irreducible and n_{+} ≤ (q − 1)_{+}.

(ii) n_{+} ≥ (q − 1)_{+}, and f(x) is a product f(x) = p_{1}(x)p_{2}(x), where p_{1}(x) and p_{2}(x) are irreducible polynomials such that deg(f_{1}(x)) ≠ deg(f_{2}(x)) and neither deg(f_{1}(x)) nor deg(f_{2}(x)) is divisible by p.

In particular, f(x) has an irreducible factor whose degree is distinct from any other irreducible factor of f(x).

Proof. Let f(x) = ∏_{1≤i≤u} p_{i}(x)^{m_{i}} be a prime factorisation of f(x) in 𝔽_{q}[x]. Let R be a defect group of the block b of kGL_{n}(q). Since p divides q − 1, R is conjugate to a Sylow p-subgroup of C_{G}(t). Then R ∩ SL_{n}(q) is a defect group of kSL_{n}(q)c. But by hypothesis, the Sylow p-subgroup of Z(SL_{n}(q)) is the (unique) defect group of kSL_{n}(q)c. Since the Sylow p-subgroup of Z(SL_{n}(q)) is a cyclic group of order d_{+}, where d = gcd(n, q − 1) and since R/R ∩ SL_{n}(q) is a cyclic group of order dividing (q − 1), it follows that R is meta-cyclic and has p-rank at most 2. Now by the prime decomposition of f(x) above, if n_{i} is the degree of p_{i}(x), 1 ≤ i ≤ u we have

C_{GL_{n}(q)}(t) ≅ ∏ GL_{m_{i}}(q^{n_{i}}).

Since p divides q − 1, the fact that R is a Sylow p-subgroup of C_{GL_{n}(q)}(t) and that R is metacyclic, forces u ≤ 2 and either m_{i} = 1 for all i,1 ≤ i ≤ u or u = 1, and m_{1} = 2.

Suppose u = 1 and that f(x) is irreducible and |R| = (q^{n} − 1)_{+} = (q − 1)_{+}n_{+}. Suppose, if possible that n_{+} > (q − 1)_{+}. Then |R| > (q − 1)d_{+}, a contradiction. Hence in this case, case (i) of the proposition holds.

Next, suppose u = 2, so that f(x) = p_{1}(x)p_{2}(x), and R ≅ R_{1} × R_{2} where R_{i} is a Sylow p subgroup of GL_{m_{i}}(q^{n_{i}} − 1), for i = 1, 2. So, we get

(q − 1)_{+}d_{+} ≥ |R| = (q^{n_{1}} − 1)_{+}(q^{n_{2}} − 1)_{+} = (q − 1)_{+}^{2}n_{1}_{+}n_{2}_{+},
d_{+} = (q − 1)_{+}n_{1}_{+}n_{2}_{+},

which implies that n_{+} ≥ (q − 1)_{+} and that n_{1} and n_{2} are not divisible by p. Now, if n_{1} = n_{2}, then n would be a p′ number, a contradiction. Finally, consider the case that u = 1 and m_{1} = 2. Then f(x) = p_{1}^{2}(x). The same argument as above will lead to a contradiction.

The last assertion is immediate from the description of f(x). □

Notation - For p(x) ∈ 𝔽_{q^{2}}[x] an irreducible polynomial, different from x, we let p_{−}(x) be the irreducible polynomial over 𝔽_{q^{2}}[x] whose roots are of the form λ^{−q}, where λ is a root of p(x).We say that p(x) is of unitary type if p(x) = p_{−}(x) and we say that f(x) is of non-unitary type otherwise.

Note that if t is a semi-simple element of GU_{n}(q), and if f(x) is the characteristic polynomial of t over 𝔽_{q^{2}}, then for any irreducible p(x) ∈ 𝔽_{q^{2}}[x], the multiplicity of p(x) as a divisor of f(x) is the same as that of p_{−}(x).

Lemma 5.7. Suppose that p|(q +1) and p|n. Suppose also that c is a block of SU_{n}(q) with central defect group. Let b be a block of GU_{n}(q) covering SU_{n}(q), let [t] be the semi-simple label of b, and let f(x) be the characteristic polynomial of t over 𝔽_{q^{2}}. One of the following holds:

(i) f(x) is irreducible, of unitary type and n_{+} ≤ (q + 1)_{+}.

