On simple endotrivial modules

Geoffrey R. Robinson, Institute of Mathematics, University of Aberdeen, Aberdeen AB24 3UE, Scotland g.r.robinson@abdn.ac.uk

Abstract

We show that when G is a finite group which contains an elementary Abelian subgroup of order p^{2} and k is an algebraically closed field of characteristic p, then the study of simple endotrivial kG-modules which are not monomial may be reduced to the case when G is quasi-simple. AMS Classification: 20C20

1 Introduction:

Let G be a finite group which contains an elementary Abelian p-subgroup of order p^{2}, and k be an algebraically closed field of prime characteristic p. The category of finite-dimensional kG-modules contains infinitely many isomorphism types of indecomposables, and this category is wild unless p = 2 and G has a Sylow 2-subgroup which is dihedral or semidihedral.

E.C. Dade introduced the notion of an endotrivial kG-module as an example of a type of kG-module which occurs naturally in some representation theoretic problems, but may be more amenable to a reasonable classification. A finite dimensional kG-module M is said to be endotrivial if the trivial module k is a direct summand of M ⊗ M^{∗} and is the the only non-projective summand of this module. Endotrivial modules have been studied quite intensively in recent years (see, for example, [1] and the references therein). Notice that if M is an endotrivial kG-module with dim_{k}(M) > 1, then no non-identity p-element of G acts trivially on M.

In Navarro-Robinson [5], a conjecture of J.F. Carlson, N. Mazza and J. Thévenaz [1] was proved, which completed the proof of the fact that if G as above is also p-solvable, then any simple endotrivial kG-module V is 1-dimensional. Here, we extend the Clifford theoretic analysis of [5] to reduce the study of simple endotrivial modules to the case of groups which are close to quasi-simple with one type of exception in which the situation is very well-controlled.

We will prove that if G is an arbitrary finite group containing an elementary Abelian p-subgroup of order p^{2}, and V is a faithful simple endotrivial kG-module, then there are just two possibilities:

i) F^{∗}(G) = Z(G)L, where L is quasi-simple of order divisible by p and L acts irreducibly on V.

ii) F^{∗}(G) = O_{p′}(G) = A is Abelian and there is a strongly p-embedded subgroup H of G such that V is induced from a 1-dimensional kH-module.

2 Some preliminaries and reductions

We first recall some now standard group-theoretic definitions and notation. A component of a finite group G is a quasi-simple subnormal subgroup. Distinct components of G centralize each other, and the central product of all components of G is denoted by E(G). The generalized Fitting subgroup of G is denoted by F^{∗}(G) and is the (central) product F (G)E(G). We always have C_{G}(F^{∗}(G)) = Z(F (G)).

The proper subgroup H of G is said to be strongly p-embedded in G if p divides |H| but H ∩ H^{g} is a p′-group for all g ∈ G\H. Strongly p-embedded subgroups play a large role in this paper, by virtue of Lemma 1, part iii), of [5].

Lemma 1: Let G be a finite group with a strongly p-embedded subgroup H. Then ∩_{g∈G}H^{g} ≤ O_{p′}(G), and equality holds if G contains a subgroup of type (p, p). Furthermore, in the latter case, O_{p′,p}(G) = O_{p′}(G).

Proof: Let N = ∩_{g∈G}H^{g}, let P ∈ Syl_{p}(H) and set Q = P ∩ N. If Q ≠ 1, then G = NN_{G}(Q) ≤ H. Hence |N| is not divisible by p. If G contains a subgroup R of type (p, p), then

O_{p′}(G) ≤ ⟨C_{G}(x) : x ∈ R^{#}⟩ ≤ H.

In that case, set N = O_{p′,p}(G). If N > O_{p′}(G), then N = O_{p′}(G)(P ∩ N) ≤ H, a contradiction as N ⊲ G and p divides |N|.

