ENDO-TRIVIAL MODULES: A REDUCTION TO p′-CENTRAL EXTENSIONS

CAROLINE LASSUEUR AND JACQUES THÉVENAZ

Abstract. We examine the behavior of the group of endo-trivial modules under inflation from a quotient modulo a normal subgroup of order prime to p. We prove that everything is controlled by a representation group of the quotient. Examples show that this inflation map is in general not surjective.

1. Introduction

Endo-trivial modules play an important role in the representation theory of finite groups in prime characteristic p. They have been classified in a number of special cases (see the recent papers [CMN15, LM15b] and the references therein). Over an algebraically closed field k of prime characteristic p, endo-trivial modules for a finite group G form an abelian group T (G), which is known to be finitely generated. The main question is to understand the structure of T (G) and of its torsion subgroup T T (G).

We analyze how T (G) and T (G/O_{p′}(G)) are related, where O_{p′}(G) denotes as usual the largest normal subgroup of G of order prime to p. Setting Q := G/O_{p′}(G) for simplicity, there is an inflation map

Inf_{Q}^{G} : T (Q) ⟶ T (G)

which is easily seen to be injective. But examples show that it is in general not surjective, so we cannot expect an isomorphism between T (G) and T (Q).

We let T T (G) be the torsion subgroup of T (G) and X(G) be the subgroup of T T (G) consisting of all one-dimensional representations, that is, X(G) ≅ Hom(G, k^{×}). We also let K(G) be the kernel of the restriction map Res_{P}^{G} : T (G) ⟶ T (P) to a Sylow psubgroup P of G. It is known that X(G) ⊆ K(G) ⊆ T T (G) and that K(G) = T T (G) in almost all cases (namely if we exclude the cases when a Sylow p-subgroup of G is cyclic, generalized quaternion or semi-dihedral). Moreover, there are numerous cases where K(G) is just equal to X(G), but not always. However, the structure of K(G) is not known in general.

Associated with Q, there is a p′-representation group Q̃ which is a central extension with kernel of order prime to p. This controls the behavior of projective representations of Q (in the sense of Schur). When Q is a perfect group, then Q̃ is unique and is also called the universal p′-central extension of Q. When Q is not perfect, then Q̃ may not be unique.

Our main result is as follows:

Date: December 31, 2015.

2010 Mathematics Subject Classification. Primary 20C20; Secondary 20C25.

Key words and phrases. Endo-trivial modules, Schur multipliers, central extensions, perfect groups.

Theorem 1.1. Let G be a finite group with p-rank at least 2 and no strongly p-embedded subgroups. Let Q̃ be any p′-representation group of the group Q := G/O_{p′}(G).

(a) There exists an injective group homomorphism

Φ_{G,Q̃} : T (G)/X(G) ⟶ T (Q̃)/X(Q̃) .

In particular, Φ_{G,Q̃} maps the class of Inf_{Q}^{G}(W) to the class of InfQ̃ ; Q(W), for any endotrivial kQ-module W.

(b) The map Φ_{G,Q̃} induces by restriction an injective group homomorphism

Φ_{G,Q̃} : K(G)/X(G) ⟶ K(Q̃)/X(Q̃) .

(c) In particular, if K(Q̃) ≅ X(Q̃), then K(G) ≅ X(G).

Examples show that Φ_{G,Q̃} is in general not surjective (see Section 7), but the theorem provides some information on K(G), for all groups such that G/O_{p′}(G) = Q, provided K(Q̃) is known for the single group Q̃. We also conjecture that Φ_{G,Q̃} induces an isomorphism on the torsion-free part of T (G) and T (Q̃) (see Section 5). Moreover, in case Q is perfect, then there is an alternative approach to Φ_{G,Q̃} which we present in Section 6.

