Contemporary Mathematics

Toward a classification of endotrivial modules

Jon F. Carlson

Abstract. This paper presents a brief survey of some recent work on a classification of endotrivial modules over a finite group. Two particular new developments are highlighted. The first is a characterization of the kernel of the restriction map of the group of endotrivial modules to certain subgroups by Balmer and a subsequent conjecture by the author and Thévenaz. The second is a program led by Lassueur and Malle to classify the simple endotrivial modules over quasi-simple groups.

1. Introduction

An endotrivial module is one whose endomorphism ring over the base field of the algebra is isomorphic to the trivial module in the stable category of modules modulo projectives. For the case that the algebra is the group algebra of a p-group, they were defined by Dade who viewed them as the building blocks of endopermutation modules which he showed are the sources of the simple modules for p-nilpotent groups [28]. Puig proved the same for p-solvable groups. For any group algebra kG, tensoring with an endotrivial module gives an equivalence of the stable category of kG-modules. Thus they form an important part of the Picard group of selfequivalences of the stable category. They are the modules whose deformation rings are universal as opposed to just versal [6]. They play an important role in the block theory of finite groups [44], and are related to Borel-Smith functions on the Burnside ring [9] and to the Picard group of line bundles on the Brown and Quillen complexes of chains of p-subgroups of a finite group [4]. There are many reasons for wanting to understand these modules.

In this paper we survey efforts to classify endotrivial modules over the modular group algebra of a finite group. For notation, let G be a finite group, and let k be a field of characteristic p > 0. For convenience we assume that k is algebraically closed. The isomorphism classes of endotrivial kG-modules in the stable category form a group which is denoted T (G). In the case that G is a p-group, the endotrivial

\begin{matrix}c \\ ◯ \end{matrix}0000 (copyright holder)

2000 Mathematics Subject Classification. 20C20.

Research partially supported by NSA grant H98230-15-1-0007.

modules have been completely classified and T (G) is known. For groups which are not p-groups, the main questions focus on the restriction homomorphism T (G) → T (S), where S is the Sylow p-subgroup of G. While the image of this map is occasionally in question, the more pressing question is how to determine the kernel K(G). This survey concentrates on two aspects of this question that have been the subject of several papers in the last few years. The reader is directed to the survey papers [11, 48] for more details of earlier works. It is not difficult to show that K(G) is isomorphic to the dual of a quotient of the normalizer of S in G, and there are some hints as to what this quotient might be. In addition, there have been several efforts to compute the group K(G) for quasi-simple groups. One method in particular involves the application of character theory to determine simple modules that are endotrivial.

Basic definitions and some fundamental results are contained in the next section. It is followed by a brief discussion of the history of the subject and a section on the role of the Green correspondence. In Sections 5 and 6, we discuss the recent results on the kernel of the restriction of the group of endotrivial module to subgroups containing the Sylow p-subgroup of G. The final section presents a brief survey of some of the work to classify the irreducible endotrivial modules over quasi-simple groups.

The author is grateful to the organizers of the conference ”Finite Simple Groups: Thirty Years of the Atlas and Beyond” at Princeton University in November of 2015, for the opportunity to present this work.

2. Definitions and preliminaries

Throughout the paper, it is assumed that G is a finite group and that k is an algebraically closed field of characteristic p. Unless otherwise stated, all modules are assumed to be finitely generated left kG-modules. The tensor product of two modules M and N is M ⊗ N = M ⊗_{k} N with G acting diagonally. Likewise Hom_{k}(M, N) is a kG-module with G acting by conjugation. Recall that if N is a projective module, then M ⊗ N is a projective module for any module M.

The stable category stmod(kG) of finitely generated modules modulo projectives has as objects all finite dimensional kG-modules. If M and N are kG-modules, then the morphisms in the stable category are given by

\underline{Hom}_{kG}(M, N) = \frac{Hom_{kG}(M, N)}{PHom_{kG}(M, N)},

where PHom_{kG}(M, N) is the subspace of kG-homomorphisms that factor through a kG-projective module. Thus a projective kG-module is isomorphic to the zero module in the stable category.