Remark: If G is a finite group with a strongly p-embedded subgroup H and O_{p′}(G) = 1 then F (G) = 1, and G has a unique minimal normal subgroup, which is simple. This is well-known, but notice that if M is a minimal normal subgroup of G, but is not simple, then G = MH by a Frattini argument, while if L, K are any two of the simple direct factors of M, then L ≤ C_{G}(K) ≤ C_{G}(P ∩ K) ≤ H. Hence M ≤ H as L was arbitrary, a contradiction.

Lemma 2: Let G be a finite group, V be an endotrivial simple kG-module, and let N be a subnormal subgroup of G of order divisible by p. Then Res_{N}^{G}(V) is simple. In particular, if N is a component of G, and V is faithful, but not 1-dimensional, then C_{G}(N) = Z(G), so N is the unique component of G.

Proof: The first claim is is clear if dim_{k}(V) = 1, so suppose that this is not the case. By induction, it suffices to treat the case that N is normal, so suppose that this is the case. Then since V is endotrivial, we may suppose that G acts faithfully on V, since the kernel of the action of G on V is a p′-group. Let U be a simple summand of Res_{N}^{G}(V) and set H = I_{G}(U) ≥ N. If H < G, then V is induced from a simple endotrivial kH-module. Since V is endotrivial, it follows from Lemma 1 of Navarro-Robinson [5] that H is strongly p-embedded, contrary to Lemma 1 above, as p divides |N|. Hence H = G.

Now there is a p′-central extension Ĝ of G such that U extends to a kĜmodule and Ĝ has a normal subgroup N̂ which may be identified with N, and V ≅ U ⊗ W, where W ≅ Hom_{kN̂}(U, V). Now both U and W are endotrivial by Lemma 1 of [5]. Since N acts trivially on W, some non-trivial p-element of Ĝ acts trivially on W, so that W must be 1-dimensional. Hence V restricts irreducibly to N, so C_{G}(N) = Z(G).

We state the next Lemma in more generality than needed here, since it may have application and interest beyond the question at hand.

Lemma 3: Let V be a primitive indecomposable endotrivial kG-module, where G is a finite group which contains an elementary Abelian subgroup Q of order p^{2}. Then O_{p′}(G) acts as scalars on V.

Proof: Let U be a simple summand of Res_{O_{p′} (G)}^{G}(V). Since V is primitive,

I_{G}(U) = G. Now there is a finite p′-central extension Ĝ such that U extends to a kĜ-module, and as such V ≅ U ⊗ W, where W = Hom_{kO_{p′} (Ĝ)}(U, V) (via u ⊗ f → uf for u ∈ U, f ∈ Hom_{kO_{p′} (Ĝ)}(U, V)). Then V is still an endotrivial

kĜ-module, so U and W are both endotrivial kĜ-modules by Lemma 1 of [5]. Since Ĝ contains a subgroup Q̂ isomorphic to Q and since U is an endotrivial simple kO_{p′}(Ĝ)Q̂-module, U must be 1-dimensional by the main result of [5].

Lemma 4: Let V be a faithful imprimitive simple endotrivial kG-module, where G is a finite group containing an elementary Abelian subgroup of order p^{2}. Then A = O_{p′}(G) is Abelian and either A = C_{G}(A) or A = Z(G). In the latter case, G has a component L of order divisible by p which acts irreducibly on V.

Proof: V is induced from a primitive simple kH-module, say U of some proper subgroup H of G. By Lemma 1 of [5], H is strongly p-embedded in G. Then A = O_{p′}(G) ≤ N = ∩_{g∈G}H^{g}. By Lemma 3 (applied within H, as U is primitive), A acts as scalars on U. Hence A is Abelian by Clifford’s Theorem. If C = C_{G}(A) > A, then |C| is divisible by p, so that C acts irreducibly on V by Lemma 2. In that case, A acts as scalars on V and A ≤ Z(G). Thus G has a component L of order divisible by p and Lemma 2 applies.

Hence we may suppose that C = A. Now, by Lemma 1 and the ensuing discussion, G/A has a corefree strongly p-embedded subgroup, and F^{∗}(G/A) is simple.

