ENDO-TRIVIAL MODULES FOR FINITE GROUPS WITH DIHEDRAL SYLOW 2-SUBGROUP

SHIGEO KOSHITANI AND CAROLINE LASSUEUR

Abstract. Let k be an algebraically closed field of characteristic p > 0 and G a finite group. We provide a description of the torsion subgroup T T (G) of the finitely generated abelian group T (G) of endo-trivial kG-modules when p = 2 and G has a dihedral Sylow 2-subgroup P. We prove that, in the case |P | ≥ 8, T T (G) ≅ X(G) the group of one-dimensional kG-modules, except possibly when G/O_{2′}(G) ≅ 𝔄_{6}, the alternating group of degree 6; in which case G may have 9-dimensional simple torsion endo-trivial modules. We also prove a similar result in the case |P | = 4, although the situation is more involved. Our results complement the tame-representation type investigation of endotrivial modules started by Carlson-Mazza-Thévenaz [10] in the cases of semi-dihedral and generalized quaternion Sylow 2-subgroups. Furthermore we provide a general reduction result, valid at any prime p, to recover the structure of T T (G) from the structure of T T (G/H), where H is a normal p′-subgroup of G.

1. Introduction

Let G be a finite group and k a field of prime characteristic p dividing the order of G. A finitely generated kG-module V is called endo-trivial if, as kG-modules,

End_{k}(V) ≅ k_{G} ⊕ Q ,

where k_{G} is the trivial kG-module and Q is a projective kG-module. The tensor product over k induces a group structure on the set of isomorphism classes of indecomposable endo-trivial kG-modules, called the group of endo-trivial modules and denoted by T (G). This group is finitely generated and it is of particular interest in modular representation theory as it forms an important part of the Picard group of self-equivalences of the stable category of finitely generated kG-modules. In particular the self-equivalences of Morita type are induced by tensoring with endo-trivial modules.

As a matter of fact, endo-trivial modules have seen a considerable interest since defined by Dade in 1978 [15] as a by-product of the Dade-Glauberman-Nagao correspondence (see [30, §5.12]). In [15] a classification of the endo-trivial modules over finite abelian p-groups is established. Since then a full classification has been obtained over finite pgroups through the joint efforts of several authors, see e.g. the survey article [37] and the references therein. Moreover many contributions towards a general classification of endotrivial modules have been obtained over the past ten years for several families of finite groups (see e.g. [7, 28, 8, 9, 6, 31, 10, 22, 13, 27] and the references therein). However, the problem of describing the structure of T (G) and its elements for an arbitrary finite group G remains open in general. In particular the problem of determining the structure of the torsion subgroup of T (G), denoted by T T (G), is a resisting part of the problem.

2000 Mathematics Subject Classification. 20C20, 20C05, 20C25.

Key words and phrases. Modular representation theory, endo-trivial modules, dihedral Sylow 2subgroup, twisted group algebras.

The first author was partially supported by the Japan Society for Promotion of Science (JSPS), Grantin-Aid for Scientific Research (C)23540007, 2011-2014 and (C)15K04776, 2015–2018. The second author acknowledges partial financial support by SNSF Fellowship for Prospective Researchers PBELP2_{−}143516.

Provided that a Sylow p-subgroup P of G is neither cyclic, nor dihedral, nor semidihedral, the group T T (G) coincides with the group

K(G) = ker ( Res_{P}^{G} : T (G) ⟶ T (P) : [V] ↦ [V ↓_{P}^{G}]) .

(See Lemma 3.3.) In particular, the group K(G) consists of the classes of the trivial source endo-trivial modules.

Our first main result holds at any prime characteristic p and relates the structure of the group K(G) to the structure of K(G/H), where H is a normal p′-subgroup of G.

Theorem 1.1. Let G be a finite group. Assume that the p-rank of G is at least 2 and that G has no strongly p-embedded subgroups. Let H ◁ G with p |̸ |H|, and set \overline{G} := G/H. If H^{2}(\overline{G}, k^{×}) = 1, then

K(G) = X(G) + InfG ; \overline{G}(K(\overline{G})) ≅ X(G) + K(\overline{G}) .

However, the main objective of this article is the determination of the structure of the group T (G), when the Sylow 2-subgroups of G are dihedral 2-groups. An investigation of endo-trivial modules in finite- and tame-representation types was started by Mazza-Thévenaz [28] in the cyclic case, Carlson-Mazza-Thévenaz [10] in the generalized quaternion and semi-dihedral cases, and continued by the authors [22] in the Klein-four case. Therefore the dihedral case was the last untreated tame-representation type case. However, we emphasize that [10] does not provide a description of the structure of the group K(G) in the semi-dihedral case, and which is still an open question.

Since the group T (G) is finitely generated (see [7, Corollary 2.5]), we may write T (G) = T T (G)⊕T F (G), where T F (G) denotes a free abelian complement of T T (G) in T (G). We recall that, when a Sylow 2-subgroup of G is dihedral of order of at least 8, the ℤ-rank, as well as generators for the torsion-free part of T (G) are known since 1980’s. More precisely, since the 2-rank of G is 2, by [7, Theorem 3.1], the ℤ-rank of T F (G) coincides with the number of G-conjugacy classes of Klein-four subgroups of G, and this number is 2 by [17, Proposition 1.48(iv)]. Then by [1, §4], we have

T F (G) ≅ ⟨[Ω^{1}(k_{G})], [M]⟩ ≅ ℤ^{2}

where Ω^{1}(k_{G}) denotes the first syzygy module of the trivial kG-module k_{G}, and M is an indecomposable direct summand of the heart of the projective cover of k_{G}. (Note that there are two such direct summands and M can be chosen to be any of them.)

As a consequence, in this article we focus our attention on the determination of the torsion subgroup T T (G) of T (G). We recall that the group X(G) of one-dimensional kGmodules endowed with the tensor product ⊗_{k} always identifies with a subgroup of T T (G). Our main result about the structure of T T (G) in the dihedral case is the following.

Theorem 1.2. Assume that G is a finite group with a dihedral Sylow 2-subgroup of order at least 8, and let T (G) be the abelian group of endo-trivial kG-modules over an algebraically closed field k of characteristic 2. Set \overline{G} := G/O_{2′}(G). Then the following hold:

(a) If \overline{G} ≇ 𝔄_{6}, then T T (G) = X(G).

(b) If \overline{G} ≅ 𝔄_{6}, then either (i) T T (G) = X(G), or (ii) if there exists an indecomposable endo-trivial kG-module V such that [V] ∈ T T (G) \ X(G), then dim_{k} V = 9, V is simple, and T T (G)/X(G) is an elementary abelian 3-group.

Remark 1.3. Case (i) of Theorem 1.2(b) happens for example for G = 𝔄_{6} (see [8, Theorem 1.2]), whereas Case (ii) happens for example for G = 3.𝔄_{6}, the triple cover of 𝔄_{6} (see Lemma 6.3). Moreover the central product C_{9} ∗ 3.𝔄_{6} provides an example where there exist classes [V] ∈ T T (G) such that [V^{⊗3}] ∈ X(G)\{[k_{G}]} (see Example 7.2).

Furthermore, in the situation of Theorem 1.2(b)(ii), any simple torsion endo-trivial kG-module V of dimension 9 originates from the triple cover 3.𝔄_{6} of the alternating group 𝔄_{6} of degree 6 in the following way. By [32, Theorem], E := E(G/ ker(V)) (the central product of all components of G/ ker(V), see [35, Definition 6.6.8])) is quasi-simple and V ↓_{E}^{G/ ker(V)} remains simple endo-trivial. Therefore, we must have E = 3.𝔄_{6} since 3.𝔄_{6} is the unique quasi-simple group with a 9-dimensional simple endo-trivial module in characteristic two by [25, Proposition 3.8 and §4].

Corollary 1.4. If G is a finite group with a dihedral Sylow 2-subgroup of order at least 8 and k is an algebraically closed field of characteristic 2, then any indecomposable torsion endo-trivial kG-module is simple, and hence lifts uniquely to an ordinary irreducible character of G.

