ENDO-TRIVIAL MODULES FOR FINITE GROUPS WITH KLEIN-FOUR SYLOW 2-SUBGROUPS

SHIGEO KOSHITANI AND CAROLINE LASSUEUR

Abstract. We study the finitely generated abelian group T (G) of endo-trivial kGmodules where kG is the group algebra of a finite group G over a field of characteristic p > 0. When the representation type of the group algebra is not wild, the group structure of T (G) is known for the cases where a Sylow p-subgroup P of G is cyclic, semi-dihedral and generalized quaternion. We investigate T (G), and more accurately, its torsion subgroup T T (G) for the case where P is a Klein-four group. More precisely, we give a necessary and sufficient condition in terms of the centralizers of involutions under which T T (G) = f^{−1}(X(N_{G}(P))) holds, where f^{−1}(X(N_{G}(P))) denotes the abelian group consisting of the kG-Green correspondents of the one-dimensional kN_{G}(P)-modules. We show that the lift to characteristic zero of any indecomposable module in T T (G) affords an irreducible ordinary character. Furthermore, we show that the property of a module in f^{−1}(X(N_{G}(P))) of being endo-trivial is not intrinsic to the module itself but is decided at the level of the block to which it belongs.

1. Introduction

In the representation theory of finite groups, endo-trivial modules have seen considerable interest in the last fifteen years. This lead to a complete classification for finite p-groups where p is a prime, as well as to the determination of the Dade groups for all finite p-groups (see, for example, [32] and the references therein). The classification of endo-trivial kG-modules is in general achieved via the description of the structure of the finitely generated group T (G), called the group of endo-trivial modules of the finite group G, where k is an algebraically closed field of characteristic p. Let us recall that by definition, a finitely generated kG-module M is called endo-trivial if there is an isomorphism

End_{k}(M) ≅ k_{G} ⊕ (proj)

of kG-modules, where k_{G} is the trivial kG-module and (proj) denotes a projective kGmodule. The structure of the group T (G) has been determined for some classes of general finite groups (see [5, 7, 25, 6, 9, 10, 23], for example), but no general solution to this problem is known.

In particular the structure of T (G) is known when the Sylow p-subgroups of G are cyclic [25], semi-dihedral and generalized quaternion [10]. Therefore if the representation type is not wild, it only remains to treat the case of dihedral Sylow 2-subgroups. In fact in this case the torsion-free rank of T (G) is known by [8, Theorem 3.1 and Corollary 3.2], so that the main problem remaining is to describe the structure of the torsion subgroup T T (G) of T (G).

2000 Mathematics Subject Classification. 20C20, 20C05.

Key words and phrases. Endo-trivial module, simple module, Green correspondence, trivial source module, Auslander-Reiten quiver.

The structure of T T (G) may be of different nature in case the Sylow 2-subgroups of G are isomorphic to the Klein-four group (that is, kG has domestic representation type) and in case the Sylow 2-subgroups are dihedral of order at least 8 (namely, kG is of tame, but not domestic, representation type), see [2, Theorem 4.4.4]. In this note, we treat the former case.

In [5, Theorem A], Carlson states that in the known cases, but for few exceptions, T T (G) = X(G), where X(G) is the subgroup of T (G) consisting of all one-dimensional kG-modules, see [25, Section 2]. We shall show that groups with the Klein-four Sylow 2-subgroup are exceptions in the sense of Carlson.

In fact, if P ∈ Syl_{2}(G) is such that P ≅ C_{2} ×C_{2}, then the indecomposable torsion endotrivial modules coincide with the trivial source endo-trivial modules, or in other words the kG-Green correspondents of the one-dimensional kN_{G}(P)-modules which are at the same time endo-trivial, that is to say T T (G) = T (G) ∩ f^{−1}(X(N_{G}(P))), where f is the Green correspondence with respect to (G, P, N). We obtain a necessary and sufficient condition in terms of the centralizers of involutions under which T T (G) = f^{−1}(X(N_{G}(P))) occurs. Namely, one of our main results of this paper is the following.

Theorem 1.1. Let p = 2, and let G be a finite group with a Klein-four Sylow 2-subgroup P ≅ C_{2} × C_{2}. Let (K, O, k) be a splitting 2-modular system for G and its subgroups. Further set N := N_{G}(P), and let f be the Green correspondence with respect to (G, P, N). Now, suppose that 1a is any one-dimensional kN-module, and B is the 2-block of G to which f^{−1}(1a), the kG-Green correspondent of 1a, belongs. Then:

(a) The kG-module f^{−1}(1a) uniquely lifts to an OG-lattice affording an irreducible character in Irr(B).

(b) Further, f^{−1}(1a) is an endo-trivial kG-module if and only if it holds that B is of principal type and that ϕ_{u}(1) = 1 for any element u ∈ P with u ≠ 1, where ϕ_{u} is the unique irreducible Brauer character of a 2-block B_{u} of C_{G}(u) such that B_{u}^{G} = B (notice that such a B_{u} is unique since B is of principal type and B_{u} is nilpotent).

(c) In particular, if f^{−1}(1a) is endo-trivial then all the kG-Green correspondents of one-dimensional kN-modules which lie in B are endo-trivial.

We emphasize that the endo-trivial modules f^{−1}(1a) afford irreducible ordinary characters, and also that endo-triviality in this case depends only on the block, see Theorem 1.1(a) and (c). From the previously known cases, see e.g. [6, 7, 8, 9, 10, 25], this does not seem to occur in general.

Remark 1.2. Keep the notation as in Theorem 1.1. There are finite groups G, both in the case that |N_{G}(P) : C_{G}(P)| = 1 (namely G is 2-nilpotent) and in the case that |N_{G}(P) : C_{G}(P)| = 3, which satisfy the condition that there exists an indecomposable trivial source kG-module f^{−1}(1a) which is not endo-trivial. In particular, we give an example where X(G) ≅ {0} and T T (G) ≅ ℤ/3ℤ while X(N_{G}(P)) ≅ ℤ/3ℤ⊕ℤ/3ℤ⊕ℤ/3ℤ. See Examples 3.3 and 3.5.

We obtain sufficient conditions under which f^{−1}(1a) in Theorem 1.1 is endo-trivial.

Corollary 1.3. Keep the notation as in Theorem 1.1.

Let u be any element of P with u ≠ 1.

(a) If f^{−1}(1a) belongs to a 2-block of G containing a one-dimensional kG-module, then f^{−1}(1a) is endo-trivial. In particular, if 1a is in the principal block of N, then f^{−1}(1a) is endo-trivial.

(b) If O_{2′}(C_{G}(u)) is abelian, then f^{−1}(1a) is endo-trivial.

In terms of the structure of the group T (G) of endo-trivial module, we have the following result, which is also one of our main results:

Theorem 1.4. Let p = 2, and let G be a finite group with the Klein-four Sylow 2-subgroup P ≅ C_{2} × C_{2}. Assume further that |N_{G}(P) : C_{G}(P)| = 3. Then the following holds:

(a) The group T (𝔄_{4}) ≅ ℤ/3ℤ ⊕ ℤ embeds in the group T (G) via inflation from N_{G}(P)/O_{2′}(N_{G}(P)) ≅ 𝔄_{4} to N_{G}(P) followed by the Green correspondence with respect to (G, P, N_{G}(P)).