Definition 2.1. A kG-module M is endotrivial if its endomorphism algebra

Hom_{k}(M, M) = End_{k}(M) ≅ k ⊕ proj

as kG-modules. That is, M is endotrivial if its k-endomorphism ring in the stable category is a trivial module.

Note that Hom_{k}(M, N) ≅ M^{∗} ⊗ N where M^{∗} ≅ Hom_{k}(M, k) is the k-dual of M. Thus End_{k}(M) ≅ M^{∗} ⊗_{k} M, and if M and N are endotrivial modules so is M ⊗ N. The group of endotrivial modules T (G) is defined to be the group with elements the isomorphism classes [M] of endotrivial modules M in the stable category. Thus [M ⊕ proj] = [M], where by ⊕ proj we mean the direct sum with some (or any) projective module. The operation on T (G) is given by [M] + [N] = [M ⊗ N]. So T (G) is an abelian group, with identity [k] and inverses given by −[M] = [M^{∗}].

It is obvious that the trivial module k is an endotrivial module. In addition, if M has dimension one, then End_{k}(M) = End_{kG}(M) is a trivial module. So M is endotrivial.

For other examples we need the following notation. If M is a kG-module, then Ω(M) is the kernel of the projective cover P → M. Also Ω^{2}(M) = Ω(Ω(M)), etc. Similarly Ω^{−1}(M) is the cokernel of the injective hull M → Q, and Ω^{−2}(M) = Ω^{−1}(Ω^{−1}(M)), etc. On the stable category, Ω is a well defined functor. Indeed, stmod(kG) is a triangulated category and the translation functor or shift functor is Ω^{−1}.

The calculus of these operations, which is not difficult to prove, is that for any kG-modules M and N and for any integers m and n, Ω^{m}(M) ⊗ Ω^{n}(N) ≅ Ω^{m+n}(M ⊗ N) ⊕ proj, and (Ω^{m}(M))^{∗} ≅ Ω^{−m}(M^{∗}). With this in mind, it can be seen that any syzygy Ω^{n}(M) of an endotrivial module M is an endotrivial module.

A couple of results are worth noting for their importance in understanding the nature of endotrivial modules. The first is in the style of Quillen’s Dimension Theorem (see Chapter 8 of [26]), showing the significance of the elementary abelian p-subgroups in the representation theory of finite groups. Its proof requires some understanding of group cohomology. See Lemma 2.9 of [22].

Theorem 2.2. A kG-module M is endotrivial if and only if its restriction M_{E} to every elementary abelian p-subgroup E is an endotrivial module. In particular, a kG-module is endotrivial if and only if its restriction to a Sylow p-subgroup S of G is endotrivial.

The first statement is a consequence of Chouinard’s theorem (see Theorem 8.2.12 of [26]), that a kG-module is projective if and only if its restriction to every elementary abelian p-subgroup is projective. It is easy to see that the restriction of an endotrivial module to any subgroup is again an endotrivial module. On the other hand if M is a kG-module whose restriction to every elementary abelian p-subgroup is endotrivial, then for any such E, Hom_{k}(M, M)_{↓E} ≅ k ⊕ P for P a projective kE-module. But P is the kernel of the trace map Hom_{k}(M, M) → k and hence it is also a kG-module. Its restriction to E must be projective because E is endotrivial. So by Chouinard’s Theorem, P is a projective module and the trace map splits.

In view of the above result and the fact that there is a classification of endotrivial modules over p-group (more details follow) a great deal of recent research has been pointed at characterizing the kernel of the restriction T (G) → T (S) where S is a Sylow p-subgroup of G. We say that a kG-module M has trivial Sylow restriction if its class is in the kernel of the restriction to S (i. e. M_{↓S} ≅ k ⊕ proj).

We note one other very useful result.

Theorem 2.3. (See Lemma 2.6 of [42].) If G has a nontrivial normal psubgroup, then any kG-module with trivial Sylow restriction has dimension one.

The proof is an easy application of the Mackey Theorem. That is, if M is an endotrivial module and Q is a nontrivial p-subgroup of the Sylow p-subgroup S of G, then M is a direct summand of the induced module k_{S}^{↑G}, but Q acts trivially on k_{S}^{↑G}, by the Mackey Theorem and hence (k_{S}^{↑G})_{↓S} has no projective summands. Thus, M_{↓S} ≅ k has dimension 1.