Proof. This follows immediately from Theorem 1.2, the fact that T T (G) consists only of classes of trivial source modules (see Lemma 3.3), and the fact that trivial source modules lift uniquely (see [30, Theorem 4.8.9(iii)]). □

Our new method, developed to treat the dihedral case of order at least 8, also allows us to finish off the classification of torsion endo-trivial modules for finite groups with Klein-four Sylow 2-subgroups, which we started in [22]. We note that our results in this case are explicit, whereas those recently obtained by Carlson and Thévenaz in [13] (where they treat the general question of computing the group K(G) for finite groups G with abelian Sylow p-subgroups) only provides an algorithmic method to identify the group K(G).

Theorem 1.5. Assume that G is a finite group with a Klein-four Sylow 2-subgroup P. Further, set T T_{0}(G) := {[V] ∈ T T (G) | V indecomposable and V ∈ B_{0}(G)} where B_{0}(G) is the principal block of G and \overline{G} := G/O_{2′}(G). Then T T_{0}(G) ≅ ℤ/3ℤ, any indecomposable kG-module V with [V] ∈ T T (G) lifts uniquely to an OG-lattice V̂ affording an ordinary irreducible character χV̂ ∈ Irr(G), and the structure of T T (G) is as follows:

(a) If \overline{G} ≅ P, then T T (G) = X(G).

(b) If \overline{G} ≅ 𝔄_{4}, then T T (G) = X(G).

(c) If \overline{G} ≅ 𝔄_{5}, then

T T (G) ≅ T T (G_{0}) = X(G_{0}) ,

where G_{0} is a strongly 2-embedded subgroup in G with G_{0}/O_{2′}(G_{0}) ≅ 𝔄_{4}. Furthermore, if V is a non-trivial indecomposable endo-trivial kG-module such that [V] ∈ T T_{0}(G), then dim_{k}(V) = 5.

(d) If \overline{G} ≅ PSL(2, q) ⋊ C_{f}, where q > 5 is a power of an odd prime such that q ≡ ±3 (mod 8) and f is an odd integer, then

T T (G) ≅ X(G) ⊕ T T_{0}(G) ≅ X(G) ⊕ ℤ/3ℤ .

Furthermore, if V is a non-trivial indecomposable endo-trivial kG-module such that [V] ∈ T T_{0}(G), then dim_{k}(V) = (q − 1)/2 when q ≡ 3 (mod 8) and dim_{k}(V) = q when q ≡ 5 (mod 8).

The first main tool used in our investigation is Gorenstein-Walter’s classification of finite groups G with dihedral Sylow 2-subgroups modulo O_{2′}(G). Moreover our methods heavily rely on a Theorem of Schur’s [30, Theorem 3.5.8] combined with two results of Navarro-Robinson [31], the first of which states that if an endo-trivial module is induced from a proper subgroup, then this subgroup must be strongly p-embedded in G, and the second of which states that simple endo-trivial modules over p-nilpotent groups of p-rank at least 2 have dimension one. This enables us to reduce our computation of T T (G) to that of T T (G/O_{2′}(G)) using Theorem 1.1. Finally, we note that our methods require to decompose torsion endo-trivial modules as tensor products of modules over non-proper twisted group algebras, although the final statements of Theorem 1.2 and Theorem 1.5 do not reflect this fact.

The paper is organized as follows. In §2 we introduce the notation, and in §3 preliminary known results on endo-trivial modules, which we will rely on. In §4 we prove Theorem 1.1. In §5 we give general results on finite groups with dihedral Sylow 2-subgroups of order at least 8 and their endo-trivial modules, and §§6–7 are devoted to the proof of Theorem 1.2. Finally in §8 we prove Theorem 1.5.

2. Notation

Throughout, unless otherwise specified we use the following notation and conventions. We let p denote a prime number and G a finite group of order divisible by p. We assume that (K, O, k) is a splitting p-modular system for all subgroups of G, that is, O is a complete discrete valuation ring of rank one such that its quotient field K is of characteristic zero, its residue field k := O/rad(O) is of characteristic p, and both K and k are splitting fields for all subgroups of G. Modules are finitely generated left modules. By an OGlattice, we mean an OG-module which is O-free of finite rank. For a ring R, we denote by R^{×} the group of units of R. We write Syl_{p}(G) for the set of all Sylow p-subgroups of G. For a p-subgroup Q of G and H ≤ G with H ≥ N_{G}(Q), we denote by f = f_{(G, Q, H)} the Green correspondence with respect to (G, Q, H), see [30, p.276]. For a positive integer n, we denote by C_{n} the cyclic group of order n, and by 𝔄_{n} the alternating group of degree n. We write Z(G) for the center of G, [G, G] for the commutator subgroup of G. If H is a normal subgroup of G, and L a subgroup of G, then we write G = H ⋊ L if G is a semi-direct product of H by L. For two kG-modules M and M′, M ⊗_{k} M′ is the tensor product over k, M^{⊗n} is the tensor product M ⊗_{k} ⋯ ⊗_{k} M of n copies of M, we write M^{∗} for the k-dual of M, that is M^{∗} := Hom_{kG}(M, k), and we write M′ | M when M′ is (isomorphic to) a direct summand of M. We denote by k_{G} the trivial kG-module. We write H ≤ G if H is a subgroup of G. In such a case, for a kG-module M and a kH-module L, we denote by M↓_{H} and L↑^{G}, respectively, the restriction of M to H and the induction of L to G.

We denote by M(G) = H^{2}(G, ℂ^{×}) the Schur multiplier of G. For a 2-cocycle α ∈ H^{2}(G, k^{×}), we denote by k^{α}G the twisted group algebra of G over k with respect to α. Then for a module M over the twisted group algebra k^{α}G and H ≤ G we write M↓_{H} or M↓_{k^{α}H}^{k^{α}G} for the restriction of M from G to H. For N ◁ G and an O(G/N)-lattice M we write Inf_{G/N}^{G}(M) for the inflation of M from G/N to G. We denote by B_{0}(G), the principal block of G, and by Irr(G) the set of all irreducible ordinary characters of G. For a p-block B of G, we also denote by Irr(B) the set of all characters in Irr(G) which belong to B. We write 1_{G} for the trivial ordinary or Brauer character of G.

We say that a kG-module M is a trivial source module if it is a direct sum of indecomposable kG-modules, all of whose sources are trivial modules, see [36, p.218]. It is known that a trivial source kG-module M lifts uniquely to a trivial source OG-lattice, which we denote by M̂, see [30, Theorem 4.8.9(iii)]. Then, we denote by χM̂ the ordinary character of G afforded by M̂. For a non-negative integer m and a positive integer n, we write n_{p} = p^{m} if p^{m}|n and p^{m+1} |̸n.

For further standard notation and terminology, we refer the reader to the books [30, 36].

3. Preliminary results

3.1. Endo-trivial modules. A kG-module V is called endo-trivial provided

End_{k}(V) ≅ V^{∗} ⊗_{k} V ≅ k_{G} ⊕ (proj)

as kG-modules where (proj) denotes a projective direct summand (possibly the zero module).

Any endo-trivial kG-module V splits as the direct sum V = V_{0} ⊕ (proj) where V_{0} := Ω^{0}(V), the projective-free part of V, is indecomposable and endo-trivial. The relation U ∼ V ⇔ U_{0} ≅ V_{0} is an equivalence relation on the class of endo-trivial kG-modules, and we let T (G) denote the resulting set of equivalence classes (which we denote by square brackets). Then T (G), endowed with the law [U] + [V] := [U ⊗_{k} V], is an abelian group called the group of endo-trivial modules of G. The zero element is the class [k_{G}] and −[V] = [V^{∗}].

Notice that if p ∤ |G|, then any kG-module is endo-trivial, but the above construction of the group T (G) is not valid any more.

The group T (G) is known to be a finitely generated abelian group, see e.g. [7, Corollary 2.5]. Therefore, we may write T (G) = T T (G) ⊕ T F (G), where T T (G) is the torsion subgroup of T (G) (hence a finite group) and T F (G) is a torsion-free complement.