(b) If P is self-centralizing, that is C_{G}(P) = P, then T (G) ≅ ℤ/3ℤ ⊕ ℤ.

(c) If u ∈ P is non-trivial and O_{2′}(C_{G}(u)) is abelian then T (G) ≅ f^{−1}(X(N_{G}(P)))⊕ℤ.

(d) If G = H × G_{0}, where H is a 2′-group and G_{0} is almost simple, that is G_{0} = L_{2}(q) ⋊ C_{n} with q ≡ ±3 (mod 8) and C_{n} is the cyclic group of odd order n, then it holds that T (G) ≅ f^{−1}(X(N_{G}(P))) ⊕ ℤ and that f^{−1}(X(N_{G}(P))) ≅ ℤ/3ℤ ⊕ ℤ/nℤ ⊕ H/[H, H].

Remark 1.5. Keep the notation as in Theorem 1.4.

(a) We note that a finite group G satisfying the assumptions of Theorem 1.4(b) can be written as G/O_{2′}(G) ≅ L_{2}(q) with q ≡ ±3 (mod 8). The groups L_{2}(q) with q ≡ ±3 (mod 8) and q > 3 are in fact the only finite simple groups with a Kleinfour Sylow 2-subgroup C_{2} × C_{2}, see [1] and [33].

(b) Also note that the case |N_{G}(P) : C_{G}(P)| = 1 is excluded from the above Theorem 1.4, as it is already treated in [9] and [28]. Actually in this case T (G) ≅ X(G) ⊕ ℤ.

This paper is divided into five sections. In §2 we introduce the notation we shall use and fundamental general facts, and also preliminaries on endo-trivial modules. §3 contains known results about endo-trivial modules for finite groups with Klein-four Sylow 2-subgroups. §4 deals with the Green correspondents of one-dimensional modules over the normalizer of Klein-four Sylow 2-subgroups. §5 contains complete and detailed proofs of the main results Theorems 1.1 and 1.4, and Corollary 1.3.

2. Notation and Preliminaries

Throughout, unless otherwise specified, we always let p denote a prime number and G a finite group such that p divides the order |G| of G. 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, and its residue field k = O/rad(O) is of characteristic p, and that K and k are splitting fields for all subgroups of G. Moreover, modules are always finitely generated, and kG-mod denotes the category of all finitely generated left kG-modules. We say that 𝔐 is an OGlattice if 𝔐 is a left OG-module such that 𝔐 is a free O-module of finite rank. We write Syl_{p}(G) for the set of all Sylow p-subgroups of G. If P ∈ Syl_{p}(G), then we denote by f the Green correspondence with respect to (G, P, N_{G}(P)), see [27, p.276]. We write ΔG for the diagonal subgroup {(g, g) ∈ G × G | g ∈ G} of G × G. For a positive integer n, we denote by C_{n} a cyclic group of order n, and by 𝔖_{n} and 𝔄_{n}, respectively, the symmetric and alternating groups of degree n. We write Z(G) for the center of G, for elements g, h ∈ G we write [g, h] := g^{−1}h^{−1}gh and g^{h} := h^{−1}gh, and we write G^{ab} := G/[G, G] for the abelianization of G. For two kG-modules M and M′, M ⊗ M′ means M ⊗_{k} M′. We denote by k_{G} or simply by k the trivial kG-module. We write H ≤ G if H is a subgroup of G. In such a case, for M ∈ kG-mod and N ∈ kH-mod, we denote by Res_{H}^{G}(M) and Ind_{H}^{G}(N), respectively, the restriction of M to H and the induced module of N to G. 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. For H, L ≤ G, we write G = H ⋊ L if G is a semi-direct product of H by L, note that H ⊴ G. By a block B of G (or OG), we mean a block ideal of OG. Then, we write KB and kB, respectively, for K ⊗_{O} B and k ⊗_{O} B. We denote by 1_{B} the block idempotent of kB or sometimes of B. We denote by Irr(G) and IBr(G) the sets of all irreducible ordinary and Brauer characters of G, respectively. For a block B of G, we also denote by Irr(B) the set of all characters in Irr(G) which belong to B. Similarly for IBr(B). We write 1_{G} for the trivial ordinary or Brauer character of G. For an OG-lattice 𝔐, we denote by K𝔐 and k𝔐, respectively, K ⊗_{O} 𝔐 and k ⊗_{O} 𝔐.

For Q ≤ G, we define the Brauer homomorphism Br_{Q}^{G} : kG → kC_{G}(Q) by setting ∑_{g∈G} α_{g}g ↦ ∑_{g∈C_{G}(Q)} α_{g}g where α_{g} ∈ k. For a block B of G with defect group P we say that B is of principal type if, for any Q ≤ P, Br_{Q}^{G}(1_{B}) is a block idempotent of kC_{G}(Q), see [16, Proposition 3.1(iii)].

For an OG-lattice L̂, we denote by L̂^{∨} the O-dual of L̂, namely, L̂^{∨} = Hom_{O}(L̂, O). Let M be a kG-module. We write M^{∗} for the k-dual of M, namely, M^{∗} = Hom_{k}(M, k). We say that M is a trivial source module if it is a direct sum of indecomposable kG-modules all of whose sources are trivial modules, see [31, p.218]. It is known that a trivial source kG-module M lifts uniquely to a trivial source OG-lattice, so denote it by M̂, see [27, Chap.4, Theorem 8.9(iii)]. Then, by χM̂ we denote the ordinary character of G afforded by M̂. For χ ∈ Irr(G) and ϕ_{u} ∈ IBr(C_{G}(u)) where u is a p-element of G, we denote by d_{χ,ϕ_{u}}^{u} the generalized decomposition number with respect to χ and ϕ_{u}, see [27, p.327]. We let X(G) denote the group of one-dimensional kG-modules endowed with the tensor product − ⊗_{k} −, and recall that X(G) ≅ (G^{ab})_{p′}. For an integer n ≥ 1 and a ring R we denote by Mat_{n}(R) the full matrix ring of n × n-matrices over R.

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 other notation and terminology, we refer to the books [27] and [31].

2.1. Endo-trivial kG-modules. We start by summing up some the basic properties of endo-trivial modules.

Lemma 2.1. Let G be a finite group, let P ∈ Syl_{p}(G) and let H ≤ G be a subgroup.

(a) If M ∈ kG-mod is endo-trivial, then M ≅ M_{0}⊕(proj) where M_{0} is indecomposable and endo-trivial. The relation M ∼ N ⇔ M_{0} ≅ N_{0} is an equivalence relation on the class of endo-trivial modules. We let T (G) denote the resulting set of equivalence classes.

(b) The set T (G), endowed with the law [M] + [N] := [M ⊗_{k} N] induced by the tensor product ⊗_{k}, is an abelian group called the group of endo-trivial modules of G. The zero element is the class [k] and −[M] = [M^{∗}], the class of the dual module.