3. A little history

For p-groups, endotrivial modules were first defined by Dade [28] who classified them in the case that G is an abelian p-group. Dade’s interest was in endopermutation modules. A kG-module M is an endopermutation module if End_{k}(M) ≅ P, where P is a permutation module. It is not difficult to show that the endotrivial modules are the building blocks of the endopermutation modules. Dade showed that for p-nilpotent groups, the sources (in the sense of J. A. Green’s vertices and sources) of the irreducible modules are endopermutation modules. This means that for these groups, the irreducible modules are direct summands of modules induced from endopermutation modules. This theorem was soon extended to p-solvable groups by Puig [45]. Puig also demonstrated other roles of the endotrivial modules in block theory (see [44, 48]).

As regards endotrivial modules, Dade proved the following.

Theorem 3.1. [29, Theorem 10.1] If G is a noncyclic abelian p-group then T (G) ≅ ℤ and is generated by [Ω(k)].

Dade’s theorem led to an effort to classify the endotrivial modules over p-groups. A milestone was the theorem of Alperin [2] establishing the rank of the torsion-free part of the group. If the the group G has p-rank 2, then the torsion-free rank of T (G) is the number of conjugacy classes of maximal elementary abelian p-subgroups. If the p-rank of G is greater than two, then the torsion-free rank of T (G) is one plus the number of conjugacy classes of maximal elementary abelian p-subgroups of rank two. Alperin’s method was to construct a collection of relative syzygies of the trivial module that generate a torsion-free subgroup of maximal possible rank. The result was generalized to arbitrary finite groups in [13], using homological methods.

A characterization of the group of endotrivial modules over an arbitrary p-group was accomplished by the author and Jacques Thévenaz in a series of papers [22, 23, 24]. The most difficult of these papers was [23], in which the torsion part of T (G) was determined. Specifically, it was shown that for G a p-group, the group T (G) is torsion-free except in the cases that G is cyclic or that p = 2 and G is quaternion or semidihedral. If G is a cyclic p-group of order greater than 2, then T (G) is generated by the class of Ω(k), but because Ω^{2}(k) ≅ k, T (G) ≅ ℤ/2ℤ. For the quaternion and semidihedral groups, there are actually exotic endotrivial modules. If G is quaternion, then Ω^{4}(k) ≅ k and T (G) ≅ ℤ/4ℤ ⊕ ℤ/2ℤ where the first summand is generated by the class of Ω(k). For the semidihedral group, T (G) ≅ ℤ ⊕ ℤ/2ℤ.

In [24], it is proved that Alperin’s relative syzygies generate the torsion-free part of the group. A different construction of the generators of the torsion-free part of T (G) is given in [10] using group cohomology. In [27], Coşkun uses Borel-Smith functions to prove again that Alperin’s relative syzygies generate the torsion-free part of T (G) (see also [9]).

Building on the classification of endotrivial modules for p-groups, Bouc [8] proved a classification of endopermutation modules over p-groups using his theory of biset functors. A classification of the endotrivial modules over other group has been studied by many others, including the author, Christine Bessenrodt, Dave Hemmer, Shigeo Koshitani, Caroline Lassueur, Frank Lübeck, Gunter Malle, Nadia Mazza, Dan Nakano, Gabriel Navarro, Geoff Robinson, Elisabeth Schulte, Jacques Thévenaz in various combinations [5, 12, 13, 14, 15, 16, 17, 18, 19, 32, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 49]. There are even a few results on the group of endotrivial modules over infinitesimal group schemes by the author and Dan Nakano [20, 21].

Among other things, from these works, the image of the restriction map T (G) → T (S), where S is the Sylow p-subgroup of G, is reasonably well understood. In only a few cases, where the torsion-free rank of T (G) is greater than one, the generators of the torsion-free part are not completely known. The torsion part of T (S) is always in the image of the restriction map [18, 42]. Hence, the main problem is:

Problem: Compute the kernel K(G) of the restriction T (G) → T (S).

That is, we wish to find all kG-modules with trivial Sylow restriction.