The two main assumptions on G in Theorem 1.1 are needed for applying a result of Koshitani and Lassueur [KL15]. However, these assumptions are not really restrictive because endo-trivial modules are completely understood in the two excluded cases: they are classified if the p-rank is 1, by [MT07, CMT13], and T (G) ≅ T (H) if G has a strongly p-embedded subgroup H, see [MT07, Lemma 2.7]. The two assumptions also allow us to prove that T (G) ≅ T (G/[G, A]), where A = O_{p′}(G), or in other words that the extension

1 ⟶ A ⟶ G ⟶ Q ⟶ 1

with kernel A of order prime to p can always be replaced by the central extension

1 ⟶ A/[G, A] ⟶ G/[G, A] ⟶ Q ⟶ 1 .

This is explained in Section 3.

2. Notation and Preliminaries

Throughout, unless otherwise specified, we use the following notation. We let k denote an algebraically closed field of prime characteristic p. We assume that groups are finite, and that all modules over group algebras are finitely generated. We set ⊗ := ⊗_{k}. If G is an arbitrary finite group and V is a kG-module, we denote by ρ_{V} : G ⟶ GL(V) the corresponding k-representation, and by π_{V} : GL(V) ⟶ PGL(V) the canonical surjection.

Assuming moreover that p | |G|, we recall that a kG-module V is called endo-trivial if there is an isomorphism of kG-modules End_{k}(V) ≅ k ⊕(proj), where k denotes the trivial kG-module and (proj) some projective kG-module, which might be zero. Any endo-trivial kG-module V splits as a direct sum V = V_{0} ⊕ (proj) where V_{0}, 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 T (G) denotes the resulting set of equivalence classes (which we denote by square brackets). Then T (G), endowed with the law [U] + [V] := [U ⊗ V], is an abelian group called the group of endo-trivial modules of G. The zero element is the class [k] of the trivial module and −[V] = [V^{∗}], the class of the dual module. By a result of Puig, the group T (G) is known to be a finitely generated abelian group, see e.g. [CMN06, Corollary 2.5].

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

Furthermore, if P is a Sylow p-subgroup of G, we set

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

In other words, the class of an indecomposable endo-trivial kG-module V belongs to K(G) if and only if V ↓_{P}^{G}≅ k ⊕ (proj), that is, in other words, V has trivial source. We have X(G) ⊆ K(G) because any one-dimensional kP -module is trivial. Moreover, K(G) ⊆ T T (G) (see [CMT11a, Lemma 2.3]), and K(G) = T T (G) unless P is cyclic, generalized quaternion, or semi-dihedral, by the main result of [CT05].

By a central extension (E, π) of Q, we mean a group extension

1 ⟶ Z ⟶ E ⟶^{π} Q ⟶ 1

with Z = Ker π central in E. Recall that (E, π) is said to have the projective lifting property (relative to k) if, for every finite-dimensional k-vector space V, every group homomorphism θ : Q ⟶ PGL(V) can be completed to a commutative diagram of group homomorphisms:

\xymatrix{1 \ar[r] & Z \ar[r] \ar[d]_{λ|_{Z}} & E \ar[r]^{π} \ar[d]_{λ} & Q \ar[r] \ar[d]_{θ} & 1 \\ 1 \ar[r] & k^{×} · Id_{V} \ar[r] & GL(V) \ar[r]^{π_{V}} & PGL(V) \ar[r] & 1}

In general, the homomorphism λ is not uniquely defined. However, by commutativity of the diagram, the following holds:

Lemma 2.1. In the above situation, if λ, λ′ : E ⟶ GL(V) are two liftings of θ to E, then there exists a degree one representation μ : E ⟶ GL(k) such that λ′ = λ ⊗ μ.