(ii) n_{+} ≥ (q + 1)_{+}, and f(x) is a product f(x) = p_{1}(x)p_{2}(x), where p_{1}(x) and p_{2}(x) are irreducible polynomials of unitary type such that deg(p_{1}(x)) ≠ deg(p_{2}(x)) and neither deg(f_{1}(x)) nor deg(f_{2}(x)) is divisible by p.

(iii) f(x) = p(x)p_{−}(x) where p(x) is irreducible of non-unitary type.

(iv) f(x) = p(x)p_{−}(x)g(x)g_{−}(x), where p(x) and g(x) are irreducible polynomials of non-unitary type such that deg(p(x)) ≠ deg(g(x)) and neither deg(p(x)) nor deg(g(x)) is divisble by p.

(v) f(x) = p(x)g(x)g_{−}(x), where p(x) is of unitary type and g(x) is of non-unitary type. In particular, t has an eigen value λ satisfying the conditions of Proposition 5.5.

Proof. The proof is entirely similar to that for the linear case. Let

f(x) = ∏_{1≤i≤u} p_{i}(x)^{m_{i}} ∏_{1≤i≤v}[(g_{i}(x)(g_{i_{−}}(x)]^{n_{i}}

be a prime factorisation of f(x) in 𝔽_{q}[x], where the p_{i}(x) are of unitary type and the g_{i}(x) are of non-unitary type.

Then, if d_{i} is the degree of p_{i}(x), 1 ≤ i ≤ u and e_{i} is the degree of g_{i}(x), 1 ≤ i ≤ v, we have (see for instance, Proposition 1A of [4])

C_{GU_{n}(q)}(t) ≅ ∏_{i} GU_{m_{i}}(q^{d_{i}}) ∏_{i} GL_{n_{i}}(q^{2e_{i}}).

Furthermore, if R is a defect group of the block b of GU_{n}(q) then since p divides q +1, R is conjugate to a Sylow p-subgroup of C_{G}(t). On the other hand, R is metacyclic of order at most (q + 1)^{2}. Now proceeding as for the general linear groups, we get that f(x) satisfies one of (i)-(v). Now, if f(x) is of types (i) , (ii) , (iii) or (iv) take for λ a root of f(x). Let λ′ be any eigen value of f(x) and let ζ ∈ 𝔽_{q^{2}} satisfy ζ^{q+1} = 1. Since 𝔽_{q^{2}}[λ^{−q′}] = 𝔽_{q^{2}}[λ] and 𝔽_{q^{2}}[λ′] = 𝔽_{q^{2}}[ζλ′], if either λ or λ^{−q′} equal ζλ′, then 𝔽_{q^{2}}[λ′] = 𝔽_{q^{2}}[λ], and the degree constraints on (i)-(iv) will yield that λ′ = λ^{(−q)^{m}} for some m. If f(x) is of type (v), take for λ a root of p(x). Let λ′ be any eigen value of f(x) and let ζ ∈ 𝔽_{q^{2}} be such that ζ^{q+1} = 1 and such that either λ or λ^{−q′} equals ζλ′. Since λ is 𝔽_{q^{2}}-conjugate to λ^{−q}, λ^{−q′} is 𝔽_{q^{2}}-conjugate to (λ^{−q′})^{−q}, hence the same is true of λ′ζ and of λ′. But p(x) is the only unitary factor of f(x), hence λ′ = λ^{(−q)^{m}} for some m. □

We now prove Theorem 1.4.

Proof. Assume the conditions of Theorem 1.4 hold. Let b be a block of G covering c and let [t] be the semi-simple label of b. Since c is ϕ-stable, ϕ^{−1}(b) also covers c. By Lemma 5.1, ϕ^{−1}(b) has semi-simple label [ϕ(t)]. Thus, by Proposition 5.3 we have that [ϕ(t)] = [tz] for some z = diag(η, η, ⋯, η) ∈ Z(G). It follows from Lemmas 5.4, 5.5, 5.6 and 5.7 that |Z(t)| is a factor of q′ − ϵ. On the other hand, by Proposition 5.3 [G : I(c)] is a factor of |Z(t)|. This means that if h ∈ G is such that det(h) = α^{q′−ϵ} for some α ∈ 𝔽_{q}, then h ∈ I(c). Now let g ∈ G and set h = ϕ(g)g^{−1}. Then det(h) = det(g)^{q′−ϵ}, hence h ∈ I. But this means exactly that I(c) ⋊ ⟨ϕ⟩ is normal in G ⋊ ⟨ϕ⟩. □