(c) Any one-dimensional kG-module is endo-trivial. Thus identifying X ∈ X(G) with its class [X] ∈ T (G), we have X(G) ≤ T (G).

(d) If M is an endo-trivial kG-module, then Res_{H}^{G}(M) = N ⊕ (proj), where N is an endo-trivial kH-module. Moreover, if H ≥ P, then a kG-module M is endo-trivial if and only if Res_{H}^{G}(M) is endo-trivial.

(e) If p | |H|, then the restriction induces a group homomorphism Res_{H}^{G}: T (G) ⟶ T (H) : [M] ↦ [Res_{H}^{G}(M)]. Moreover if H ≥ N_{G}(P), then Res_{H}^{G} is injective and if M is an indecomposable endo-trivial kG-module, then [Res_{H}^{G}(M)] = [f_{H}(M)] where f_{H} denotes the Green correspondence with respect to (G, P, H).

(f) The group T (G) is finitely generated, so that we may write T (G) = T T (G)⊕T F (G) where T T (G) is the torsion subgroup and T F (G) is a torsion-free subgroup.

(g) If T T (P) is trivial, then T T (N_{G}(P))= ker (Res_{P}^{N_{G}(P)}) ≅ X(N_{G}(P)).

(h) If M is endo-trivial, then dim_{k}(M) ≡ ±1 (mod |G|_{p}) if p is odd; and dim_{k}(M) ≡ ±1 (mod \frac{1}{2}|G|_{2}) if p = 2. Moreover if M is indecomposable with trivial source, then dim_{k}(M) ≡ 1 (mod |G|_{p}).

(i) Endo-trivial kG-modules have the Sylow p-subgroups as their vertices and lie in p-blocks of full defect.

Proof. For (a) and (b) see [8, Definition 2.2.]. For (c) see [25, Section 2]. For (d) see [7, Proposition 2.6]. Statement (e) is [25, Lemma 2.7(1)]. For (f), see [7, Corollary 2.5]. For (g), see [25, Lemma 2.6] and [8, Theorem 2.3(d)(ii)]. The first statement of (h) is [23, Lemma 2.1] and the congruence for trivial source modules is obvious. Finally, (i) is straightforward from (h). □

Constructively one gets new endo-trivial modules from old ones by applying the Heller operator. If M is a kG-module, we let Ω(M) denotes the kernel of a projective cover P (M) ↠ M of M, and Ω^{−1}(M) denotes the cokernel of M ↣ I(M), where I(M) is an injective hull of M. Inductively Ω^{n}(M) := Ω(Ω^{n−1}(M)) and Ω^{−n}(M) := Ω^{−1}(Ω^{−n+1}(M)) for all integers n > 1 and Ω^{0}(M) is the projective-free part of M. If M is endo-trivial, then so are the modules Ω^{n}(M) for every n ∈ ℤ. In particular the modules Ω^{n}(k) are endo-trivial for every n ∈ ℤ. In T (G) and we have [Ω^{n}(k)] = n[Ω(k)] for all n ∈ ℤ so that T (G) ≥ ⟨[Ω(k)]⟩ ≅ ℤ.

Remark 2.2. The torsion-free rank of T (G) is non-zero if the p-rank of G is at least 2, and can be calculated explicitly by [8, Theorem 3.1]: it depends only on the number of conjugacy classes of maximal elementary abelian p-subgroups of rank 2.

The problem of determining the finite group T T (G) in general is much harder. According to Lemma 2.1(e) this is linked to the computation of the Green correspondence. If P ∈ Syl_{p}(G) and N := N_{G}(P), then we denote by f^{−1}(X(N)) the set consisting of the kG-Green correspondents of the elements of X(N). Again the tensor product ⊗_{k} induces a group structure on f^{−1}(X(N)) such that f^{−1}(X(N)) ≅ X(N) (see [21, Proposition 4.1(d)]), but we emphasise that f^{−1}(X(N)) is not contained in T (G) in general. In other words, the kG-Green correspondent of a one-dimensional kN_{G}(P)-module is not necessarily endo-trivial, although it is a trivial source module. However, if the p-rank of G is at least 2 and P is not semi-dihedral, then f^{−1}(X(N)) ∩ T (G) = ker(Res_{P}^{G}) = T T (G). This follows from Lemma 2.1(g) and the fact that T T (P) ≅ {0} in all such cases (see [32]).

In order to detect whether a module in f^{−1}(X(N)) is endo-trivial we have the following character-theoretic criterion.

Theorem 2.3 ([22, Theorem 2.2]). Let V be an indecomposable trivial source kG-module with k-dimension prime to p. Then V is endo-trivial if and only if χV̂ (u) = 1 for any non-trivial p-element u ∈ G.

Lemma 2.4. Let G and H be two finite groups with p | |H|. Assume that M is an indecomposable trivial source endo-trivial kG-module. Then, M ⊗ k_{H} is endo-trivial as a k[G × H]-module if and only if dim_{k}(M) = 1.

Proof. Since M is endo-trivial, it holds that p |̸ dim_{k}(M) by Lemma 2.1(h) and that M^{∗} ⊗ M = k_{G} ⊕ L for a projective kG-module L. Hence,

(M ⊗ k_{H})^{∗} ⊗ (M ⊗ k_{H}) = M^{∗} ⊗ k_{H}^{∗} ⊗ M ⊗ k_{H}
= (M^{∗} ⊗ M) ⊗ (k_{H} ⊗ k_{H}) = (k_{G} ⊕ L) ⊗ k_{H}
= (k_{G} ⊗ k_{H}) ⊕ (L ⊗ k_{H}).

Since p||H|, L ⊗ k_{H} is not projective as k[G × H]-module.

Now, let G_{0} be a subgroup of G. Then,

Ind_{G_{0}}^{G}(k_{G_{0}}) ⊗ k_{H} = (kG ⊗_{kG_{0}} k_{G_{0}}) ⊗ k_{H}
≅ k[G × H] ⊗_{k[G_{0}×H]} (k_{G_{0}} ⊗ k_{H}) = Ind_{G_{0}×H}^{G×H}(k_{G_{0}×H})

This shows that M ⊗ k_{H} is an indecomposable trivial source k[G × H]-module, see [19, Proposition 1.1]. So let χ be an ordinary character of G × H afforded by M ⊗ k_{H}. Take any p-elements u ∈ G and v ∈ H with u ≠ 1 and v ≠ 1. Then, we know by Theorem 2.3 that

χ(u, v) = χM̂(u)·1_{H}(v) = 1·1 = 1,
χ(u, 1) = χM̂(u)·1_{H}(1) = 1·1 = 1 and
χ(1, v) = χM̂(1)·1_{H}(v) = χM̂(1) = dim_{k}(M).

Therefore, again by making use of Theorem 2.3, the assertion follows. □

Lemma 2.5. Let B be a block of G with a maximal B-Brauer pair (P, e), and let N_{G}(P, e) be the stabilizer of (P, e), see [31, p.346].

(a) If further we have N_{G}(P, e) = N_{G}(P), then e is the unique block of C_{G}(P) with e^{G} = B (block induction).