By results of Schur (slightly generalized for dealing with the case of characteristic p), given a finite group Q, there always exists a central extension (E, π) of Q, with kernel M_{k}(Q) := H^{2}(Q, k^{×}), which has the projective lifting property. A p′-representation group of Q (or a representation group of Q relative to k) is a central extension (Q̃, π) of Q of minimal order with the projective lifting property. In this case M_{k}(Q) ≅ Ker π ≤ [Q̃, Q̃]. We recall that M_{k}(Q) ≅ H^{2}(Q, ℂ^{×})_{p′}, the p′-part of the Schur multiplier of Q, and that in general a group Q with X(Q) ≠ 1 may have several nonisomorphic p′-representation groups. Furthermore, fixing a p′-representation group (Q̃, π) of Q, the abelian group M_{k}(Q) becomes isomorphic to its k^{×}-dual via the transgression homomorphism

tr : Hom(M_{k}(Q), k^{×}) ⟶ H^{2}(Q, k^{×})

defined by tr(φ) = [φ∘α], where the cocycle α ∈ Z^{2}(Q, M_{k}(Q)) is in the cohomology class corresponding to the central extension 1 ⟶ M_{k}(Q) ⟶ Q̃ ⟶^{π} Q ⟶ 1. For further details and proofs we refer the reader to [NT89, Chap. 3, §5] and [CR90, §11E].

If V, W are two finite-dimensional k-vector spaces, then the tensor product of linear maps induces a tensor product − ⊗ − : PGL(V) × PGL(W) ⟶ PGL(V ⊗ W) via π_{V} (α) ⊗ π_{W} (β) := π_{V ⊗W} (α ⊗ β) for any α ∈ GL(V) and any β ∈ GL(W). Therefore, if μ : Q ⟶ PGL(V) and ν : Q ⟶ PGL(W) are group homomorphisms, we may define a group homomorphism μ ⊗ ν : Q ⟶ PGL(V ⊗ W) via (μ ⊗ ν)(q) := μ(q) ⊗ ν(q) for every q ∈ Q. We shall use the following well-known results throughout:

Lemma 2.2. Let 1 ⟶ A ⟶ G ⟶^{π} Q ⟶ 1 be an arbitrary group extension.

(a) Whenever V is a kG-module such that ρ_{V} (A) ⊆ k^{×}·Id_{V}, the group homomorphism ρ_{V} : G → GL(V) induces a uniquely defined group homomorphism θ_{V} : Q ⟶ PGL(V) such that the following diagram commutes:

\xymatrix{1 \ar[r] & A \ar[r] \ar[d]_{ρ_{V} |_{A}} & G \ar[r]^{π} \ar[d]_{ρ_{V}} & Q \ar[r] \ar[d]_{θ_{V}} & 1 \\ 1 \ar[r] & k^{×} · Id_{V} \ar[r] & GL(V) \ar[r]^{π_{V}} & PGL(V) \ar[r] & 1}

(b) If V, W are kG-modules such that ρ_{V} (A) ⊆ k^{×} · Id_{V} and ρ_{W} (A) ⊆ k^{×} · Id_{W}, then ρ_{V ⊗W} (A) ⊆ k^{×} · Id_{V ⊗W} and we have θ_{V ⊗W} = θ_{V} ⊗ θ_{W}.

Proof. (a) Choose a set-theoretic section s : Q ⟶ G for π and define θ_{V} := π_{V} ∘ ρ_{V} ∘ s. Since ρ_{V} (A) ⊆ k^{×} · Id_{V}, the map θ_{V} is a group homomorphism making the diagram commute. Clearly θ_{V} is uniquely defined since π is an epimorphism.

(b) This is a straightforward computation.

□

3. Endo-trivial modules and central extensions

We now fix G to be a finite group of order divisible by p, we set A := O_{p′}(G) and Q := G/A, and we denote by π_{G} : G ⟶ Q the quotient map. Moreover, we let (Q̃, πQ̃) be a fixed p′-representation group of Q.

Since A is a p′-subgroup of G, inflation induces an injective group homomorphism

Inf_{Q}^{G} : T (Q) ⟶ T (G) , [V] ⟶ [Inf_{Q}^{G}(V)] .