6. Proof of Theorem 1.2

Notation- We keep the notation of Theorem 1.2. In addition, if the simple quotient of N is P SL_{n}(q), let K_{0} := GL_{n}(q) and let N_{0} := SL_{n}(q). If the simple quotient of N is P SU_{n}(q), let K_{0} := GU_{n}(q) and let N_{0} := SU_{n}(q). Set Z := Z(K_{0}).

If q = q^{′p} and K_{0} = GL_{n}(q), we define ϕ : K_{0} → K_{0} to be the automorphism (a_{ij}) → (a_{ij}^{q′}). If q = q^{′p} and K_{0} = GU_{n}(q), we define ϕ : K_{0} → K_{0} be the automorphism (a_{ij}) → (a_{ij}^{q′})^{t^{−1}}. Note that ϕ is an automorphism of K_{0} of order p, N_{0} is ϕ-stable, and since K_{0}/N_{0} is cyclic any subgroup of K_{0} containing N_{0} is also ϕ-stable.

If K_{0} = GL_{n}(q) we set ϵ = 1 and if K_{0} = GU_{n}(q), we set ϵ = −1.

For an abelian group H, we will let H_{+} denote the Sylow p-subgroup of H and let H_{−} denote the Hall p′-subgroup of H.

We first prove the following result detailing the structure of G.

Proposition 6.1. With the notation and assumptions of Theorem 1.2, suppose that W is not self dual. Then there exists a subgroup Z_{0} of Z(N_{0}) containing the Sylow p-subgroup of Z(N_{0}) such that N is isomorphic to N_{0}Z_{+}/Z_{+}Z_{0}. Further, q = q^{′p} for some prime power q′, the index of N_{0}Z_{+}/Z_{+}Z_{0} in K_{0}/Z_{+}Z_{0} is divisible by p and letting M be the unique subgroup of K_{0}/Z_{+}Z_{0} containing N_{0}Z_{+}/Z_{+}Z_{0} as a subgroup of index p, there exists an isomorphism, G ≅ M ⋊ ⟨ϕ⟩ sending N to N_{0}Z_{+}/Z_{+}Z_{0}.

Proof. We make a series of reductions.

6.2. O_{p}(G) = 1.

Proof. If not, then W is isomorphic to Inf_{P/O_{p}(G)}^{P}W′ for some endo-permutation module for the cyclic group P/O_{p}(G), and clearly W is self-dual.

6.3. p is a divisor of gcd(q − ϵ, n) and q = q^{′p} for some prime power q′.

Proof. By Lemma 4.3, G/N is isomorphic to a subgroup of Out(N). Since N is quasi-simple, Out(N) is in turn isomorphic to a subgroup of Out(N/Z(N)). In particular, Out(N/Z(N)) has an elementary abelian subgroup of order p^{2}. Since p is odd, the result is immediate from the nature of the outer automorphism groups of P SL_{n}(q) and of P SU_{n}(q) [5, Theorem 2.5.1].

6.4. N is isomorphic to N_{0}/Z_{0}, where Z_{0} is a central subgroup of N_{0} containing the Sylow p-subgroup of Z(N_{0}).

Proof. By 6.3, and since p is odd, N/Z(N) is not one of the groups P SL_{2}(4), P SL_{3}(2), P SL_{3}(4), P SL_{4}(2), or P SL_{2}(9), P SU_{4}(2), P SU_{6}(2) or P SU_{4}(3). Hence, the exceptional part of the Schur multiplier of N/Z(N) is trivial (see [5, Table 6.1.3]). By [5, Table 6.1.2], N_{0} is a universal covering group of N/Z(N) . By[5, Corollary 5.1.5]), N is a quotient of N_{0} by a central subgroup, say Z_{0}. Finally, since Z(N) is assumed to be a p′-group, it follows that Z_{0} contains the Sylow p-subgroup of Z(N_{0}).