We let X(G) denote the group of one-dimensional kG-modules endowed with the tensor product ⊗_{k}, and recall that X(G) ≅ (G/[G, G])_{p′}. Then by identifying a one-dimensional module with its class in T (G), we consider X(G) as a subgroup of T (G).

Furthermore, define T_{0}(G) := {[V] ∈ T (G) | V indecomposable and V ∈ B_{0}(G)}. Then T_{0}(G) ≤ T (G) by [11, Proposition 9.1]. Denote by T T_{0}(G) the torsion subgroup of T_{0}(G), and by X_{0}(G) the set of one-dimensional kG-modules belonging to B_{0}(G). Clearly X_{0}(G) ≤ T T_{0}(G) ≤ T T (G).

Remark 3.1. Because the dimension of projective kG-modules is divisible by |G|_{p}, if M is an endo-trivial kG-module, then dim_{k}(V) ≡ ±1 (mod |G|_{p}) if p is odd; and dim_{k}(V) ≡ ±1 (mod \frac{1}{2}|G|_{2}) if p = 2. Moreover if V is indecomposable with trivial source, then dim_{k}(V) ≡ 1 (mod |G|_{p}). In particular, indecomposable endo-trivial kG-modules have the Sylow p-subgroups of G as their vertices, and hence lie in p-blocks of full defect.

Lemma 3.2. Let H be a subgroup of G, and let P be a Sylow p-subgroup of G.

(a) If V is an endo-trivial kG-module, then V ↓_{H} is endo-trivial. Moreover, if H ≥ P, then V is endo-trivial if and only if V ↓_{H} is endo-trivial.

(b) If p | |H|, then restriction induces a group homomorphism

Res_{H}^{G} : T (G) ⟶ T (H) : [V] ↦ [V ↓_{H}] .

If, moreover, H ≥ N_{G}(P), then Res_{H}^{G} is injective, and for any [V] ∈ T (G) with V indecomposable, Res_{H}^{G}([V]) = [f_{H}(V)], where f_{H} := f_{(G, P, H)}.

(c) If H ◁ G such that p |̸ |H| and V is a k(G/H)-module, then Inf_{G/H}^{G}(V) is endotrivial if and only if V is endo-trivial. Moreover the inflation from G/H to G induces an injective group homomorphism

Inf_{G/H}^{G} : T (G/H) ⟶ T (G) : [V] ↦ [Inf_{G/H}^{G}(V)] .

In particular, we may consider T T (G/H) ≤ T T (G).

Proof. Parts (a) and (b) are given by [7, Proposition 2.6], and part (c) is given by [26, Lemma 3.2(1)]. □

Let P be a Sylow p-subgroup of G. We fix the notation

K(G) := ker (Res_{P}^{G} : T (G) ⟶ T (P)) .

In fact, in most cases, the torsion subgroup of T (G) is equal to K(G), and has the following characterizations, which we will use throughout.

Lemma 3.3. Let P be a Sylow p-subgroup of G.

(a) If for any x ∈ G, P ∩ x^{−1}P x is non-trivial, then K(G) = X(G). In particular K(N_{G}(P)) = X(N_{G}(P)).

(b) The group K(G) is exactly the set of classes of indecomposable trivial source endotrivial kG-modules, and

K(G) = { [V] ∈ T (G) | ∃ a module M ∈ X(N_{G}(P)) with V_{0} = f^{−1}(M)} ,

where f := f_{(G, P, N_{G}(P))}. In particular, K(G) ≤ T T (G) and we may consider K(G) as a subgroup of K(N_{G}(P)) = X(N_{G}(P)) via the injective homomorphism Res_{N_{G}(P)}^{G}.

(c) Furthermore, provided P is neither cyclic, nor semi-dihedral, nor generalized quaternion, then K(G) = T T (G).

Proof. (a) This follows from [28, Lemma 2.6].

(b) By definition K(G) consists of the classes of indecomposable trivial source endotrivial kG-modules. Hence the first claim is straightforward from (a) together with Lemma 3.2(b). Next the number of isomorphism classes of indecomposable trivial source kG-modules with vertex P is finite, hence K(G) is a finite group, so that we must have K(G) ≤ T T (G). The last claim follows from Lemma 3.2(b).

(c) The claim is given by [9, Lemma 2.3]. □

Our objective in this article is to consider groups with dihedral Sylow 2-subgroups only, therefore in this case Lemma 3.3(c) allows us to identify T T (G) with K(G).

Finally in order to detect whether a trivial source module is endo-trivial, we have the following character-theoretic criterion.

Theorem 3.4 ([24, Theorem 2.2]). Let V be an indecomposable trivial source kG-module. Then V is endo-trivial if and only if χV̂(u) = 1 for any non-trivial p-element u ∈ G.

3.2. Strongly p-embedded subgroups. Recall that a subgroup H of G is said to be strongly p-embedded in G if H ≨ G, p | |H| and p |̸ |H ∩ x^{−1}Hx| for any x ∈ G \ H. Note that any strongly p-embedded subgroup of G contains the normalizer in G of a Sylow p-subgroup. Moreover the operations of induction and restriction induce equivalences of the stable module categories stmod(kH) and stmod(kG) if H is a strongly p-embedded subgroup of G.

Lemma 3.5 ([28, Lemma 2.7(2)]). Let H be a strongly p-embedded subgroup of G, and let P be a Sylow p-subgroup of H. Then Res_{H}^{G} : T (G) ⟶ T (H) is an isomorphism. Moreover, the inverse map is induced by induction, and, more precisely, on the indecomposable endo-trivial modules by the Green correspondence f_{H} := f_{(G, P, H)}, namely

T (G) = { [W ↑^{G}] | [W] ∈ T (H)}
= { [f_{H}^{−1}(W)] | W is an indecomposable endo-trivial kH-module} .

In particular K(G) ≅ K(H).

The following result due to G. Navarro and G.R. Robinson is essential for our purpose because the structure of finite groups which have strongly p-embedded subgroups is in a sense very restricted.

Lemma 3.6. Let H be a proper subgroup of G, and assume that V and W are kG- and kH-modules, respectively, with V = W ↑^{G}. Then the following are equivalent:

(1) V is endo-trivial.

(2) W is endo-trivial and H is a strongly p-embedded subgroup of G.

Proof. The implication (1) implies (2) is given by [29, Lemma 1(iv)]. The converse is straightforward from Lemma 3.5. □

4. Recovering K(G) via inflation from a normal p’-subgroup

Before starting our investigation of endo-trivial modules over finite groups with dihedral Sylow 2-subgroups, we develop a general method enabling us to recover the subgroup K(G) of T (G) using inflation from a quotient by a normal p′-subgroup of G.

Lemma 4.1 (Schur). Let F be an algebraically closed field of any characteristic. Assume that H ◁ G and Y is a simple F H-module such that Y is G-invariant. Further, let Ŷ be an F^{α}G-module, where F^{α}G is the twisted group algebra of G over F defined by a 2-cocycle α ∈ H^{2}(G/H, F^{×}), such that Ŷ is an extension of Y. If X is an F G-module such that

X↓_{H} ≅ Y ⊕ ⋯ ⊕ Y,

then there exists an F^{α^{−1}}(G/H)-module Z such that

X ≅ Ŷ ⊗_{F} Z

as F G-modules.

Proof. This is essentially Schur’s theorem, see [30, Theorem 3.5.8(i)]. More specifically, although [30, Theorem 3.5.8(i)] is stated for a module X which is simple, its proof only requires the assumption that X↓_{H} ≅ Y ⊕ ⋯ ⊕ Y in order to obtain the conclusion that there exists an F^{α^{−1}}(G/H)-module Z such that X ≅ Ŷ ⊗_{F} Z. □

Lemma 4.2. Assume that G has no strongly p-embedded subgroups and V is an indecomposable endo-trivial kG-module. If H ◁ G such that p |̸ |H| and L is a composition factor of V ↓_{H}, then L is G-invariant.