This is because the inflation of a projective module remains projective when the kernel A is a p′-group. We emphasize that endo-trivial kG-modules cannot be recovered from endo-trivial kQ-modules, as in general the inflation map Inf_{Q}^{G} is not an isomorphism (see Section 7).

Hypothesis 3.1. Assume G is a finite group fulfilling the following two conditions:

(1) the p-rank of G is greater than or equal to 2; and

(2) G has no strongly p-embedded subgroups.

Lemma 3.2. Suppose that G satisfies Hypothesis 3.1.

(a) If V is an indecomposable endo-trivial kG-module, then V ↓_{A}^{G} ≅ Y ⊕ ⋯ ⊕ Y, where Y is a one-dimensional kA-module.

(b) If V is an indecomposable endo-trivial kG-module, then ρ_{V} (A) ⊆ k^{×} · Id_{V}.

(c) The inflation map

Inf_{G/[G,A]}^{G} : T (G/[G, A]) ⟶ T (G)

is a group isomorphism.

Proof. (a) Since G satisfies Hypothesis 3.1, any composition factor Y of V ↓_{A}^{G} is Ginvariant, by [KL15, Lemma 4.3]. Therefore V ↓_{A}^{G} ≅ Y ⊕ ⋯ ⊕ Y and [KL15, Theorem 4.4] proves that dim Y = 1 (using a result of Navarro and Robinson [NR12]).

(b) This is a restatement of (a).

(c) Since [G, A] is a normal p′-subgroup of G, the inflation map Inf_{G/[G,A]}^{G} is a welldefined injective group homomorphism. In order to prove that it is surjective, it suffices to prove that [G, A] acts trivially on any indecomposable endo-trivial kG-module V. But by (b) we have

ρ_{V} ([G, A]) ⊆ [ρ_{V} (G), ρ_{V} (A)] ⊆ [ρ_{V} (G), k^{×} · Id_{V}] = {Id_{V} } .

Hence [G, A] acts trivially on V. □

Part (c) of Lemma 3.2 is a reduction to the case of central extensions. Explicitly, for the study of endo-trivial modules, we may always replace the given extension

1 ⟶ A ⟶ G ⟶ Q ⟶ 1

and consider instead the central extension

1 ⟶ A/[G, A] ⟶ G/[G, A] ⟶ Q ⟶ 1 .

We shall in fact not use this reduction for the proof of our main result, but rather apply directly Lemma 3.2(b).

Lemma 3.3. Let (Q̃, πQ̃) be a p′-representation group of Q. Then X(Q̃) = InfQ̃ ; Q(X(Q)), hence X(Q̃) ≅ X(Q).

Proof. We apply the fact, mentioned in Section 2, that Ker πQ̃ ⊆ [Q̃, Q̃]. This implies that any one-dimensional representation of Q̃ has Ker πQ̃ in its kernel, hence is inflated from Q̃/ Ker πQ̃ ≅ Q.

Another way of seeing the same thing is to associate to the central extension

1 ⟶ M_{k}(Q) ⟶ Q̃ ⟶^{πQ̃} Q ⟶ 1

the Hochschild-Serre five-term exact sequence

1⟶ Hom(Q, k^{×}) ⟶^{Inf} Hom(Q̃, k^{×}) ⟶^{Res} Hom(M_{k}(Q), k^{×}) ⟶^{tr} H^{2}(Q, k^{×}) ⟶^{Inf} H^{2}(Q̃, k^{×}) .

Since the transgression map tr is an isomorphism, the first map Inf must be an isomorphism as well. □

4. Proof of Theorem 1.1

Keep the notation of the previous section. Moreover, given an endo-trivial kG-module V such that ρ_{V} (A) ⊆ k^{×} · Id_{V}, we let

θ_{V} : Q ⟶ PGL(V)

denote the induced homomorphism constructed in Lemma 2.2(a). The projective lifting property for the central extension (Q̃, πQ̃) allows us to fix a representation

ρ_{VQ̃} : Q̃ ⟶ GL(V)

lifting θ_{V} to Q̃. We denote by VQ̃ the corresponding kQ̃-module.