3 Proof of the main Theorem

Theorem: Let G be a finite group containing an elementary Abelian subgroup of order p^{2}, and let V be a faithful simple endotrivial kG-module. Then one of the following occurs:

i) E = E(G) is quasi-simple and acts irreducibly on V.

ii) O_{p′}(G) = A is Abelian, C_{G}(A) = A and V is induced from a 1-dimensional module of a strongly p-embedded subgroup H of G.

PROOF: If V is primitive, then O_{p′}(G) acts as scalars on V by Lemma 3. Hence G must have a component L of order divisible by p and case i) holds by Lemma 2.

We may thus suppose that V is imprimitive, and by Lemma 2, that G has no component of order divisible by p. Then V is induced from a primitive simple endotrivial kM-module W for some strongly p-embedded subgroup M of G. If W is 1-dimensional, we are done, so suppose that dim_{k}(W) > 1. Set A = O_{p′}(G). By Lemma 4, since G has no component of order divisible by p, we must have A = F^{∗}(G) and C_{G}(A) = A. Also A ≤ M.

Applying the earlier Lemmas to M in its action on W, we know that the kernel K of the action of M on W is a p′-group, that O_{p′}(M)/K acts as scalars on W, and that M/K has a unique component which acts irreducibly on W.

Set Ḡ = G/A, etc.. Then M̄ is still strongly p-embedded in Ḡ. Also A/K is central in M/K, so that M/K is a p′-central extension of M̄. Since M/K has a single component (of order divisible by p) and O_{p}(M/K) = 1, we know that M̄ also has a unique component and O_{p}(M̄) = 1.

The finite simple groups of order divisible by p which contain an elementary Abelian subgroup of order p^{2} and have a strongly p-embedded subgroup are carefully listed in 7.6.1 and 7.6.2 of (Gorenstein, Lyons and Solomon, [3]). Since O_{p}(M̄) = 1 most of these possibilities are excluded in the present situation. Also, since M̄ has a unique component of order divisible by p, the possibility that E(Ḡ) ≅ A_{2p} is also excluded. The only possibility that remains is that p = 5, E(Ḡ) ≅ F i_{22} and M̄ ∩ E(Ḡ) ≅ Aut(O_{8}^{+}(2)).

By Lemma 2, W is a (faithful) simple endotrivial module of dimension greater than 1 for a component, L say, of M/K. This component L is isomorphic to either O_{8}^{+}(2) or a double cover of O_{8}^{+}(2). From [4] (also from [2]) no such group has a non-trivial absolutely irreducible module in characteristic 5 whose dimension is congruent to ±1 (mod 25). This contradicts the fact that W is endotrivial of dimension greater than 1, so that dim_{k}(W)^{2} ≡ 1 (mod 25).

4 Concluding Remarks

For the sake of completeness, we prove that simple monomial endotrivial kGmodules of dimension greater than one exist for some groups G which are not quasi-simple.

LEMMA 5: Let X be a finite group of order divisible by p. Suppose that O_{p′}(X) = 1 and that X has a strongly p-embedded subgroup Y. Then there is a finite group G such that F^{∗}(G) = A is an Abelian p′-group with G/A ≅ X and there is a simple endotrivial kG-module which is induced from a 1-dimensional module of a proper subgroup of G.

PROOF: We might as well suppose that Y is a maximal subgroup of X, and we do so. Let q be a prime which does not divide |X| and set F = GF(q). Then the permutation module Ind_{Y}^{X}(F) has a non-trivial irreducible summand, say A. By Frobenius reciprocity, Y fixes a non-identity element of the Abelian group A, but X fixes no such element. By Brauer’s permutation Lemma, Y fixes a nontrivial linear character, say λ, of A, but X fixes no such linear character of A. Let G be the semidirect product AX. Then H = AY is a strongly p-embedded maximal subgroup of G and H is the inertial subgroup I_{G}(λ).