Proof. Set G̃ := T_{G}(L), the stabilizer of L in G, and let B be the block of kG to which V belongs. Let B̃ be the block of kG̃ such that B̃ is the Fong-Reynolds correspondent of B, see [30, Theorem 5.5.10]. Then, B and B̃ are Morita equivalent. Write Ṽ for the kG̃-module in B̃ corresponding to V via this Morita equivalence. Then, V = Ṽ↑^{G}. Thus Lemma 3.6 yields that Ṽ is endo-trivial. If G̃ ≠ G, then it follows from Lemma 3.6 that G̃ is strongly p-embedded in G, which is a contradiction. □

Theorem 4.3. Assume that the p-rank of G is at least 2 and that G has no strongly p-embedded subgroups. Let H ◁ G with p |̸ |H|, and set \overline{G} := G/H. Let V be an indecomposable endo-trivial kG-module such that [V] ∈ K(G). Then the following hold:

(a) There exists a 2-cocycle α ∈ H^{2}(\overline{G}, k^{×}) such that

V = 1b ⊗_{k} W ,

where 1b is a one-dimensional k^{α}G-module, W is a k^{α^{−1}} \overline{G}-module, and W is considered as a k^{α^{−1}}G-module with H ≤ ker W via inflation from \overline{G} to G. Moreover, if P ∈ Syl_{p}(G) and \overline{P} := HP/H = (H ⋊ P)/H, then both α↓\overline{P} and α↓_{H⋊P} are trivial.

(b) Keep the notations α, 1b and W as in (a). If n := |α| is the order of α, then 1c := (1b)^{⊗n} is a one-dimensional kG-module, and W^{⊗n} is a trivial source endotrivial k\overline{G}-module. Namely, [1c] ∈ X(G), [W^{⊗n}] ∈ InfG ; \overline{G}(K(\overline{G})) and

[V^{⊗n}] = [1c] + [W^{⊗n}] ∈ X(G) + InfG ; \overline{G}(K(\overline{G})) .

Proof. (a) Let L be a composition factor of V ↓_{H}. Then, by Lemma 4.2, L is G-invariant. Let B and b be the blocks of kG and kH, respectively, to which V and L belong. So clearly B covers b. Denote by θ ∈ Irr(b) the ordinary irreducible character corresponding to L, and hence Irr(b) = {θ}.

Since B covers b, Irr(b) = {θ}, θ is G-invariant and H is a p′-group, it follows from [30, Lemma 5.5.8(ii)] that

V ↓_{H} ≅ L ⊕ ⋯ ⊕ L .

Now, since L is a G-invariant simple kH-module, we know from another Theorem of Schur [30, Theorem 3.5.7] that there exist a 2-cocycle α ∈ H^{2}(\overline{G}, k^{×}) and a k^{α}G-module L̂ such that L̂↓_{H} = L. Hence by Lemma 4.1, there exists a k^{α^{−1}} \overline{G}-module W such that

V = L̂ ⊗_{k} W ,

where W is considered as a k^{α^{−1}}G-module with H ≤ ker W via inflation from \overline{G} to G.

Next we claim that both α↓\overline{P} and α↓_{H⋊P} are trivial. Indeed α↓\overline{P} is trivial by [30, Proof of Theorem 3.5.11(ii)], and therefore so is α↓_{H⋊P} when we consider α ∈ H^{2}(G, k^{×}). This implies that

V ↓_{H⋊P} = L̂↓_{k(H⋊P)} ⊗_{k} W ↓_{k(H⋊P)} ,

where all three modules are modules over the (proper non-twisted) group algebra k(H⋊P). Then, by Lemma 3.2(a), V ↓_{H⋊P} is endo-trivial. Hence, both L̂↓_{k(H⋊P)} and W ↓_{k(H⋊P)} are endo-trivial by [31, Lemma 1(iii)]. In addition, since L is simple and L̂ is an extension of L, L̂↓_{k(H⋊P)} is simple as well. Thus, as the the p-rank of G is assumed to be at least 2, [31, Theorem] yields dim L = dim L̂ = 1. So, we set 1b := L̂ and the assertion follows.

(b) First since V is an indecomposable endo-trivial kG-module with [V] ∈ K(G), we have [V^{⊗n}] ∈ K(G). Then, since |α| = n, we have by (a) that

V^{⊗n} = 1c ⊗_{k} W^{⊗n} ,

where 1c := (1b)^{⊗n} is a one-dimensional (non-twisted proper) kG-module and W^{⊗n} is a (non-twisted proper) k\overline{G}-module, which we may regard as a kG-module via the inflation from \overline{G} to G. Since V^{⊗n} is endo-trivial, again by [31, Lemma 1(iii)], both 1c and W^{⊗n} are endo-trivial kG-modules, and thus W^{⊗n} is also endo-trivial as a k\overline{G}-module by Lemma 3.2(c). Now in T (G) we have

[V^{⊗n}] = [1c] + [W^{⊗n}] ,

where [1c] ∈ X(G) ≤ K(G). Therefore it remains to prove that [W^{⊗n}] ∈ InfG ; \overline{G}(K(\overline{G})). But this is clear. Indeed, since [W^{⊗n}] = [V^{⊗n}] − [1c] ∈ K(G) , W^{⊗n} must be a trivial source kG-module (not necessarily indecomposable if n > 1) by Lemma 3.3(b) and therefore so is W^{⊗n} seen as a k\overline{G}-module, that is, [W^{⊗n}] ∈ InfG ; \overline{G}(K(\overline{G})). □

As a corollary we obtain Theorem 1.1 of the introduction.

Proof of Theorem 1.1. Since H^{2}(\overline{G}, k^{×}) = 1, the integer n in Theorem 4.3(b) is equal to 1. Hence the claim follows by identifying K(\overline{G}) with InfG ; \overline{G}(K(\overline{G})). □

5. Groups with dihedral Sylow 2-subgroups

Throughout this section we assume that P be a dihedral Sylow 2-subgroup of G of order at least 8. Gorenstein and Walter proved in [18] (see also [16, Theorem on p.462]) that \overline{G} := G/O_{2′}(G) is isomorphic to either

(1) P,

(2) the alternating group 𝔄_{7}, or

(3) a subgroup of PΓL(2, q) containing PSL(2, q), where q is a power of an odd prime.

Hypotheses 5.1. For the purposes of our computations, we split case (3) above in further subcases and say that G satisfies the hypothesis: (D1) if \overline{G} ≅ P; (D2) if \overline{G} ≅ 𝔄_{7}; (D3) if \overline{G} ≅ PSL(2, 9) ≅ 𝔄_{6}; (D4) if \overline{G} ≅ PGL(2, 9) ≅ 𝔄_{6}.2_{2}; (D5) if \overline{G} ≅ PSL(2, q), where q is a power of an odd prime with q ≠ 9, and q ≡ ±1

(mod 8); (D6) if \overline{G} ≅ PGL(2, q), where q is a power of an odd prime with q ≠ 9; (D7) if \overline{G} ≅ PSL(2, q) ⋊ C_{f}, where q is a power of an odd prime with q ≠ 9, q ≡ ±1

(mod 8), and f > 1 is odd; (D8) if \overline{G} ≅ PGL(2, q) ⋊ C_{f}, where q is a power of an odd prime with q ≠ 9, and f > 1

is odd.

The splitting of case (3) into (D3)-(D8) follows from the fact that the structure of PΓL(2, q), where q = r^{m} is a power of an odd prime r, is well-known: PΓL(2, q) ≅ PGL(2, q) ⋊ Gal(𝔽_{q}/𝔽_{r}), where Gal(𝔽_{q}/𝔽_{r}) is cyclic of order m. Moreover [35, Chapter 6 (8.9)] shows that f is odd.

Lemma 5.2. There are no strongly 2-embedded subgroups in G.