4. The kernel of restriction and Green correspondents

From this point on in this narrative, we concentrate efforts to discover the kernel of the restriction map T (G) → T (S). That is, what are the kG-modules with trivial Sylow restriction? The kernel is denoted K(G). One tool for understanding the kernel is the Green correspondence. This survey does not presume to give a thorough explanation of the Green correspondence, but rather only explores its implications to the subject at hand. The interested reader should consult [47, Chapter 3] for more details.

Theorem 4.1. Let H be a subgroup of G that contains a Sylow p-subgroup S. If M is an indecomposable endotrivial kG-module, then M_{↓H} ≅ U ⊕ proj for U an indecomposable endotrivial kH-module, and M is a direct summand of the induced module U^{↑G} ≅ kG ⊗_{kH} U.

The theorem, as stated, is mostly true of any kG-module M. That is, under the assumption that H contains a Sylow subgroup of G, the restriction to H of any kG-module M has a direct summand whose induction back to a kG-module has M as a direct summand. The point here is that if M is an indecomposable endotrivial module then its restriction to H is also endotrivial and hence has a unique (up to isomorphism) indecomposable direct summand U that is endotrivial. The emphasis here is that M is a direct summand of the induction of an endotrivial kH-module.

One of the implications of the theorem is that if M is an indecomposable endotrivial module whose restriction to a subgroup H that contains S is the direct sum of a trivial module and a projective module (i. e. M_{↓H} ≅ k ⊕ proj), then M is a direct summand of k(G/H) = k_{H}^{↑G} the permutation module with point stabilizer H.

In addition, if it is assumed that H contains the normalizer in G of the Sylow subgroup S, then we can invoke the Green correspondence [30, Theorem 1] to obtain an even more precise statement. Namely, M is the unique endotrivial module that is a direct summand of U^{↑G}. Specifically, we have the following.

Theorem 4.2. Suppose that H is a subgroup of G and that H contains the normalizer of a Sylow p-subgroup of G. Let U be an endotrivial kH-module. Then U^{↑G} ≅ M ⊕W, where every direct summand of W is induced from a proper subgroup of S. In other words, there is only one direct summand of U^{↑G} that has a chance of being an endotrivial module and it is called the Green correspondent of U. If M is endotrivial then M_{↓H} ≅ U ⊕ proj.

Thus the problem of characterizing endotrivial modules can be rephrased as below. We recall Theorem 2.3, which says that if G has a normal p-subgroup, then an indecomposable module representing any class in K(G) has dimension one. Thus, in the situation of the above theorem, if H has a normal p-subgroup, then the module U has dimension one.

Problem (rephrased): Suppose that Q is any characteristic subgroup of S. Then N = N_{G}(Q) contains N_{G}(S). Which one dimensional kN-modules have Green correspondents that are endotrivial?

Nothing in this section is new or difficult if one understands the basics of the Green correspondence. However, it is important for the next development.

5. The kernel of restriction to the Sylow subgroup

Paul Balmer [3] found a method for computing the kernel of the restriction map T (G) → T (H) in the case that H ⊆ G is a subgroup that contains a Sylow psubgroup of G. The basic idea can be expressed as follows. Given a module M in Kernel{T (G) → T (H)}, and g ∈ G with p dividing |H ∩ gHg^{−1}|, we can form the diagram in the stable category:

\xymatrix{M \ar[r]^{conjugation by g} \ar[d]_{_{res_{G,H∩gHg^{−1}}}} & M \ar[d]^{_{res_{G,g^{−1}Hg∩H}}} \\ k \ar[r]_{a_{g}} & k}

where “res” means restriction from and to the indicated subgroups, and by “conjugation by g”, we mean the isomorphism that takes M to g ⊗ M. The restricted module M_{H∩gHg^{−1}} is isomorphic to the trivial module in the stable category. Thus the conjugation induces an isomorphism on the lower row of the diagram, and that isomorphism is multiplication by a nonzero scalar a_{g}.

Balmer defines an object that he calls a “weak H-homomorphism”. Then he shows that the assignment g → a_{g} (g → 1, if p fails to divide the order of H ∩gHg^{−1}) is a weak H-homomorphism. The definition is as follows. For notation, k^{×} is the multiplicative group of nonzero elements of k.