Let U be an irreducible kA-module with Brauer character λ. Now A = [A, Y] × C_{A}(Y), so that U extends to a 1-dimensional kH-module Ũ with [A, Y] in its kernel. We still have H = I_{G}(U). Let V be a composition factor of Ind_{H}^{G}(Ũ). Then V covers the kA-module U, so that V is induced from a simple kI_{G}(U)-module. Hence [G : H] divides dim_{k}(V). Thus Ind_{H}^{G}(Ũ) = V is simple.

Now Res_{H}^{G}(V) = Ũ ⊕ T for some projective kH-module T, as H is strongly p-embedded in G. Thus Res_{H}^{G}(V ⊗V^{∗}) = k ⊕W for some projective kH-module W. By Green correspondence, V ⊗ V^{∗} = k ⊕ S for some projective kG-module S. Thus V is endotrivial.

For the purpose of eliminating possible components L of finite groups G which contain an elementary Abelian subgroup of order p^{2}, yet have a faithful endotrivial kG-module which is not monomial, the following may be useful:

Lemma 6: Let L be a finite quasi-simple group of order divisible by p which is a component of a finite group G containing an elementary Abelian p-subgroup of order p^{2} and having a faithful simple endotrivial kG-module, say V. Let Q be a Sylow p-subgroup of G and P = Q ∩ L. Then there is faithful simple kL-module U whose dimension is congruent to ±1 (mod |P |) if p is odd, or congruent to ±1 (mod \frac{|P |}{2}) if p = 2. If P is cyclic, then |P | > p and there is a faithful simple kL-module U with dim_{k}(U) ≡ ±1 (mod p|P |).

PROOF: Notice that the hypotheses imply that Z(L) is a p′-group. In particular, notice that (by the Brauer-Suzuki theorem), P is not generalized quaternion. We set U = Res_{L}^{G}(V), which is simple by Lemma 2 (and is endotrivial). Since dim_{k}(V)^{2} ≡ 1 (mod |Q|), the stated congruences hold in all cases (notice that P ≠ Q by hypothesis when P is cyclic). Also, L contains an elementary Abelian subgroup of order p^{2} unless p is odd and P is cyclic. In that exceptional case, L̄ = L/Z(L) admits an outer automorphism α of order p, since F^{∗}(G) = LZ(G) by Lemma 2 (and since O_{p}(G) = 1). By the classification of finite simple groups, |P | > p in that case- we outline one argument: by a Frattini argument, we may suppose first that α normalizes NL̄(P̄). Another Frattini-type argument (using a Hall p′-subgroup of NL̄(P̄)) allows us to suppose that α normalizes a Hall p′-subgroup H̄ of NL̄(P̄). If |P | = p, then α is central in ⟨α⟩NL̄(P̄⟨α⟩), which is a Sylow p-normalizer of L̄⟨α⟩. Hence C_{⟨α⟩L̄}(α) controls strong p-fusion in ⟨α⟩L̄. Then [L̄, α] = 1, since O_{p′}(L̄⟨α⟩) = 1 and the odd analogue of Glauberman’s Z^{∗}-theorem holds. This is a contradiction, as α induces a non-trivial outer automorphism of L̄. Hence |P | > p and the proof is complete.

Acknowledgements: We are grateful to C.W. Parker for helpful exchanges and to M. Geck for computational assistance.

References

[1] J. F. Carlson, N. Mazza and J. Thévenaz, Endotrivial modules for p-solvable groups, to appear in Transactions of the American Mathematical Society.

[2] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.4.12,2008,(http://www.gap-system.org).

[3] D. Gorenstein, R. Lyons and R. Solomon, The Classification of the Finite Simple Groups, Number 3, American Mathematical Society Mathematical Surveys and Monographs, Volume 40, Number 3, AMS, Providence, 1998.

[4] C. Jansen, K. Lux, R.Parker and R.A.Wilson An atlas of Brauer characters, (with an appendix by T. Breuer and S. Norton), London Mathematical Society Monographs. New Series, 11. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1995.

[5] G. Navarro and G.R. Robinson, On endo-trivial modules for p-solvable groups, preprint, July 2010.