Proof. This follows from the Bender-Suzuki Theorem [2, Satz 1] (cf. [34]) and also a result of Gorenstein-Walter [18], see [16, Theorem on p.462], too. □

Lemma 5.3. Set h := |H^{2}(\overline{G}, k^{×})|. Then

h = \begin{cases}3 & if \overline{G} ∈ \{𝔄_{6}, 𝔄_{7}, PGL(2, 9)\}, \\ 1 & otherwise. \end{cases}

Proof. Since k has characteristic 2, it follows from [40, Proposition 3.2] (see also [21, Lemma 5] and [30, Lemma 3.5.4(ii)]) that H^{2}(\overline{G}, k^{×}) ≅ M(\overline{G})_{2′}, where M(\overline{G})_{2′} is the 2′-part of M(\overline{G}) = H^{2}(\overline{G}, ℂ^{×}).

If (D1) holds, then h = 1 by [30, Theorem 2.7.4]. If (D2) or (D3) holds, then h = 3 by [14, p.10 and p.4, respectively].

Assume (D4) holds. Since \overline{G}/PSL(2, 9) is cyclic and PSL(2, 9) is perfect, we know by [19, Theorem 3.1] that |M(\overline{G})| | |M(PSL(2, 9))| = 6. So h = 1 or h = 3. Then [39, A_{6} (mod 2)] yields h = 3.

Next assume (D5) holds. It is known that if q ≠ 9 is a power of an odd prime, then |M(PSL(2, q))| = 2 by a result of R. Steinberg in [20, Theorem 4.9.1(ii)]. Hence, we have h = 1.

Next, consider the particular case that q = 3 when (D6) or (D8) holds (note that q ≠ 3 if (D7) holds). Assume first that (D6) occurs. Then, in the former case, \overline{G} ≅ PGL(2, 3) ≅ 𝔖_{4} and by Schur’s result [20, Theorem 4.3.8(i)] we have that |M(𝔖_{4})| = 2, so that h = 1. Assume that (D8) occurs. It follows from [19, Theorem 3.1(i)] that

|M(\overline{G})|||(M(𝔖_{4})| · |M(𝔖_{4}/[𝔖_{4}, 𝔖_{4}])| = 2 × 2 = 4.

Hence we have h = 1.

Finally assume that (D6), (D7) or (D8) holds with q > 3. Then

\overline{G}/PSL(2, q) ≅ \begin{cases}C_{2} & if (D6) holds, \\ C_{f} & if (D7) holds, \\ C_{2f} & if (D8) holds, \end{cases}

which, in particular, is cyclic in all three cases. Thus, as PSL(2, q) is perfect, it follows from [19, Theorem 3.1(i)] that |M(\overline{G})| | |M(PSL(2, q))| = 2. Therefore we obtain h = 1. □

6. Torsion endo-trivial modules in the dihedral case: the basic examples

We now turn to the description of T T (G) for groups G with dihedral Sylow 2-subgroup of order at least 8. First we investigate the case when O_{2′}(G) is trivial and prove that torsion endo-trivial modules are always one-dimensional in this case. Throughout this section we use the notations G, P and \overline{G} as in §5.

Proposition 6.1. If O_{2′}(G) = 1, then T T (G) = K(G) = X(G).

Proof. By assumption, we have G = \overline{G}, thus we may go through the possibilities for G according to Hypotheses 5.1.

If (D1) holds, i.e. G = P, then T T (G) = {[k_{G}]} by [12, Theorem 5.4]. If (D2) holds, i.e. G = 𝔄_{7}, then T T (G) = {[k_{G}]} by [8, Theorem B(a)].

Now assume that G satisfies one of (D3) to (D8). Set N := N_{G}(P). As P is dihedral of order at least 8, its automorphism group is a 2-group, so that

N = P C_{G}(P) = P × O_{2′}(C_{G}(P)) .

In particular, if G satisfies (D3), (D4), (D5) or (D6), then N = P, and hence X(N) = {[k_{N}]}, so that Lemma 3.3(b) yields T T (G) = K(G) = {[k_{G}]} = X(G).

Finally assume that G satisfies (D7) or (D8). Then N = P × C_{f}, so that X(N) ≅ C_{f}. But clearly X(G) ≅ C_{f} as well, so that the kG-Green correspondents of the onedimensional kN-modules are all one-dimensional. Hence

T T (G) = K(G) = X(G) ≅ C_{f}

by Lemma 3.3(b). □

As a consequence, we see that any torsion endo-trivial module of a finite group with dihedral Sylow 2-subgroup of order at least 8 which lies in the principal block has to be one-dimensional.

Corollary 6.2. There is an isomorphism of groups T T_{0}(G) ≅ X_{0}(G).

Proof. Since O_{2′}(G) acts trivially on the principal block, Lemma 3.2(c) yields

T T_{0}(G) = Inf_{G/O_{2′}(G)}^{G} (T T_{0}(G/O_{2′}(G))) .

Now, by Proposition 6.1, T T_{0}(G/O_{2′}(G)) = X_{0}(G/O_{2′}(G)). The claim follows. □

Next we consider the triple covers of 𝔄_{6}, 𝔄_{7} and PGL(2, 9) whose Schur multipliers have non-trivial 2′-parts as seen in Lemma 5.3. The next lemma also shows that T T (G) is not isomorphic to T T (G/O_{2′}(G)) via inflation in general.

Lemma 6.3. (a) Let H := 3.𝔄_{6}, then T T (H) = {[k_{H}], [9_{1}], [9_{2}]} ≅ ℤ/3ℤ, where 9_{1} and

9_{2} are mutually dual 9-dimensional, faithful, simple and trivial source kH-modules.

(b) Let G := 3.𝔄_{7}, then T T (G) = T T_{0}(G) = X(G) = {[k_{G}]}.

(c) Let R := 3.PGL(2, 9), then T T (R) = T T_{0}(R) = X(R) = {[k_{R}]}.

In the following proof, an ordinary irreducible character of degree d of a group G is denoted by χ_{d}, whereas irreducible Brauer characters are denoted by their degrees, and are identified with the corresponding simple kG-modules. Moreover ordinary irreducible characters are labelled according to [39, Decomposition Matrices]. We note that the above result about 3.𝔄_{6} and 3.𝔄_{7} appears in [27, Proposition 6.1], where it was obtained via a MAGMA computation [4], while we give here a character-theoretic proof.

Proof. First note that we may identify H with a subgroup of G and let P ∈ Syl_{2}(H), so that P ∈ Syl_{2}(G) as well. Then P ≅ D_{8} (the dihedral group of order 8), N := N_{H}(P) = N_{G}(P) ≅ C_{3} × P, and N ≤ H ≤ G. In particular, X(N) ≅ ℤ/3ℤ. By Lemma 3.3(b), we need to determine whether the kH- and kG-Green correspondents of the two non-trivial elements 1a, 1a′ ∈ X(N) are endo-trivial. So, set 𝔣_{H} := f_{(H, P, N)} and 𝔣_{G} := f_{(G, P, N)}. But 1a′ = 1a^{∗}, and endo-triviality is preserved by passage to the k-dual, hence we only need to determine whether 𝔣_{H}^{−1}(1a) and 𝔣_{G}^{−1}(1a) are endo-trivial modules.

Note that the group 3.𝔄_{6} has a maximal subgroup M ≅ C_{3} × 𝔖_{4} (indeed the product is direct because M(𝔖_{4}) ≅ C_{2}) containing N, so that N ≤ M ≤ H ≤ G. Hence

T T (M) = X(M) ≅ ℤ/3ℤ

and we denote by 1b and 1b^{∗} the two non-trivial elements of X(M), which we identify with their ordinary characters.

(a) We may assume that 𝔣_{H}^{−1}(1a) | 1b↑_{M}^{H}. Then we calculate that

χ_{1b}↑_{M}^{H} = χ_{6_{1}} + χ_{9_{2}} .