Definition 5.1. Assume that the subgroup H contains a Sylow p-subgroup S of G. A function u : G → k^{×} is a weak H-homomorphism if it satisfies the three conditions:

(a) if h ∈ H, then u(h) = 1,

(b) for g ∈ G, if |H ∩^{g}H| has order prime to p, then u(g) = 1, and

(c) if a, b ∈ G and p divides |H ∩^{a}H ∩^{ab}H| then u(ab) = u(a)u(b).

In addition, for f any weak H-homomorphism, it can be shown that the image of the map ψ(f) : k(G/H) → k(G/H) given by

ψ(f)(gH) = \frac{1}{|G : H|} ∑_{x∈G/H} f(g)gxH.

is an endotrivial kG-module in the kernel of the restriction map T (G) → T (H). Here we are viewing k(G/H) as the permutation module with point stabilizer H. Balmer proves that the weak homomorphism associated to the endotrivial module Image(ψ(f)) is again f. The ultimate result is the following.

Theorem 5.2. [3, Theorems 2.8, 2.9] Let A(G, H) be the set of all weak Hhomomorphisms. Then A(G, H) is a group under multiplication and is isomorphic to the kernel of the restriction map T (G) → T (H).

An approximation to the group A(G, H) is easily programmed for the computer. In many cases of quasi-simple groups, the method quickly demonstrates that A(G, H) is trivial. Or more to the point, when H = S, it shows that K(G) = A(G, S) is trivial. The approximation is recursive and requires a computational system that can compute Sylow subgroups and normalizers and intersections of subgroups. The author has implemented this program in the system Magma [7].

To make the computation easier, we assume that H = S, the Sylow p-subgroup of G. We define a sequence of functions ρ_{G,S}^{i} = ρ^{i}, for i ≥ 1, from the collection of nontrivial p-subgroups of S to the set of subgroups of G. For i = 1, and any Q ⊆ S, Q ≠ {1}, let

ρ^{1}(Q) = [N_{G}(Q), N_{G}(Q)]S′,

where [H, H] denotes the commutator subgroup of a group H and S′ is a Sylow p-subgroup of N_{G}(Q). For i > 1, inductively, let ρ^{i}(Q) be the subgroup generated by N_{G}(Q) ∩ ρ^{i−1}(Q′) for all Q′, nontrivial p-subgroups of S. For each i and each such Q, we have that ρ^{i−1}(Q) ⊆ ρ^{i}(Q) ⊆ N_{G}(Q). It is not difficult to show that at each step ρ^{i}(G) is in the kernel of every weak S-homomorphism. As a result we have the following.

Theorem 5.3. [25, Corollary 4.4] Let S be a Sylow subgroup of G. If for some i, ρ^{i}(S) = N_{G}(S), then K(G) = Kernel{T (G) → T (S)} is trivial.

The theorem is used by the author, Mazza and Nakano [16] to prove such results as that T (SL(2e, q)) ≅ ℤ for q not divisible by p, where e is the least integer such that p divides q^{e} − 1 and e > 1. This is an important step in the determination of the endotrivial modules for groups G of Lie type A in the case that p is not the defining characteristic of G.

6. Another viewpoint

Balmer’s method emphasizes that K(G) is written into the structure of the group G. Considered from the standpoint of the Green correspondence, this is not a surprise. Indeed, let N = N_{G}(S). Recall that if M is an endotrivial module whose class is in the kernel of the restriction T (G) → T (S), then M_{↓N} ≅ χ ⊕ proj where χ is a one dimensional character (kN-module) and M is the Green correspondent of χ.

Let U be the set of all one dimensional characters of kN whose Green correspondents are endotrivial. Let W be the intersection of the kernels of elements of U. Then an easy exercise shows that

K(G) ≅ (N_{G}(S)/W)^{∗} = Hom(N_{G}(S)/W, k^{×}).

That is, (N_{G}(S)/W)^{∗} is the dual group of N_{G}(S)/W. Thus the question becomes: What is the subgroup W? Given the results of the last section, it seems natural to conjecture the following.