(b) In particular, if the Brauer correspondent of B in N_{G}(P) has a one-dimensional kN_{G}(P)-module, then e is the unique block of C_{G}(P) with e^{G} = B.

Proof. Set N := N_{G}(P) an H := N_{G}(P, e).

(a) We use the same notation e to mean a block and its block idempotent. In general, e is a block idempotent even of kH, see [31, Exercise (40.2)(b)], and hence of kN by the assumption. Namely, e is the block idempotent of the Brauer correspondent of B in N. Hence, Brauer’s 1st Main Theorem implies e = Br_{P}^{G}(1_{B}). Now, let e′ be any block idempotent of kC_{G}(P) with (e′)^{G} = B (block induction). Then, by [27, Chap.5, Theorem 3.5(ii)], e′ = Br_{P}^{G}(1_{B})·e′ = ee′, which yields e′ = e.

(b) By the result of Fong-Reynolds [27, Chap.5, Theorem 5.10(ii)], we obtain kNb ≅ Mat_{|N:H|}(kHe) as k-algebras where b is the block idempotent of the Brauer correspondent of B in N. Since b has a one-dimensional kN-module, we have |N : H| = 1, so the assertion follows by (a). □

2.2. Puig equivalences.

Lemma 2.6. Let G and H be finite groups, and let 𝔐 and 𝔑, respectively, be an (OG, OH)- and (OH, OG)-bilattices. Then, for any R ∈ {K, k}, we have

R ⊗_{O} (𝔐 ⊗_{OH} 𝔑) ≅ (R ⊗_{O} 𝔐) ⊗_{KH} (R ⊗_{O} 𝔑) as (RG, RG)-bimodules.

Proof. This follows by direct calculations, see e.g. [31, Exercise (9.7)] and [20, I Lemma 14.5(3)]. □

Lemma 2.7. Let G and G′ be finite groups, and let B and B′, respectively, be block algebras of OG and OG′ with the same defect group P (and hence P ⩽ G ∩ G′). Assume that B and B′ are Puig equivalent. Namely, there exist an (OG, OG′)-lattice 𝔐 and an (OG′, OG)-lattice 𝔑 such that the pair (𝔐, 𝔑) induces a Morita equivalence between B-mod and B′-mod, 𝔑 = 𝔐^{∨}, and 𝔐 is an indecomposable trivial source lattice with vertex ΔP as an O[G × G′]-lattice. Then, if V is an indecomposable trivial source kGmodule in B with vertex Q such that Q ⩽ P, it holds that k𝔑 ⊗_{kG} V is an indecomposable trivial source kG′-module in B′ with vertex Q (if we replace Q by g′^{−1}Qg′ for some element g′ ∈ G′).

Proof. This follows from exactly the same argument given in the proof of [18, Lemma A.3 (i), (iii)], though G′ is not necessarily a subgroup of G. See also [30, Remark 7.5] and [24, Theorem 4.1]. □

3. Endo-trivial modules for groups with Klein-four Sylow subgroups

Hypothesis 3.1. Henceforth we assume p = 2, so that k is an algebraically closed field of characteristic 2. We assume further that G has a Klein-four Sylow 2-subgroup P ≅ C_{2} × C_{2}. Set N := N_{G}(P) and N̄ := N/O_{2′}(N). Recall that Aut(P) ≅ 𝔖_{3}, (see [15, Chap.7, Theorem 7.1]), so that one of the following holds:

(i) |N_{G}(P) : C_{G}(P)| = 1 and there are three conjugacy classes of involutions. By Burnside transfer theorem and its converse, this happens if and only if G is 2nilpotent, that is N̄ ≅ C_{2} × C_{2} ; or

(ii) |N_{G}(P) : C_{G}(P)| = 3 and all involutions are conjugate in G. In this case N̄ ≅ 𝔄_{4}.

Next we sum up what is known on the structure of the group T (G) in previously published articles.

Lemma 3.2. Let G be a finite group having a Sylow 2-subgroup P ≅ C_{2} × C_{2}. (a) T (C_{2} × C_{2}) = ⟨[Ω(k)]⟩ ≅ ℤ;

(b) T (G) = T T (G) ⊕ ℤ, where ℤ ≅ ⟨[Ω(k)]⟩ and

T T (G) = ker(Res_{P}^{G}) = f^{−1}(X(N_{G}(P))) ∩ T (G)

consists of the classes of the indecomposable endo-trivial modules with a trivial source, or alternatively the classes of the indecomposable endo-trivial kG-modules whose kN_{G}(P)-Green correspondent is one-dimensional;

(c) T (𝔄_{5}) ≅ T (𝔄_{4}) = X(𝔄_{4}) ⊕ ⟨[Ω(k)]⟩ ≅ ℤ/3ℤ ⊕ ℤ, where X(𝔄_{4}) ≅ ℤ/3ℤ consists of the three one-dimensional k𝔄_{4}-modules, which we denote by k_{𝔄_{4}}, k_{ω}, k\overline{ω};

(d) If G = O_{2′}(G) ⋊ P, then T (G) ≅ X(G) ⊕ ℤ.

Proof. Part (a) was proved by Dade [13, Theorem 9.13]. For (b), it is straightforward from [8, Theorem 3.1] that T F (G) ≅ ℤ generated by [Ω(k)]. Now since T T (P) = {[k]} by (a), clearly we must have T T (G) = ker(Res_{P}^{G}). The rest follows directly from Lemma 2.1(d),

(e) and (g). For (c) it follows from (b) and Lemma 2.1(h) that T (𝔄_{4}) = X(𝔄_{4}) ⊕ ⟨[Ω(k)]⟩ and X(𝔄_{4}) ≅ (𝔄_{4}/[𝔄_{4}, 𝔄_{4}])_{2′} ≅ ℤ/3ℤ. In addition T (𝔄_{5}) ≅ T (𝔄_{4}) since 𝔄_{4} is strongly 2-embedded in 𝔄_{5}, see [8, Theorem 4.2]. Part (d) was conjectured in [9] and proven in [28]. □

Thanks to Lemma 3.2(b), it remains to understand the group T T (G) in general. In the next section we describe a necessary and sufficient condition under which we have

T T (G) = f^{−1}(X(N_{G}(P))) .

However, we point out that even in the 2-nilpotent case (see Lemma 3.2(d)) it may happen that the Green correspondence f^{−1} does not preserve endo-trivial modules, that is T T (G) ≨ f^{−1}(X(N_{G}(P))). The following are typical examples of such situations.

Example 3.3. Let P be a Klein four-group with generators u and v, that is ⟨u⟩ × ⟨v⟩ ≅ C_{2} × C_{2}. Let ℓ be an odd prime, and H denote an extra-special group of order ℓ^{3} and exponent ℓ, given by the presentation

H = ⟨a, b, c | a^{ℓ} = b^{ℓ} = c^{ℓ} = 1, [a, c] = 1 = [b, c], [a, b] = c⟩ .