Lemma 4.1. Let V be an endo-trivial kG-module such that ρ_{V} (A) ⊆ k^{×} · Id_{V}. Then VQ̃ is an endo-trivial kQ̃-module.

Proof. We have to work with two group extensions

1 ⟶ A ⟶ G ⟶^{π_{G}} Q ⟶ 1  and  1 ⟶ M ⟶ Q̃ ⟶^{πQ̃} Q ⟶ 1 ,

where M := M_{k}(Q). Both A and M have order prime to p.

Let P ∈ Syl_{p}(G), set \overline{P} := AP/A ∈ Syl_{p}(Q), and let ı_{P} : P ⟶ AP be the inclusion map, so that φ := π_{G} ∘ ı_{P} : P ⟶ \overline{P} is an isomorphism. Next choose P̃ ∈ Syl_{p}(Q̃) such that MP̃/M = \overline{P} ∈ Syl_{p}(Q). Let ıP̃ : P̃ ⟶ MP̃ be the inclusion map, so that ψ := πQ̃ ∘ ıP̃ : P̃ ⟶ \overline{P} is an isomorphism. Consider now the two commutative diagrams:

\xymatrix{P \ar[r]^{ϕ} \ar[d]_{ρ_{V}} & \overline{P} \ar[d]_{θ_{V}} & and & P̃ \ar[r]^{ψ} \ar[d]_{ρQ̃} & \overline{P} \ar[d]_{θ_{V}} \\ GL(V) \ar[r]^{π_{V}} & PGL(V) &  & GL(V) \ar[r]^{π_{V}} & PGL(V)}

where we write ρQ̃ := ρ_{VQ̃} for simplicity. Since ϕ and ψ are isomorphisms, for any u ∈ \overline{P}, we have

π_{V} ρ_{V} ϕ^{−1}(u) = θ_{V} (u) = π_{V} ρQ̃ψ^{−1}(u) .

We claim that if two elements u_{1}, u_{2} ∈ GL(V) have p-power order and satisfy π_{V} (u_{1}) = π_{V} (u_{2}), then u_{1} = u_{2}. Postponing the proof of the claim, we deduce that

ρ_{V} ϕ^{−1}(u) = ρQ̃ψ^{−1}(u) ,

because they have p-power order. This means that the representations (ρ_{V})_{|P} and (ρQ̃)_{|P̃}, transported via isomorphisms to representations of \overline{P}, are equal. Now, a module is endotrivial if and only if its restriction to a Sylow p-subgroup is (see [CMN06, Proposition 2.6]). Moreover, this property is preserved when transported via group isomorphisms. Since V is endo-trivial, so is V ↓_{P}, hence so is VQ̃ ↓P̃, and it follows that VQ̃ is endo-trivial.

We are left with the proof of the claim. If π_{V} (u_{1}) = π_{V} (u_{2}), then u_{1} = αu_{2} where α ∈ k^{×}. For some large enough power p^{n}, we have u_{1}^{p^{n}} = u_{2}^{p^{n}} = 1. Therefore we obtain

1 = u_{1}^{p^{n}} = (αu_{2})^{p^{n}} = α^{p^{n}}u_{2}^{p^{n}} = α^{p^{n}} .

But there are no nontrivial p-th roots of unity in k^{×}, so we get α = 1, hence u_{1} = u_{2}. □

Proposition 4.2. Assume G satisfies Hypothesis 3.1. Then there is an injective group homomorphism

Φ_{G,Q̃} : T (G)/X(G) ⟶ T (Q̃)/X(Q̃)

defined by Φ_{G,Q̃}([V] + X(G)) := [VQ̃] + X(Q̃) for any indecomposable endo-trivial kGmodule V. Moreover, for any endo-trivial kQ-module W, the homomorphism Φ_{G,Q̃} maps

the class of Inf_{Q}^{G}(W) to the class of InfQ̃ ; Q(W).