Question 1. [25, Question 5.5] Is it possible that

W = ρ^{∞}(S)?

where ρ^{∞}(S) is the limit of the sequence ρ^{1}(S) ⊆ ρ^{2}(S) ⊆ . . ..

It is straightforward to show that ρ^{∞}(S) ⊆ W. As a consequence, we have that K(G) is isomorphic to a quotient of N_{G}(S)/ρ^{∞}(S). The basic idea of the proof is the following. Suppose that M is an endotrivial module with trivial Sylow restriction and that M_{↓N} ≅ χ ⊕ proj. Suppose farther that g ∈ N is also in the commutator subgroup [N′, N′] where N′ = N_{G}(Q) for some nontrivial subgroup Q ⊆ S. Then M_{↓N′} ≅ μ ⊕ proj for some linear character μ of N′. Then g is in the kernel of μ since it is in the commutator subgroup. But because Q is not the trivial group, χ and μ must coincide on the intersection N ∩ N′ by the Krull-Schmidt Theorem. Hence, g which is in this intersection, is also in the kernel of χ.

There is at least one situation where the above question has an affirmative answer, and additionally, the convergence of the sequence ρ^{1}(S), ρ^{2}(S), … is very quick.

Theorem 6.1. [25, Theorem 5.1] Suppose that a Sylow p-subgroup S of G is abelian, then ρ^{2}(S) = W, and K(G) ≅ N_{G}(S)/ρ^{2}(S).

A major ingredient in the proof is Burnside’s Fusion Theorem which says that, assuming S is abelian, the normalizer of S controls p-fusion in S. That is, two subgroups of S are conjugate in G if and only if they are conjugate in N = N_{G}(S). The result is the primary reason that ρ^{2}(S) = ρ^{∞}(S). With this basis, the proof of the theorem consists of showing that the one dimensional characters of N can be extended (uniquely) to weak S-homomorphisms. Balmer’s Theorem 5.2 completes the proof.

Some effort has been made by the author and Thévenaz to generalize the theorem to cases where S is not abelian. However, there are significant problems associated with removing the abelian assumption in Theorem 6.1. First, Burnside’s Fusion Theorem no longer holds. The substitute would likely be Alperin’s Fusion Theorem (see [47]), which is far more complicated. The other problem is that the statement about the convergence in Theorem 6.1 is not true without the assumption that S is abelian. That is, if S is not abelian, then it is not necessarily the case that ρ^{2}(S) = ρ^{∞}(S). A counterexample is the group of Lie type G_{2}(5) in characteristic 3. Here ρ^{2}(S) ≠ ρ^{3}(S) = ρ^{∞}(S).

Using the theorem, the calculation of K(G) for many groups G can be easily obtained on a computer. A simple program in the computing language Magma [7] computes ρ^{∞}(G). For example, it can be shown that K(G) is a Klein four group if G is isomorphic to the Mathieu group M_{11} and p = 3. Also, K(G) is cyclic of order 2 if G is the Suzuki simple group and p = 5. These calculations take only a few seconds of computing time.

For another easy example, suppose that G is a perfect group whose Sylow 2subgroup S is a Klein four group. And assume that N_{G}(S)/C_{G}(S) ≅ C_{3} is a cyclic group of order 3. The alternating group A_{5} is an example of such a group. Then ρ^{1}(S) is contained in C_{G}(S) since N_{G}(S)/C_{G}(S) is abelian. Moreover an element of order 3 in N_{G}(S) that is not in C_{G}(S) is also not in ρ^{2}(S), because it does not normalize any nontrivial subgroup of order 2. Therefore, K(G) has order at least 3. In the case of A_{5} it has order exactly 3.

A variation on this theme is the following theorem from [16, Theorem 3.1].

Theorem 6.2. Suppose that Q is a characteristic subgroup of a Sylow p-subgroup of G. Let N = N_{G}(Q). Suppose that N is generated by a collection g_{1}, …, g_{t} such that for each i = 1, …, t, either

(1) g_{i} ∈ [N, N], or

(2) there exists a nontrivial subgroup R ⊆ Q such that g_{i} ∈ [N_{G}(R), N_{G}(R)].