Then build G := H ⋊ P to be the semi-direct product where P acts on H as follows: v acts trivially, that is [H, v] = 1, a^{u} = a^{−1}, b^{u} = b^{−1}, c^{u} = c. We have H = O_{2′}(G) and N_{G}(P) = C_{G}(P) = Z × P where Z :=< c >≅ C_{ℓ} is the center of H.

By Lemma 3.2(d), T (G) ≅ X(G) ⊕ ℤ and, by Lemma 2.1(e), it embeds as a subgroup of T (N_{G}(P)) ≅ X(N_{G}(P)) ⊕ ℤ via the restriction map Res_{N_{G}(P)}^{G} : T (G) ⟶ T (N_{G}(P)). Now

X(G) = (G^{ab})_{2′} = [((H)^{ab})_{P} × P^{ab}]_{2′} = (H/Z)_{P} =< aZ, bZ >_{P} = 1

whereas X(N_{G}(P)) = (N_{G}(P)^{ab})_{2′} = Z_{P} = Z ≅ C_{ℓ} . (The subscript P denotes taking the co-invariants with respect to the action of P, that is for a group U which is a P -set then U_{P} := U/ < u^{p}u^{−1} | u ∈ U, p ∈ P >). In consequence T T (G) ≨ T T (N_{G}(P)) ≅ f^{−1}(X(N_{G}(P))).

Example 3.4. Let us reconsider Example 3.3 and restrict ourselves to the situation ℓ := 3, and let P, u, v and G be the same as in Example 3.3. Set H := 3_{+}^{1+2}, namely, H is an extra-special group of order 3^{3} with exponent 3, see [11, p.xx]. Then, G is 2-nilpotent and H = O_{2′}(G). Set Z := Z(H) and N := N_{G}(P), so that Z ≅ C_{3} and

G = (H ⋊ ⟨u⟩) × ⟨v⟩ = H ⋊ P = C_{G}(v), N = Z × P.

Set Z := ⟨z⟩. There exist exactly two characters in Irr(H) with degree 3. Take one of them and denote it by θ ∈ Irr(H), namely θ(1) = 3. Since z^{u} = z and z^{v} = z, we have T_{G}(θ) = G, where T_{G}(θ) is the inertial group of θ in G. Since G is 2-nilpotent, it follows from Morita’s theorem [26, Theorem 2] (see [17, Lemma 2]) and [14] that there is a block B of OG with

kB ≅ Mat_{θ(1)}(kP) = Mat_{3}(kP), as k-algebras.

Namely, θ is covered by B, and B is a non-principal block with defect group P. Let b be a block of ON with b^{G} = B. Since N = Z × P and this is abelian, kb ≅ kP as k-algebras. Hence, the unique simple kN-module W in b is of dimension one. Set V := f^{−1}(W). Note that OP is a source algebra of B at least by making use of [12, Theorem 1.1]. Hence, by the proof of Case 1 of Proposition 4.3(a), V is the unique simple kG-module in B. Set χ := χV̂. Then, by Proposition 4.3(a), χ ∈ Irr(B). It follows from the character table of G that χ(1) = 3, χ(u) = 1, χ(v) = 3 and χ(uv) = 1. Then, it follows from Theorem 2.3 that V is not endo-trivial (or since χ(1) = dim_{k}(V) ≢ 1 (mod 4) it follows from Lemma 2.1(h) that V is not endo-trivial).

We also obtain the same conclusion using Theorem 1.1. Namely, since G = C_{G}(v), we get ϕ_{v} = ϕ where ϕ is the unique irreducible Brauer character in B and ϕ_{v} is the same as in Lemma 4.3(b). Since kB ≅ Mat_{3}(kP), ϕ_{v}(1) = ϕ(1) = 3, which means ϕ_{v}(1) ≠ 1. Therefore, Theorem 1.1 yields that V is not endo-trivial,

As a matter of fact, kG has the following block decomposition:

kG ≅ kP ⊕ 4 × Mat_{2}(k⟨v⟩) ⊕ 2 × Mat_{3}(kP) as k-algebras,

by making use of [26, Theorem 2] (see [17, Lemma2]) and [14].

Example 3.5. We now give an example of a finite group G, which is not 2-solvable, has a Sylow 2-subgroup P ≅ C_{2} ×C_{2}, and such that there exists a non-trivial one-dimensional kN_{G}(P)-module 1a such that its kG-Green correspondent f^{−1}(1a) is not endo-trivial.

We rely here on computations using the algebra software MAGMA [4]. We know from the ATLAS [11, p.207] that the sporadic simple group (F i_{24})′, which is the commutator subgroup of the Fischer group F i_{24}, has a maximal subgroup N(3B) with N(3B) ≅ 3_{+}^{1+10} : (U_{5}(2) : 2). Then, we compute that N(3B) has a maximal subgroup G such that |N(3B) : G| = 2^{8}·3^{4} = 20 736, |G| = 2^{2}·3^{12}·5·11 = 116 917 020, and has G a Sylow 2-subgroup P with P ≅ C_{2} × C_{2}. Furthermore, |N_{G}(P)| = 2^{2}·3^{6} and |C_{G}(P)| = 2^{2}·3^{5}.

Using MAGMA to compute the character tables of G and N := N_{G}(P), we gather the following information. On the one hand, we know that G has exactly three irreducible characters χ_{1}, χ_{2}, χ_{3} ∈ Irr(G) such that χ_{1}(u) = χ_{2}(u) = χ_{3}(u) = 1 for any u ∈ P − {1}. Therefore we conclude from Theorem 1.1(a), Corollary 1.3(a), and Theorem 2.3 that χ_{1}, χ_{2}, χ_{3} are the characters afforded by three non-isomorphic trivial source endo-trivial modules V_{1} := k_{G}, V_{2}, V_{3} in the principal block B := B_{0}(G) of G, and moreover that these are the only trivial source endo-trivial modules of kG, up to isomorphism. On the other hand, there are exactly twenty-seven non-isomorphic simple kN-modules, say 1_{1} := k_{N}, 1_{2}, ⋯, 1_{27} and X(N) ≅ ℤ/3ℤ ⊕ ℤ/3ℤ ⊕ ℤ/3ℤ. So, we have twenty-seven non-isomorphic indecomposable trivial source kG-modules f^{−1}(1_{i}) with vertex P for i = 1, ⋯, 27.

More precisely, the three non-isomorphic endo-trivial kB-modules V_{1}, V_{2} and V_{3} are all simple, and their k-dimensions are 1, 5, 5, so that the principal block B is Puig equivalent to O𝔄_{4}. Moreover, by Brauer’s 1st Main Theorem and the character table of G, we know that there are exactly eight other blocks, say B_{2}, ⋯, B_{9}, of G with full defect and all of them have kG-modules of the form f^{−1}(1a), that is the Green correspondents of one-dimensional kN-modules, but none of these are endo-trivial modules.

As a result, this example shows that X(G) ≅ {0}, T T (G) ≅ ℤ/3ℤ and X(N_{G}(P)) ≅ ℤ/3ℤ ⊕ ℤ/3ℤ ⊕ ℤ/3ℤ.