Thus dim_{k}(End_{kH}(1b↑_{M}^{H})) = 2 by making use of Scott’s Theorem in [33] (see also [23, II Theorem 12.4(i) and I Lemma 14.5]). Then it follows from the 2-decomposition matrix of H [39, A_{6}(mod 2)] that

1b↑_{M}^{H} = 9_{2} ⊕ ( \begin{matrix}3c \\ 3d \end{matrix} )

as kH-modules, where 3c and 3d are the two non-isomorphic simple modules of dimension 3 belonging to the 2-block containing 9_{2}, and ( \begin{matrix}3c \\ 3d \end{matrix} ) is a uniserial kH-module with composition factors 3c and 3d. Therefore 𝔣_{H}^{−1}(1b) = 9_{2}, which affords χ_{9_{2}} ∈ Irr(H). Hence 𝔣_{H}^{−1}(1b) is endo-trivial by Theorem 3.4 as it takes value 1 on any non-trivial 2-element of H, see [14, p.5]. As a consequence, T T (H) ≅ X(N) ≅ ℤ/3ℤ.

(b) Now passing to G, 𝔣_{G}^{−1}(1a) | f_{M}^{−1}(1b)↑_{H}^{G}, where f_{M} := f_{(H, P, M)} (recall that, on the other hand, 𝔣_{G} denotes the Green correspondence with respect to (G, P, N)). We calculate that

χ_{9_{2}}↑_{H}^{G} = χ_{15_{4}} + χ_{24_{1}} + χ_{24_{3}} .

But χ_{24_{1}}, χ_{24_{3}} ∈ Irr(G) being defect-zero characters, we obtain that 𝔣_{G}^{−1}(1a) affords χ_{15_{4}} ∈ Irr(G), which does not take value 1 on the unique conjugacy class of involutions, see [14, p.10]. Thus, by Theorem 3.4, 𝔣_{G}^{−1}(1a) and 𝔣_{G}^{−1}(1a′) are not endo-trivial modules, and we must have T T (G) = {[k_{G}]}.

(c) By [39, A_{6}.2_{2} (mod 2)], R has a unique block of full defect, namely the principal block. Therefore it follows from Remark 3.1 that T T (R) = T T_{0}(R) and we conclude from Corollary 6.2 and the proof of Proposition 6.1 that T T (R) = X(R) = {[k_{R}]}. □

Corollary 6.4 (See Remark 1.3.). Set H := O_{2′}(G) and \overline{G} := G/H. Assume further that \overline{G} ≅ 𝔄_{6} and G has a unique component E := E(G) ≅ 3.𝔄_{6}. Then, the following hold:

(a) G = HE, [H, E] = 1 (and hence G is a central product of H and E), and moreover H ∩ E = Z(E) ≅ C_{3} and Z(G) ≤ H = C_{G}(E).

(b) Let W be a simple kE-module such that [W] ∈ K(E) and dim(W) = 9 as in Lemma 6.3(a). Then, there exists a simple and trivial source kG-module V such that V ↓_{E} ≅ W ⊕ ⋯ ⊕ W (m summands) for a positive integer m. Further, V is endo-trivial if and only if W extends to V. In particular, if H^{2}(G/E, k^{×}) = 1, then there exists a 9-dimensional simple kG-module whose class is in K(G).

Proof. (a) Since E ◁ G and G/H is non-abelian simple, we have G = HE. Then,

𝔄_{6} ≅ G/H = HE/H ≅ E/(H ∩ E) = (3.𝔄_{6})/(H ∩ E).

Hence C_{3} ≅ H ∩ E = Z(E). We have also Z(G) ≤ H since Z(G) ◁ G and G/H is non-abelian simple. Further, if G = HC_{G}(E), then [P, E] = 1, a contradiction, so that C_{G}(E) ≤ H since C_{G}(E) ◁ G and G/H is non-abelian simple.

Next, we claim that H ≤ C_{G}(E). Take any h ∈ H, and let ϕ_{h} be an element of Aut(E) given by y ↦ h^{−1}yh for y ∈ E. Since |Z(E)| = 3, ϕ_{h} acts trivially on Z(E), and hence we can consider that ϕ_{h} ∈ Aut(E/Z(E)) = Aut(𝔄_{6}). Since |h| is odd and |Out(𝔄_{6})| = 4 by [14, p.4], we know that ϕ_{h} is an inner automorphism of 𝔄_{6}, and hence ϕ_{h} is an inner automorphism of E. This implies that there is an element y_{0} ∈ E with hy_{0} ∈ C_{G}(E). Since we know already that C_{G}(E) ≤ H, we have y_{0} ∈ H, so that y_{0} ∈ H ∩E = Z(E) ≤ C_{G}(E). Therefore h ∈ C_{G}(E).

(b) The first part follows easily from (a) and the Clifford Theorem. Then, the second part follows from Theorem 3.4. The final part follows immediately. □

7. Torsion endo-trivial modules in the dihedral case: Proof of Theorem 1.2

We now turn to the general case and prove Theorem 1.2 of the introduction. Throughout this section we use the notations G, P and \overline{G} as in §5, and further set H := O_{2′}(G) (hence \overline{G} := G/H) and \overline{P} := HP/H ≅ P.

Proposition 7.1. Let V be an indecomposable endo-trivial kG-module such that [V] ∈ K(G). Then the following hold:

(a) If \overline{G} ≇ 𝔄_{6}, then dim_{k} V = 1.

(b) If \overline{G} ≅ 𝔄_{6} and dim_{k} V ≠ 1, then dim_{k} V = 9, V is simple, [V^{⊗3}] ∈ X(G).

Proof. Since G has no strongly 2-embedded subgroups by Lemma 5.2, Theorem 4.3(a) yields that there exist a 2-cocycle α ∈ H^{2}(\overline{G}, k^{×}), a one-dimensional k^{α}G-module 1b and a k^{α^{−1}} \overline{G}-module W, which we may regard as a k^{α^{−1}}G-module via inflation from \overline{G} to G, such that

V = 1b ⊗_{k} W .

Now, we go through the possibilities for \overline{G} according to Hypotheses 5.1 and compute dim_{k}(V) in each case.

To start with, if G satisfies one of the hypotheses (D1), (D5), (D6), (D7), or (D8), then H^{2}(\overline{G}, k^{×}) = 1 by Lemma 5.3. Therefore Theorem 1.1 yields

K(G) = X(G) + InfG ; \overline{G}(K(\overline{G})) .

Besides K(\overline{G}) = X(\overline{G}) by Proposition 6.1. In consequence K(G) = X(G) and it follows that dim_{k}(V) = 1 in all cases.

Hence we may assume that G satisfies one of (D2), (D3), or (D4). If α is trivial, then by the same argument as above we obtain dim_{k}(V) = 1. Therefore we assume from now on that α is non-trivial and it follows from Lemma 5.3 that |α| = 3. Then there exists a non-split central extension

(Z) : 1 → Z̃ → G̃ → \overline{G} → 1

where Z̃ ≅ C_{3}, and we write G̃ = 3.\overline{G}, the triple cover of \overline{G}. Then it follows from Theorems of Schur [30, p.214, and Theorems 3.5.21 and 3.5.22] that the module W over k^{α^{−1}} \overline{G} corresponds to an indecomposable kG̃-module W̃ such that W̃ = W as k-vector spaces, and moreover, if P̃ ∈ Syl_{2}(G̃), then W ↓_{k\overline{P}}^{k^{α^{−1}} \overline{G}} ≅ W̃↓_{kP̃} as kP̃- (and also as k\overline{P} -) modules via the canonical isomorphism P̃ ≅ \overline{P} ≅ P. We claim that [W̃] ∈ K(G̃). Indeed, since α↓_{H⋊P} is trivial by Theorem 4.3(a) and its proof, we have

V ↓_{H⋊P} = (1b)↓_{k(H⋊P)} ⊗_{k} W ↓_{k(H⋊P)}

as a tensor product of (non-twisted) k(H ⋊ P)-modules. By Lemma 3.3(b), [V ↓_{H⋊P}] ∈ K(H ⋊ P) since [V] ∈ K(G) and Res_{H⋊P}^{G} is a group homomorphism. Therefore, since K(H ⋊ P) is a subgroup of T (H ⋊ P), we have

[W ↓_{k(H⋊P)}] = [V ↓_{H⋊P}] − [(1b)↓_{k(H⋊P)}] ∈ K(H ⋊ P) = T T (H ⋊ P) ,

where the latter equality of groups holds by Lemma 3.3(c). Thus, as H acts trivially on W, it follows from Lemma 3.2(c) that

[W ↓_{k((H⋊P)/H)}] ∈ T T ((H ⋊ P)/H) = T T (\overline{P}) = K(\overline{P}) ,

where, again, the latter equality of groups holds by Lemma 3.3(c). But K(\overline{P}) = \{[k\overline{P}]\} by definition since \overline{P} is a 2-group. Thus, [W ↓\overline{P}] = [k\overline{P}], and hence [W̃↓P̃] = [kP̃], so that by Lemma 3.2(a), W̃ itself is a trivial source endo-trivial kG̃-module, that is, [W̃] ∈ K(G̃).