It follows from 6.3 that |K_{0}/Z_{+}Z_{0} : N_{0}Z_{+}/Z_{+}Z_{0}| is divisible by p and that ϕ is defined. Let M be the unique subgroup of K_{0}/Z_{+}Z_{0} containing N_{0}Z_{+}/Z_{+}Z_{0} as a subgroup of index p.

6.5. There exists an isomorphism, G ≅ M ⋊ ⟨ϕ⟩ sending N to N_{0}Z_{+}/Z_{+}Z_{0}.

Proof. Since Z_{0} contains Z_{+} ∩ N_{0}, the inclusion of N_{0} in K_{0} induces an isomorphism between N_{0}/Z_{0} and N_{0}Z_{+}/Z_{+}Z_{0}. Henceforth, we will identify N with the subgroup N_{0}Z_{+}/Z_{+}Z_{0} of K_{0}/Z_{+}Z_{0}.

Let G′ = M ⋊ ⟨ϕ⟩. Then, clearly O_{p}(G′) = 1 and G′/N is elementary abelian of order p^{2}. Now, since p is odd it follows from the structure of the outer automorphism group of P SL_{n}(q) and of P SU_{n}(q) that Out(N) has metacyclic Sylow p-subgroups. On the other hand, for odd p, metacyclic p-groups have at most one elementary abelian subgroup of order p^{2} (see for instance [7, Lemma 2.1]). Thus, using the notation of Lemma 4.4, π ∘ γ_{G}(G) and π ∘ γ_{G′}(G′) are conjugate subgroups of Out(N). The claim follows from Lemma 4.4. □

Notation- If N/Z(N) ≅ P SL_{n}(q), let τ : GL_{n}(q) → GL_{n}(q) be the transpose automorphism. If N/Z(N) ≅ P SU_{n}(q), let τ : GU_{n}(q) → GU_{n}(q) be the automorphism which raises every entry to the q-th power.

We now prove Theorem 1.2.

Proof. Suppose if possible that W is not self-dual. By Proposition 6.1, we may assume that G = M ⋊⟨ϕ⟩, where Z_{0} and M are as in the statement of Proposition 6.1. We identify N with N_{0}Z_{+}/Z_{+}Z_{0} as before.

Let c be the block (necessarily of defect 0) of kN containing the simple kN-module U and let c_{0} be the unique block of kN_{0}Z_{+} whose image under the canonical surjection of kN_{0}Z_{+} onto kN is c. Let I be the inertial subgroup of c in K_{0}/Z_{0}Z_{+} and let I_{0} be the the inertial subgroup of c_{0} in K_{0}.

6.6. For any g ∈ K_{0}, g^{ϕ}g^{−1} ∈ I_{0}. Further, K_{0}_{+} ≤ I_{0}.

Proof. Since Z_{+} is a central p-group, c_{0} is also a block of kN_{0}. Furthermore, since c is a block of defect 0 of kN_{0}Z_{+}/Z_{+}Z_{0}, as block of kN_{0}, c_{0} has central defect group. Thus, I_{0} is the inertial subgroup in K_{0} of a central defect block of kN_{0}. By hypothesis, U is G-stable, from which it follows that c_{0} is ϕ-stable. The first assertion follows from Theorem 1.4 applied to K_{0}, N_{0} and c_{0} and the second follows from Proposition 5.3.

Set J = N_{0}I_{0}_{−}Z_{+}/Z_{+}Z_{0} and set L = M_{0}I_{0}_{−}/Z_{+}Z_{0} ⋊⟨ϕ⟩, where M_{0} is the inverse image of M in K_{0}. Then we have the following diagram of group inclusions:

[figure: (I_{0}/Z_{+}Z_{0}) ⋊ ⟨ϕ⟩; L = M_{0}I_{0}_{−}/Z_{+}Z_{0} ⋊ ⟨ϕ⟩; G = M ⋊ ⟨ϕ⟩; J = N_{0}I_{0}_{−}Z_{+}/Z_{+}Z_{0}; N = N_{0}Z_{+}/Z_{+}Z_{0}]

Note that J and N are normal in (I_{0}/Z_{+}Z_{0}) ⋊ ⟨ϕ⟩, that (I_{0}/Z_{+}Z_{0}) ⋊ ⟨ϕ⟩/J is a p-group and that J/N is isomorphic to a quotient of I_{0}_{−}/I_{0}_{−} ∩ N_{0}Z_{+}, hence is a cyclic p′-group, thus by Lemma 4.1 (applied with H = (I_{0}/Z_{+}Z_{0}) ⋊ ⟨ϕ⟩) there is a block f of kJ such that f = bf and such that f is (I_{0}/Z_{+}Z_{0}) ⋊ ⟨ϕ⟩-stable.