A similar example can be obtained from a maximal subgroup H of G of order 481 140 = 2^{2} · 3^{7} · 5 · 11 in which case we obtain X(H) ≅ ℤ/3ℤ, T T (H) ≅ ℤ/3ℤ ⊕ ℤ/3ℤ and X(N_{H}(P)) ≅ ℤ/3ℤ ⊕ ℤ/3ℤ ⊕ ℤ/3ℤ.

4. Green correspondents of one dimensional module over N_{G}(P)

Lemma 4.1. Let G := 𝔄_{5} and let B_{0} := B_{0}(O𝔄_{5}) be the principal block algebra of O𝔄_{5}. Then, the indecomposable trivial source kG-modules in B_{0} with vertex P are k_{G} and two uniserial modules

\begin{matrix}2a \\ k_{G} \\ 2b \end{matrix} ,  \begin{matrix}2b \\ k_{G} \\ 2a \end{matrix} ,

where 2a, 2b are nonisomorphic simple kG-modules in B_{0} of k-dimension two. These three indecomposable modules lift uniquely to trivial source OG-lattices which afford ordinary characters 1_{G}, χ_{5} and χ_{5}, respectively, where χ_{5} ∈ Irr(B_{0}) is of degree five.

Proof. This follows from direct calculations of the Green correspondents f^{−1}(1a) of three simple kN_{G}(P)-modules 1a of k-dimension one (recall that N_{G}(P) = 𝔄_{4}). Then, the assertions follow by [27, Chap.4, Problem 10]. □

Lemma 4.2. Let B be a block of OG with a defect group P ≅ C_{2} × C_{2} such that B is one of OP, O𝔄_{4} or B_{0}(O𝔄_{5}), and hence G is P, 𝔄_{4} or 𝔄_{5}, respectively. Assume χ ∈ Irr(B) such that

χ ∈ \begin{cases}\{1_{P} \}, & if B = OP, \\ \{1_{𝔄_{4}}, χ_{ω}, χω̄\}, & if B = O𝔄_{4}, \\ \{1_{𝔄_{5}}, χ_{5}\}, & if B = B_{0}(O𝔄_{5}), \end{cases}

where χ_{ω} and χω̄ are two different non-trivial ordinary characters of 𝔄_{4} of degree one, and χ_{5} is the same as in Lemma 4.1. Then, for any u ∈ P with u ≠ 1, it holds

d_{χ,ϕ_{u}}^{u} = 1

where ϕ_{u} is an irreducible Brauer character of C_{G}(u) such that B_{u}^{G} = B for a block B_{u} of C_{G}(u) which satisfies IBr(B_{u}) = {ϕ_{u}} (note that B_{u} is nilpotent).

Proof. For the case B = OP, the claim is obvious.

Next assume that B = O𝔄_{4}. Then the generalized 2-decomposition matrix is the following:

ϕ_{1} ϕ_{ω} ϕω̄ ϕ_{u}
χ_{1} 1 . .  1
χ_{ω} . 1 .  1
χω̄ . . 1 1
χ_{3} 1 1 1 −1

where IBr(B) = {ϕ_{1}, ϕ_{ω}, ϕω̄}, Irr(B) = {χ_{1}, χ_{ω}, χω̄, χ_{3}} and IBr(B_{u}) = {ϕ_{u}}. Hence the assertion holds.

Suppose that B = B_{0}(O𝔄_{5}). Then, the generalized 2-decomposition matrix is

ϕ_{1} ϕ_{2a} ϕ_{2b} ϕ_{u}
χ_{1} 1 . .  1
χ_{3a} 1 1 . −1
χ_{3b} 1 . 1 −1
χ_{5} 1 1 1  1

where IBr(B) = {ϕ_{1}, ϕ_{2a}, ϕ_{2b}}, Irr(B) = {χ_{1}, χ_{3a}, χ_{3b}, χ_{5}} and IBr(B_{u}) = {ϕ_{u}}. Thus, this implies the assertion. □

Proposition 4.3. Let G be a finite group with a Sylow 2-subgroup P such that P ≅ C_{2} × C_{2}, and set N := N_{G}(P). Let V := f^{−1}(1a) be the kG-Green correspondent of a one-dimensional kN-module 1a, and set χ := χV̂. Then, the following holds:

(a) χ is irreducible, namely, χ ∈ Irr(G).

(b) Let b be the block of ON to which 1a belongs, and set B := b^{G} (block induction). Then, for any u ∈ P with u ≠ 1, and for any block B_{u} of OC_{G}(u) with B_{u}^{G} = B, we have that B_{u} is nilpotent and that

d_{χ,ϕ_{u}}^{u} = 1

where ϕ_{u} is the unique irreducible Brauer character in B_{u}, that is IBr(B_{u}) = {ϕ_{u}}.

Proof. (a) Let b and B be as in (b) above. Clearly, P is a defect group of b and hence of B as well. It is known that V belongs to B, see [27, Chap.5, Corollary 3.11]. Recall that all indecomposable trivial source kG-modules in B with vertex P are the Green correspondents of simple kN-modules in b, see [27, Chap.4, Problem 10] and [20, II Lemma 10.3].

Now, it follows from [12, Theorem 1.1] that B is Puig equivalent to one of the three block algebras B_{0}(O𝔄_{5}), O𝔄_{4} or OP, where B_{0}(O𝔄_{5}) is the principal block algebra of O𝔄_{5}. Let G′ be one of the three groups 𝔄_{5}, 𝔄_{4} or P, and let B′ be one of the three block algebras B_{0}(O𝔄_{5}), O𝔄_{4} or OP, depending on the situation. Since 1a is of dimension one, 1a is an indecomposable trivial source kN-module with vertex P.

Since B and B′ are Puig equivalent, there exist an (OG, OG′)-lattice 𝔐 and an (OG′, OG)-lattice 𝔑 such that 𝔑 = 𝔐^{∨} (O-dual of 𝔐), 𝔐 is an indecomposable trivial source lattice with vertex ΔP as an O[G × G′]-lattice and the pair (𝔐, 𝔑) induces a Morita equivalence between B and B′.

Note that the (RB, RB′)-bimodule R𝔐 induces a Morita equivalence between RB and RB′ for R ∈ {K, k}, see [31, Exercise (9.7)].

Case 1: B′ = OP. Hence G′ = P in this case. Recall that 𝔐 takes a trivial source simple module to a trivial source simple module with the same vertex by Lemma 2.7. Let S be the unique simple kG-module in B. Then, clearly, k𝔑 ⊗_{kP} k_{P} = S. So, S is a trivial source simple kG-module in B with vertex P as remarked above. Thus, S = f^{−1}(1a) since f^{−1}(1a) is the unique indecomposable trivial source kG-module in B with vertex P by Lemma 2.7. Namely, we may assume V = S. Now, just as in Lemma 2.6,

k ⊗_{O} (𝔑 ⊗_{OG} V̂) = (k ⊗_{O} 𝔑) ⊗_{kG} (k ⊗_{O} V̂) = k𝔑 ⊗_{kG} V = k_{P} = k ⊗_{O} O_{P} .