Now assume that G satisfies (D4), i.e. \overline{G} ≅ PGL(2, 9). In this case, G̃ = 3.PGL(2, 9). Since W̃ is an indecomposable endo-trivial module, Lemma 6.3(c) yields W̃ = kG̃. Thus

dim_{k} V = dim_{k}(1b ⊗_{k} W) = dim_{k} W = dim_{k} W̃ = 1 .

Next assume that G satisfies (D2), i.e. \overline{G} ≅ 𝔄_{7}. In this case, G̃ = 3.𝔄_{7}. By the above, the kG̃-module W̃ is indecomposable endo-trivial such that [W̃] ∈ K(G̃). Thus, Lemma 6.3(b) yields W̃ = kG̃, so that dim_{k} V = dim_{k} W = dim_{k} W̃ = 1.

Finally assume that G satisfies (D3), i.e. \overline{G} ≅ 𝔄_{6}, so that G̃ = 3.𝔄_{6}. Since W̃ is indecomposable endo-trivial and [W̃] ∈ K(G̃), Lemma 6.3(a) implies that W̃ ∈ {kG̃, 9_{1}, 9_{2}} (where 9_{1} and 9_{2} are as in Lemma 6.3(a)) and therefore is a simple module. Hence, again, we compute

dim_{k} V = dim_{k}(1b ⊗_{k} W) = dim_{k} W = dim_{k} W̃ ∈ {1, 9} .

Next, we claim that V is a simple module as well. We may assume, without loss of generality, that W̃ = 9_{1}, which lies in one of the two non-principal 2-blocks of full defect of 3.𝔄_{7}. So let B and B̃ be the blocks of kG and kG̃, respectively, to which V and W̃ belong. Then, B and B̃ correspond to each other by the Morita equivalence given by the result of Morita in [21, Lemma 2] (see also [29] and [30, Theorem 5.7.4]), and V and W̃ correspond to each other via this Morita equivalence. Hence V is a simple kG-module since W̃ is simple.

Now, by Theorem 4.3(b), we have that V^{⊗3} = (1b)^{⊗3} ⊗_{k} W^{⊗3} is the tensor product of a (non-twisted) kG-module (1b)^{⊗3} and a (non-twisted) kG-module W^{⊗3} such that [W^{⊗3}] ∈ InfG ; \overline{G}(K(\overline{G})). Since \overline{G} = 𝔄_{6}, we have K(\overline{G}) = \{[k\overline{G}]\} by Proposition 6.1. Therefore [W^{⊗3}] = [k_{G}], and hence [V^{⊗3}] = [(1b)^{⊗3}] ∈ X(G).

□

Proof of Theorem 1.2. Since the Sylow 2-subgroups of G are dihedral of order at least 8, Lemma 3.3(c) yields T T (G) = K(G). Thus the claims follow from Proposition 7.1. □

Example 7.2. We now give an example of a group G with a dihedral Sylow 2-subgroup P of order 8, for which there exist classes [V] ∈ T T (G) such that

[V^{⊗3}] ∈ X(G)\{[k_{G}]} .

We put ourselves in the situation of Corollary 6.4 and consider the following example. Namely, G is a central product defined by

G := C_{9} ∗ 3.𝔄_{6} ≅ (C_{9} × 3.𝔄_{6})/C_{3} ,

so that \overline{G} := G/O_{2′}(G) ≅ 𝔄_{6}, but H^{2}(\overline{G}, k^{×}) ≅ C_{3}. It easily follows that X(G) ≅ ℤ/3ℤ and X(N_{G}(P)) ≅ ℤ/9ℤ. Then, using GAP [5], we compute the following. First, inducing the nine linear characters of N_{G}(P) to G, we find that the kG-Green correspondents of the associated kN_{G}(P)-modules afford the three linear characters corresponding to X(G) ≅ ℤ/3ℤ (this part is obvious) and six 9-dimensional ordinary characters χ_{9_{1}}, …, χ_{9_{6}} reducing modulo 2 to 9-dimensional simple kG-modules 9_{1}, …, 9_{6}, respectively (all lying in pairwise distinct blocks). Then Theorem 3.4 ensures that these modules are all endotrivial since their ordinary characters take value one on any non-trivial 2-element of G. Hence we conclude that

T T (G) = K(G) ≅ X(N_{G}(P)) ≅ ℤ/9ℤ.

Finally we see that the characters χˆ9_{i} ⊗_{K} χˆ9_{i} ⊗_{K} χˆ9_{i} for each 1 ≤ i ≤ 6 have no trivial constituents, so that [(9_{i})^{⊗3}] ∈ X(G)\{[k_{G}]} for each 1 ≤ i ≤ 6.

8. Groups with Klein-four Sylow 2-subgroups revisited

The purpose of this section is to prove Theorem 1.5. Throughout this section we assume that P is a Klein-four Sylow 2-subgroup of G, that is P ≅ C_{2} × C_{2}, and let N := N_{G}(P). Furthermore, we use the notations H := O_{2′}(G), \overline{G} := G/H and \overline{P} := (HP)/H = (H ⋊ P)/H ≅ P.

Lemma 8.1. One of the following holds:

(1) \overline{G} ≅ P.

(2) \overline{G} ≅ 𝔄_{4} ≅ PSL(2, 3),

(3) \overline{G} ≅ PSL(2, q) ⋊ C_{f}, where q = r^{m} is a power of an odd prime r with 3 < q ≡ ±3 (mod 8), f is an odd integer, and C_{f} ≤ Gal(𝔽_{q}/𝔽_{r}) ≅ C_{m}.

Proof. This follows from [38, Theorem I], see also [35, Proof of Theorem 6.8.7, (8.9) in Chapter 6 and Theorem 6.8.11]. □

Hypotheses 8.2. For the purposes of our computations, we split case (3) above in further subcases and say that G satisfies the hypothesis: (K1) if \overline{G} ≅ P; (K2) if \overline{G} ≅ 𝔄_{4} ≅ PSL(2, 3); (K3) if \overline{G} ≅ 𝔄_{5} ≅ PSL(2, 5); (K4) if \overline{G} ≅ PSL(2, q) ⋊ C_{f}, where q = r^{m} is a power of an odd prime r with 3 < q ≡ 3

(mod 8) and f is odd with f|m (namely C_{f} ≤ Gal(𝔽_{q}/𝔽_{r}) ≅ C_{m}); (K5) if \overline{G} ≅ PSL(2, q) ⋊ C_{f}, where q = r^{m} is a power of an odd prime r with 5 < q ≡ 5

(mod 8) for an odd f with f|m (namely C_{f} ≤ Gal(𝔽_{q}/𝔽_{r}) ≅ C_{m}).

Lemma 8.3. The following two conditions are equivalent:

(1) G has a strongly 2-embedded subgroup.

(2) \overline{G} ≅ 𝔄_{5}. Moreover, if (1) holds, then any strongly 2-embedded subgroup G_{0} of G is of the form HN where N := N_{G}(P) for a suitable choice of P ∈ Syl_{2}(G), so that G_{0}/O_{2′}(G_{0}) ≅ 𝔄_{4}.

Proof. This follows from the Bender-Suzuki Theorem [35, Theorem 6.4.2(2)]. □

Lemma 8.4. The group H^{2}(\overline{G}, k^{×}) is trivial.