Further, we see that the conditions of Lemma 4.2 hold, hence for some W′, (P, W′) is a vertex-source pair for the unique simple module, say V, of kLf and (P, W′) is also a vertex-source pair of the kG-module U. We may assume without loss that W′ = W.

Note that K_{0} acts by conjugation on J.

6.7. The group K_{0}_{−} acts transitively on the K_{0}-orbit of f.

Proof. By choice of f, I_{0}/Z_{+}Z_{0}, and hence I_{0} stabilizes f. On the other hand, by 6.6 I_{0} contains K_{0}_{+}. The claim follows.

6.8. K_{0}_{−} normalizes L.

Proof. Let g ∈ K_{0}_{−}. By 6.6, g^{ϕ}g^{−1} ∈ I_{0}, but since g ∈ K_{0}_{−}, in fact g^{ϕ}g^{−1} ∈ I_{0}_{−}, proving the claim.

6.9. The simple kL-module V is automorphically dual.

Proof. Let e be a block of K_{0}/Z_{+}Z_{0} such that ef^{∨} ≠ 0. Then ^{τ} e^{τ} f^{∨} ≠ 0. But by Lemma 3.1 and its proof, ^{τ} e = e^{∨} and e^{∨}f ≠ 0, from which it follows that f and ^{τ} f^{∨} are covered by the same block of K_{0}, namely e^{∨}. By 6.7 we get that ^{τ} f^{∨} = ^{g}f for some g ∈ K_{0}_{−}. Now, by definition, τ and ϕ commute as automorphisms of K_{0}, hence the action of τ on K_{0} extends to an automorphism of K_{0} ⋊ ⟨ϕ⟩, and the group L is clearly invariant under this automorphism.

Let ω : L → L be the map x → g^{−1}τ(x)g, x ∈ L. By 6.8, ω is well defined and is an automorphism of L. The claim follows by setting ψ = ω^{−1}.

Now, as observed above (P, W) is a vertex source pair of the kL-module V. So, by 6.9 and by Proposition 1.3 (applied to the simple kL module V), it follows that W is self-dual.

□

References

[1] C. Bonnafé, Sur les caractères des groupes réductifs finis á centre non connexe : applications aux groupes spéciaux linaires et unitaires, Astérisque 306(2006).

[2] E.C. Dade, Endo-permutation modules over p-groups,II Ann. of Math 108 (1978), 317–346.

[3] F. Digne and J. Michel Representations of finite groups of Lie type, London Mathematical Society Student Texts 21 (1991).

[4] P. Fong and B. Srinivasan, The blocks of finite general linear and unitary groups, Invent. Math. 69 (1982) 109–153.

[5] D.Gorenstein, R. Lyons and R. Solomon, The classification of finite simple groups 3, Mathematical Surveys and Monographs 40, American Mathematical Society (1998).

[6] R. Gow, On the Schur indices of characters of finite classical groups , J. London Math. Soc (2) 24 (1981), no. 1, 135-147.

[7] N. Mazza, The Dade group of a metacyclic p-group, J. Algebra 266(2003), 102-111.

[8] N. Mazza, Endo-permutation modules as sources of simple modules, J. Group Theory 6 (2003), 477-497.

[9] G.Navarro, P.H. Tiep and A.Turull, Brauer characters with cyclotomic field of values, Journal of Pure and Applied Algebra 212 (2008),628–635.

[10] A. Salminen, On the sources of simple modules in nilpotent blocks, (preprint).

[11] A. Salminen, On the sources of simple modules in nilpotent blocks II, (preprint).

[12] J. Thevenaz, Endo-permutation modules: A guided tour, Group representation theory(M. Geck, D. Testerman,J,. Thevenaz), EPFL press, (2007), 115-147.

[13] M. Wonenburger, Transformations which are products of two involutions , J. Math. Mech. 16 (1966) 327-338.

Department of Mathematics, University of Aberdeen, Aberdeen, AB243UE

  E-mail address: kessar@maths.abdn.ac.uk