Because of the uniqueness of the lifting from k to O for trivial source modules (see [27, Chap.4, Theorem 8.9(iii)]), 𝔑 ⊗_{OG} V̂ = O_{P}. This implies that

K ⊗ (𝔑 ⊗ OGV̂) = K ⊗_{O} O_{P} = K_{P} =: χ_{1_{P}} .

Since K ⊗_{O} (𝔐 ⊗_{OP} 𝔑) ≅ (K ⊗_{O} 𝔐) ⊗_{KP} (K ⊗_{O} 𝔑) as (KG, KG)-bimodules by Lemma 2.6, these yield

χV̂ := K ⊗_{O} V̂ = K ⊗_{O} ((𝔐 ⊗_{OP} 𝔑) ⊗_{OG} V̂)
= (K ⊗_{O} 𝔐) ⊗_{KP} (K ⊗_{O} (𝔑 ⊗_{OG} V̂)) = K𝔐 ⊗_{KP} K_{P} .

Therefore, the ordinary character χV̂ of G afforded by V is irreducible, see [31, Exercise (9.7)].

Case 2: B′ = O𝔄_{4}. Hence G′ = 𝔄_{4} in this case. First, since P ⊴ 𝔄_{4}, all indecomposable trivial source k𝔄_{4}-modules with vertex P are the three simple k𝔄_{4}-modules of k-dimension one, which we denote by k_{𝔄_{4}}, k_{ω} and kω̄. Thus, as in Case 1, all the indecomposable trivial source kG-modules in B with vertex P are exactly k𝔐 ⊗_{k𝔄_{4}} k_{𝔄_{4}}, k𝔐 ⊗_{k𝔄_{4}} k_{ω}, and k𝔐 ⊗_{k𝔄_{4}} kω̄. In fact, these are all the simple kG-modules in B as well.

Now, since |N_{G}(P)/C_{G}(P)| = 3, b is also Puig equivalent to O𝔄_{4} by looking at the 2-decomposition matrices of B_{0}(O𝔄_{5}), O𝔄_{4} and OP. Namely, b has exactly three simple kN-modules, which we denote by W =: W_{0}, W_{1} and W_{2}. As remarked above, all indecomposable trivial source kG-modules in B with vertex P are f^{−1}(W_{i}) for i = 0, 1, 2. Thus it holds

IBr(B) = {k𝔐 ⊗_{k𝔄_{4}} k_{𝔄_{4}}, k𝔐 ⊗_{k𝔄_{4}} k_{ω}, k𝔐 ⊗_{k𝔄_{4}} kω̄}
= {f^{−1}(W), f^{−1}(W_{1}), f^{−1}(W_{2})}

as sets. Since all simple k𝔄_{4}-modules are liftable as we know by looking at the 2decomposition matrix of 𝔄_{4}, it holds that all the three simple kG-modules in B are trivial source modules with vertex P, and they uniquely lift from k to O, and these afford irreducible ordinary characters, as in Case 1.

Case 3: B′ = B_{0}(O𝔄_{5}). Hence G′ = 𝔄_{5} in this case. Recall that N_{𝔄_{5}}(P) = 𝔄_{4} by looking at the 2-decomposition matrices. Since 3 = ℓ(B) = ℓ(b), and since P ⊴ N, b is Puig equivalent to O𝔄_{4}. So, we can set IBr(b) := {W =: W_{0}, W_{1}, W_{2}}. Obviously, W_{i} for i = 0, 1, 2 are the all indecomposable trivial source kN-modules in b with vertex P. Thus, similarly to the previous argument, all the indecomposable trivial source kG-modules in B with vertex P are f^{−1}(W_{i}) for i = 0, 1, 2.

On the other hand, all the indecomposable trivial source modules in B_{0}(O𝔄_{5}) with vertex P are k_{𝔄_{5}}, 5a and 5b, where 5a and 5b are the uniserial k𝔄_{5}-modules in Lemma 4.1, and we know also that k_{𝔄_{5}} ↔ χ_{1}, 5a ↔ χ_{5} and 5b ↔ χ_{5} (with notation as in Lemma 4.1). Hence, all the indecomposable trivial source kG-modules in B with vertex P are

{f^{−1}(W), f^{−1}(W_{1}), f^{−1}(W_{2})} = {k𝔑 ⊗_{k𝔄_{5}} k_{𝔄_{5}}, k𝔑 ⊗_{k𝔄_{5}} 5a, k𝔑 ⊗_{k𝔄_{5}} 5b}

as sets. Therefore, by making use of [31, Exercise (9.7)] again, as above, all of these three indecomposable kG-modules uniquely lift from k to O and the afforded ordinary characters are all irreducible.

(b) From the proof of (a), B is Puig equivalent to a block algebra B′ which is one of OP, O𝔄_{4} or B_{0}(O𝔄_{5}). Let χ′ ∈ Irr(B′) and ϕ′_{u} ∈ IBr(B′_{u}) be the characters corresponding to χ and ϕ_{u}, respectively, via the Puig equivalence. We know that on the side of B′, the generalized 2-decomposition number is one, namely, d_{χ′,ϕ′_{u}}^{u} = 1. We know also that any Puig equivalence preserves generalized decomposition numbers, see [31, Proposition (43.10)]. Therefore, we finally get the assertion. □

5. Proofs of Theorem 1.1, Corollary 1.3 and Theorem 1.4

We finally turn to the proofs of the main results.

Proof of Theorem 1.1. (a) This follows from Proposition 4.3(a).

(b) Let V and χ be as in Proposition 4.3. Take any element u ∈ P with u ≠ 1, and let B_{u}^{(1)}, ⋯, B_{u}^{(n)} be all the blocks of C_{G}(u) with (B_{u}^{(i)})^{G} = B for 1 ≤ i ≤ n. Then, these blocks are nilpotent since C_{G}(u) is 2-nilpotent, so let ϕ_{u}^{(i)} be the unique irreducible Brauer character in B_{u}^{(i)} for each i. Then, by Proposition 4.3(b), d_{χ,ϕ_{u}^{(i)}}^{u} = 1 for any i. Hence, by Brauer’s 2nd Main Theorem

χ(u) = ∑_{i=1}^{n} d_{χ,ϕ_{u}^{(i)}}^{u}·ϕ_{u}^{(i)}(1) = ∑_{i=1}^{n} ϕ_{u}^{(i)}(1).

This shows that χ(u) = 1 if and only if n = 1 and ϕ_{u}^{(1)}(1) = 1. Now, notice that the assumption and Lemma 2.5 imply that there is exactly one block of C_{G}(P) whose block induction to G is B. Thus, since P ≅ C_{2} × C_{2}, B is of principal type if and only if n = 1 for any u ∈ P with u ≠ 1 (note that n is determined by u).

Assume first that V is endo-trivial. Note that V is an indecomposable trivial source kG-module and p |̸ dim_{k}(V) by Lemma 2.1(h). Hence, by Theorem 2.3, χ(u) = 1 for any u ∈ P with u ≠ 1. Therefore the above argument implies that B is of principal type and ϕ_{u}^{(1)}(1) = 1 for any u ∈ P with u ≠ 1.