Proof. Set h := |H^{2}(\overline{G}, k^{×})|. It follows from [40, Proposition 3.2] (see also [21, Lemma 5] and [30, Lemma 3.5.4(ii)]) that |H^{2}(\overline{G}, k^{×})| = |M(\overline{G})|_{2′} since k has characteristic 2.

If (K1) holds, then h = 1 by [30, Theorem 2.7.4]. Otherwise, by Lemma 8.1, we have \overline{G} ≅ PSL(2, q) ⋊ C_{f}, where q is a power of an odd prime with q ≡ ±3 (mod 8), and f is odd. First, note that |M(PSL(2, q))| = 2, see [20, Theorem 4.9.1(ii)]. If q = 3, then \overline{G} ≅ PSL(2, 3) ≅ 𝔄_{4} and f = 1, so that the assertion holds.

So we may assume q > 3, and hence PSL(2, q) is non-abelian simple as is well known. Since \overline{G}/PSL(2, q) is cyclic and PSL(2, q) is perfect, it follows from [19, Theorem 3.1(i)] that |M(\overline{G})| | |M(PSL(2, q))| = 2, so that h = 1. □

Lemma 8.5. Suppose that θ ∈ Irr(H) and that V is an indecomposable kG-module such that V ↓_{H} contains θ as a constituent. If \overline{G} ≅ 𝔄̸_{5} and V is endo-trivial, then θ is Ginvariant.

Proof. If \overline{G} ≅ 𝔄̸_{5}, then by Lemma 8.3, G has no strongly 2-embedded subgroups. Therefore the claim follows from Lemma 4.2. □

Proposition 8.6. If H = 1, then one of the following five cases holds:

(a) If (K1) holds, then K(G) = {[k_{G}]}.

(b) If (K2) holds, then K(G) = X(G) = {[k], [1_{ω}], [1_{ω^{2}}]} ≅ ℤ/3ℤ, where 1_{ω} and 1_{ω^{2}} are the two non-trivial one-dimensional kG-modules.

(c) If (K3) holds, then K(G) = {[k], [5i], [(5i)^{∗}]} ≅ K(𝔄_{4}) ≅ ℤ/3ℤ, and where 5i, (5i)^{∗} are uniserial, trivial source, and endo-trivial kG-modules in B_{0}(G), both affording the unique irreducible character of degree 5, χ_{5} ∈ Irr(G). (See [22, Lemma 4.1].)

(d) If (K4) holds, then K(G) ≅ X(G) ⊕ T T_{0}(G), where X(G) ≅ ℤ/fℤ and T T_{0}(G) = ⟨[(q − 1)/2]⟩ ≅ ℤ/3ℤ for (q − 1)/2 a simple trivial source endo-trivial kG-module of dimension (q − 1)/2 affording an irreducible character χ_{(q−1)/2} ∈ Irr(B_{0}(G)).

(e) If (K5) holds, then K(G) ≅ X(G) ⊕ T T_{0}(G), where X(G) ≅ ℤ/fℤ and T T_{0}(G) = ⟨[V]⟩ ≅ ℤ/3ℤ for a trivial source endo-trivial kG-module V such that V is uniserial of length 3 with composition factors ((q−1)/2)a, k_{G}, ((q−1)/2)b, where ((q−1)/2)a and ((q − 1)/2)b are non-isomorphic simple kG-modules in B_{0}(G) of dimension (q − 1)/2, and V affords the Steinberg character St_{G} ∈ Irr(B_{0}(G)) of degree q.

Proof. (a) is clear since G ≅ C_{2} × C_{2} is a 2-group. Next K(𝔄_{4}) ≅ K(𝔄_{5}) ≅ ℤ/3ℤ by [8, Theorem 4.2 and its proof]. Moreover K(𝔄_{4}) ≅ X(𝔄_{4}) by Lemma 3.3(a) because C_{2} × C_{2} is normal in 𝔄_{4}, and the structure of the modules in K(𝔄_{5}) is given by [22, Lemma 4.1]. Hence (b) and (c) hold.

If (K4) or (K5) holds, then K(G) ≅ ℤ/fℤ ⊕ ℤ/3ℤ with X(G) ≅ ℤ/fℤ and T T_{0}(G) ≅ ℤ/3ℤ by [22, Theorem 1.4(d) and its proof]. Finally the structure of a generator [V] of T T_{0}(G) with V indecomposable is obtained as follows. By [22, Proposition 4.3(a)], the lift of V to characteristic zero affords an irreducible character χ_{V} ∈ Irr(B_{0}(G)), so that using Theorem 3.4, by investigation of the generic character table of PSL(2, q) (see e.g. [3, Table 5.4]) we see that χ_{V} (1) = (q − 1)/2 if (K4) holds, and χ_{V} (1) = q if (K5) holds. Then the composition factors of V follow from the 2-decomposition matrix of PSL(2, q), see [3, Table 9.1]. □

Proof of Theorem 1.5. First T T_{0}(G) ≅ ℤ/3ℤ by [22, Theorem 1.4 and its proof], and any indecomposable torsion endo-trivial module lifts to an irreducible ordinary character by [22, Theorem 1.1(a)]. In addition, since P ≅ C_{2} × C_{2}, T T (G) = K(G) by Lemma 3.3(c).

Assume that (K1) holds, then G is 2-nilpotent, and [31, Theorem] yields K(G) = X(G), as was conjectured in [9, Conjecture 3.6]. Hence (a) holds.

Assume that (K2) holds, then G is solvable by the Feit-Thompson Theorem, so that K(G) = X(G) by [9, Theorem 6.2(2)]. Hence (b) holds.

Assume that (K3) holds. Then, by Lemma 8.3, G has a strongly 2-embedded subgroup G_{0} such that G_{0}/O_{2′}(G_{0}) ≅ 𝔄_{4}. Therefore K(G) ≅ K(G_{0}) by Lemma 3.5 and K(G_{0}) ≅ X(G_{0}) by (b). The non-trivial indecomposable torsion endo-trivial kG-modules in B_{0}(G) have dimension 5 by Proposition 8.6(c). Hence (c) holds.

Assume that (K4) or (K5) holds. Then G has no strongly 2-embedded subgroups, and H^{2}(G, k^{×}) = 1 by Lemmas 8.3 and 8.4. Therefore, by Theorem 1.1,

K(G) = X(G) + K(\overline{G}) ,

where we identify K(\overline{G}) with InfG ; \overline{G}(K(\overline{G})). In addition, by Proposition 8.6(d) and (e), we have K(\overline{G}) = X(\overline{G}) ⊕ T T_{0}(\overline{G}) ≅ ℤ/fℤ ⊕ ℤ/3ℤ. Therefore

K(G) = X(G) + X(\overline{G}) + T T_{0}(\overline{G}) = X(G) + T T_{0}(G)

since InfG ; \overline{G}(T T_{0}(\overline{G})) = T T_{0}(G) and InfG ; \overline{G}(X(\overline{G})) ≤ X(G). But X(G) ∩ T T_{0}(G) = \{[k_{G}]\} by Proposition 8.6(d) and (e), thus

K(G) ≅ X(G) ⊕ T T_{0}(G) ≅ X(G) ⊕ ℤ/3ℤ .

Hence (d) holds. □

Acknowledgements. The authors are grateful to Ron Solomon for answering a question on finite groups, to Gunter Malle for useful comments on a preliminary version of this text, and to Atumi Watanabe and Rebecca Waldecker for useful discussions. The first author is grateful to the Swiss National Foundation of Sciences for supporting his visit at the TU Kaiserslautern in September 2014 through the second author’s Fellowship for Prospective Researchers PBELP2_{−}143516, and also to Gunter Malle for supporting his visit at the TU Kaiserslautern in May 2015.

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Department of Mathematics, Graduate School of Science, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba, 263-8522, Japan.

 E-mail address: koshitan@math.s.chiba-u.ac.jp

FB Mathematik, TU Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany.

 E-mail address: lassueur@mathematik.uni-kl.de