Then the only indecomposable kG-module with trivial Sylow restriction is the trivial module.

The point of the proof is that if an element of N satisfies the given conditions, then it must be in the kernel of the Green correspondent kN-module of any endotrivial module with trivial Sylow restriction. This theorem was used several times in the classification of endotrivial modules over group of Lie type A in [16], and is at the heart of an effort by the same authors to compute the group of endotrivial module for groups of other Lie types in nondefining characteristics.

Note added in proof: In a recent reprint, Jesper Grodal has proved that Question 1 has an affirmative answer.

7. Simple endotrivial modules

Another recent development in the subject is the effort to classify the simple endotrivial modules for finite groups. This research has been led mostly be Caroline Lassueur and Gunter Malle. It is motivated in part by a result of Geoff Robinson [46] which asserts that if the Sylow p-subgroup of G has p-rank at least two, and M is a faithful irreducible endotrivial kG-module, then either M remains irreducible on restriction to some normal quasi-simple subgroup or M is induced from a one dimensional module on some strongly p-embedded subgroup. (A subgroup H is strongly p-embedded if p divides the order of H and if, for every element g that is not in H, H ∩ H^{g} has order prime to p.) Hence the problem of finding simple endotrivial modules reduces to looking at quasi-simple groups.

The study was initiated in a paper [35] by Lassueur, Malle and Elisabeth Schulte in which they laid out the basic techniques. One of their fundamental results [35, Theorem 1.3] is a generalization of the lifting theorem of Alperin [2]. For this we suppose that we have a p-modular system (K, O, k). In this setting, O is a complete discrete valuation ring with residue class field k of characteristic p and with field of fractions K of characteristic 0. Lassueur, Malle and Schulte show that if M is an endotrivial kG-module, then there is an endotrivial OG-lattice (free module over O) whose reduction modulo p is M. This result was first proved by Alperin in the case that G is a p-group.

The point is that if M is a simple endotrivial kG module, then M lifts to a simple OG-lattice M̂. That is, K ⊗_{O} M̂ is a simple KG-module and is determined entirely by its character. The fact, that the module is endotrivial, puts several constraints on that character. For example, if χ is the character of such an M̂, then |χ(g)| = 1 for all p-singular elements g. Hence, from this information and other clues it is possible to determine when an irreducible character of a finite group might be that of a lift of a simple endotrivial module.

In a couple of recent papers it has been shown that most complex irreducible characters for finite simple classical groups vanish on some p-singular element. The result is that there are few simple endotrivial modules. For example, Lassueur and Malle prove [34, Theorem 3.1] that any central factor group of SL(n, q), with n ≥ 3, p not dividing q and noncyclic Sylow p-subgroup, has no faithful simple endotrivial modules. Frank Lübeck and Malle deal with types B, C and D in [39, Theorem 5.4], completing the proof that in all cases of quasi-simple classical groups, not of very small rank and not having cyclic Sylow p-subgroups, except for G ≅ 𝕊p_{8}(2) in characteristic 5, there are no faithful irreducible endotrivial modules.

In another direction, there are some general theorems. The following was conjectured in [35] and proved in [34].

Theorem 7.1. [34, Theorem 1.1] Suppose that G is a finite quasi-simple group with a faithful simple endotrivial module in characteristic p. Then G has p-rank at most 2.

The connection between K(G), the kernel of the restriction map from T (G) to T (S), and the structure of the simple endotrivial modules is not well understood. It seems that in many cases where there exist nontrivial elements of K(G) of dimension greater than one, the elements are in fact simple modules or K(G) is generated by classes of simple modules. For example, if G = A_{7} the alternating group on seven letters, and p = 3, then K(G) ≅ ℤ/4ℤ. The elements of order 4 are the classes of irreducible modules of dimension 10. The squares of these elements is a class represented by an indecomposable endotrivial module having dimension 28. This module is not irreducible.

In addition, Theorem 7.1 seems to hold more generally. That is, in the known examples, it is only when G has p-rank 2 that there exist nontrivial indecomposable modules with trivial Sylow restriction and with dimension greater than one.

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Department of Mathematics, University of Georgia, Athens, GA 30602, USA

E-mail address: jfc@math.uga.edu