Proof. First, Lemma 4.1 allows us to define a map ϕ : T (G) ⟶ T (Q̃)/X(Q̃) by setting ϕ([V]) := [VQ̃] + X(Q̃) for any [V] ∈ T (G) such that ρ_{V} (A) ⊆ k^{×} · Id_{V}. The definition of ϕ([V]) does not depend on the choice of VQ̃, for if ρ_{V ′ ; Q̃} is a second lifting of θ_{V} to Q̃, then

by Lemma 2.1 there exists X ∈ X(Q̃) such that V ′ ; Q̃ ≅ VQ̃ ⊗ X, hence ϕ([VQ̃]) = ϕ([V ′ ; Q̃]).

Next, let V, W be two indecomposable endo-trivial kG-modules. Part (b) of Lemma 3.2 implies that ρ_{V ⊗W} (A) = (ρ_{V} ⊗ ρ_{W})(A) ⊆ k^{×} · Id_{V ⊗W}. Thus, by Lemma 2.2(b), θ_{V ⊗W} = θ_{V} ⊗ θ_{W}, and it is easy to verify that ρ_{VQ̃} ⊗ ρ_{WQ̃} lifts θ_{V} ⊗ θ_{W} to Q̃. Therefore, by

Lemma 2.1, there exists X ∈ X(Q̃) such that (V ⊗ W)Q̃ ≅ VQ̃ ⊗ WQ̃ ⊗ X. This shows that ϕ is a group homomorphism.

It is clear that Ker ϕ = X(G), since by construction dim VQ̃ = dim V for any indecomposable endo-trivial kG-module V. As a result, ϕ induces the required homomorphism Φ_{G,Q̃}.

Finally, if W is any endo-trivial kQ-module, then the kQ̃-module constructed from V = Inf_{Q}^{G}(W) is easily seen to be the inflated module VQ̃ = InfQ̃ ; Q(W), because the map θ_{V} : Q → PGL(V) comes from a group homomorphism Q → GL(V). This shows that the class of Inf_{Q}^{G}(W) is mapped to the class of InfQ̃ ; Q(W) under the map Φ_{G,Q̃}, proving the additional statement. □

Corollary 4.3. Assume G satisfies Hypothesis 3.1. If Q̃_{1} and Q̃_{2} are two non-isomorphic p′-representation groups of Q, then

Φ_{Q̃_{1},Q̃_{2}} : T (Q̃_{1})/X(Q̃_{1}) ⟶ T (Q̃_{2})/X(Q̃_{2})

is an isomorphism.

Proof. Let V be an indecomposable kQ̃_{1}-module. By construction

Φ_{Q̃_{1},Q̃_{2}}([V] + X(Q̃_{1})) = [W] + X(Q̃_{2}) ,

where W := VQ̃_{2} is a kQ̃_{2}-module such that ρ_{W} lifts θ_{V} : Q ⟶ PGL(V) to Q̃_{2}. But then ρ_{V} lifts θ_{W} = θ_{V} to Q̃_{1}, so that by construction

Φ_{Q̃_{2},Q̃_{1}}([W] + X(Q̃_{2})) = [V] + X(Q̃_{1}) .

In other words, Φ_{Q̃_{1},Q̃_{2}} ∘ Φ_{Q̃_{2},Q̃_{1}} = Id. Similarly Φ_{Q̃_{2},Q̃_{1}} ∘ Φ_{Q̃_{1},Q̃_{2}} = Id. □

Corollary 4.4. Assume G satisfies Hypothesis 3.1. The map Φ_{G,Q̃} induces by restriction an injective group homomorphism

Φ_{G,Q̃} : K(G)/X(G) ⟶ K(Q̃)/X(Q̃) .