Conversely, suppose that the two conditions hold. Then, again by the above argument, χ(u) = ϕ_{u}(1) = 1 for any u ∈ P with u ≠ 1. Therefore, by Theorem 2.3, V is endo-trivial.

(c) Let V be as in the proof of (b). By (b), the fact that V is endo-trivial depends only on the block to which V belongs. Hence the assertion holds. □

Proof of Corollary 1.3. (a) Set V := f^{−1}(1a), and let B be a block of G to which V belongs. By the assumption there is a one-dimensional kG-module V′ in B. Clearly V′ is endo-trivial by Lemma 2.1(c). Since f(V′) = Res_{N}^{G}(V′) and this is of dimension one, set 1b := f(V′). Obviously, V′ = f^{−1}(1b). Since Theorem 1.1(b) holds for f^{−1}(1b), Theorem 1.1(c) yields that V is endo-trivial. The latter part is easy.

(b) Set V := f^{−1}(1a), and let B be a block of G to which V belongs. Let B_{u} be the same as in Theorem 1.1. Then, B has full defect, and hence so dose B_{u}. Since C_{G}(u) is 2-nilpotent, it follows from Morita’s theorem [17, Lemma 2] that

kB_{u} ≅ Mat_{θ(1)}(kP) as k-algebras

for some θ ∈ Irr(O_{2′}(C_{G}(u))). Hence, the assumption implies that θ(1) = 1, so that kB_{u} ≅ kP as k-algebras, which means ϕ_{u}(1) = 1. Thus, Theorem 1.1(b) yields the assertion. □

We finally turn to the structure of the group T (G) of endo-trivial modules and prove Theorem 1.4. We start by showing that when |N_{G}(P) : C_{G}(P)| = 3 the torsion subgroup T T (G) is neither trivial, nor equal to X(G) .

Proof of Theorem 1.4. As before set N := N_{G}(P) amd N̄ := N/O_{2′}(N). (a) We have N̄ ≅ 𝔄_{4} by assumption that |N : C_{G}(P)| = 3, and by Lemma 3.2,

T (𝔄_{4}) = X(𝔄_{4}) ⊕ ⟨[Ω(k)]⟩ ≅ ℤ/3ℤ ⊕ ℤ .

Now both the inflation map InfN ; N̄ : T (N̄) ⟶ T (N) : [M] ↦ [InfN ; N̄(M)] and the restriction map Res_{N}^{G} : T (G) ⟶ T (N) are injective group homomorphisms (see Lemma 2.1(e)). Moreover, as X(𝔄_{4}) = {k_{𝔄_{4}}, k_{ω}, kω̄} consists of modules all lying in the principal 2-block of 𝔄_{4}, the three modules in f^{−1}(InfN ; N̄(X(𝔄_{4}))) ≅ ℤ/3ℤ lie in the principal 2-block of G and they are endo-trivial by Corollary 1.3. Thus it follows from Lemma 3.2(b) that ℤ/3ℤ ⊕ ℤ ≤ T (G).

(b) Since P is self-centralizing and |N : C_{G}(P)| = 3, we have N_{G}(P) ≅ 𝔄_{4}. Thus the claim is a direct consequence of (a).

(c) This is a direct consequence of Corollary 1.3.

(d) By Lemma 3.2(b), we have T (G) = T T (G)⊕ℤ, where T T (G) = f^{−1}(X(N))∩T (G). Clearly X(G) ≅ (G^{ab})_{2′} = C_{n} × H^{ab} is a subgroup of T T (G) by Lemma 2.1(c). Moreover N ≅ (𝔄_{4} ⋊ C_{n}) × H, so that X(N) ≅ (N^{ab})_{2′} ≅ (𝔄_{4} ⋊ C_{n})^{ab} × H^{ab} ≅ (𝔄_{4})^{ab} × C_{n} × H^{ab}. But by (a) and its proof, f^{−1}((𝔄_{4})^{ab}) ≅ ℤ/3ℤ is a subgroup of T T (G). This shows that T T (G) = f^{−1}(X(N)) ≅ ℤ/3ℤ ⊕ ℤ/nℤ ⊕ H^{ab}. □

Remark 5.1. An alternative proof for Corollary 1.3(a) can be obtained via the position of endo-trivial modules in the stable Auslander-Reiten quiver Γ_{s}(kG) of kG as follows.

Let us first look at the principal block of G. By Webb’s Theorem [34, Theorem D], the component Θ(k_{G}) of the trivial module in Γ_{s}(kG) is isomorphic to the component Θ(k_{𝔄_{4}}) of the trivial k𝔄_{4}-module in Γ_{s}(k𝔄_{4}) via inflation from N̄ to N followed by the Green correspondence f with respect to (G, P, N). But we see from a theorem of Webb and Okuyama ([2, §4.17], [34], [29]) that the three one-dimensional k𝔄_{4}-modules k, k_{ω}, k\overline{ω} all belong to Θ(k_{𝔄_{4}}) ≅ ℤÃ_{5}. Furthermore by a result of Bessenrodt [3, Theorem 2.6], all modules in Θ(k_{G}) are endo-trivial because k_{G} is. In consequence the Green correspondents f^{−1}(InfN ; N̄(k_{ω})) and f^{−1}(InfN ; N̄(k_{ω})) are endo-trivial as well, as required.

Similarly if kB is a non-principal block of kG containing a one-dimensional kG-module 1𝔟, which is endo-trivial by Lemma 2.1(c), then again by [3, Theorem 2.6], all modules in the component Θ(1𝔟) of 1𝔟 in Γ_{s}(kG) are endo-trivial. Moreover [3, Theorem 2.6] also says that Θ(1𝔟) ≅ ℤÃ_{5}. But from the structure of a ℤÃ_{5}-component there exists two non-isomorphic indecomposable kB-modules M ≇ 1𝔟 ≇ N such that any module lying in Θ(1𝔟) is of the form Ω^{r}(1𝔟), or Ω^{m}(M), or Ω^{n}(N) for some r, m,n ∈ ℤ. This and Lemma 3.2(b) show that kB contains exactly three torsion endo-trivial, that is kG-modules of the form 1𝔟 = f^{−1}(1a), f^{−1}(1a′), f^{−1}(1a′′) for three non-isomorphic onedimensional kN-modules 1a, 1a′, 1a′′ in the Brauer correspondent b of the block B.

Acknowledgements. The authors thank the referee for useful comments on the preliminary version of this paper. For this research the first author was partially supported by the Japan Society for Promotion of Science (JSPS), Grant-in-Aid for Scientific Research (C)23540007, 2011–2014. The second author gratefully acknowledges financial support by ERC Advanced Grant 291512 and SNF Fellowship for Prospective Researchers PBELP2_{−}143516. This research started when the second author was visiting Chiba University March 2014, which was supported by the Japan Society for Promotion of Science (JSPS), Grant-in-Aid for JSPS-Fellows 24.2274, via Dr. Moeko Takahashi, Chiba University, whom both authors are grateful to. The authors are sincerely grateful to Erwan Biland for enlightening discussions.

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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