In particular, if K(Q̃) ≅ X(Q̃), then K(G) ≅ X(G).

Proof. Let P ∈ Syl_{p}(G) and let V be an indecomposable endo-trivial kG-module. As in

the proof of Lemma 4.1, the two modules V ↓_{P}^{G} and VQ̃ ↓Q̃ ; P̃ are isomorphic, provided we view them as modules over the group \overline{P} via the isomorphisms P ≅ \overline{P} and P̃ ≅ \overline{P}. It follows that V has a trivial source if and only if VQ̃ has. Therefore Φ_{G,Q̃} restricts to an injective group homomorphism

Φ_{G,Q̃} : K(G)/X(G) ⟶ K(Q̃)/X(Q̃) .

The special case follows. □

Proposition 4.2 together with Corollary 4.4 prove Theorem 1.1.

5. Conjecture on the torsion-free part

We keep the notation of the previous sections. Let T F (G) = T (G)/T T (G), the torsionfree part of the group of endo-trivial modules. Since X(G) ⊆ T T (G), the map

Φ_{G,Q̃} : T (G)/X(G) ⟶ T (Q̃)/X(Q̃)

induces an injective group homomorphism

Ψ_{G,Q̃} : T F (G) ⟶ T F (Q̃) .

We know that Φ_{G,Q̃} is in general not surjective, but we conjecture that Ψ_{G,Q̃} is surjective.

Conjecture 5.1. (a) The map Inf_{Q}^{G} : T F (Q) ⟶ T F (G) is an isomorphism.

(b) The map Ψ_{G,Q̃} : T F (G) ⟶ T F (Q̃) is an isomorphism.

Note that (b) follows from (a), by applying (a) to both Inf_{Q}^{G} : T F (Q) ⟶ T F (G) and

InfQ̃ ; Q : T F (Q) ⟶ T F (Q̃) and composing, because the map Ψ_{G,Q̃} : T F (G) → T F (Q̃) is the identity on modules inflated from Q.

Part (a) of Conjecture 5.1 is in fact a consequence of any of the two conjectures made in [CMT11b]. First, Conjecture 10.1 in [CMT11b] asserts that, if a group homomorphism ϕ : G → G′ induces an isomorphism between the corresponding p-fusion systems, then ϕ should induce an isomorphism T F (G′) →^{∼} T F (G). In the special case where ϕ is the quotient map ϕ : G → Q = G/O_{p′}(G), it is well-known that the fusion systems are isomorphic, so we would obtain the isomorphism T F (Q) →^{∼} T F (G) of Conjecture 5.1 above. This special case is explicitly mentioned at the end of Section 10 in [CMT11b].

Conjecture 9.2 in [CMT11b] asserts that the group T F (G) should be generated by endotrivial modules lying in the principal block. Since O_{p′}(G) acts trivially on any module lying in the principal block of G, such a module is inflated from Q, so the inflation map Inf_{Q}^{G} : T F (Q) ⟶ T F (G) in Conjecture 5.1 above should be an isomorphism.

Example 7.3 below illustrates a method allowing one to prove that the maps in Conjecture 5.1 are isomorphisms in specific cases.

6. The perfect case

When the group Q = G/O_{p′}(G) is perfect, there is an alternative approach to the construction of the injective group homomorphism of Theorem 1.1(a) using universal central extensions.

Recall that a universal p′-central extension of an arbitrary finite group Q is by definition a central extension

1 ⟶ M_{p′} ⟶ Q̃ ⟶^{πQ̃} Q ⟶ 1

with M_{p′} = Ker πQ̃ of order prime to p and satisfying the following universal property: For any central extension

1 ⟶ Z ⟶ E ⟶^{π} Q ⟶ 1

with Z = Ker π of order prime to p, there exists a unique group homomorphism ϕ : Q̃ → E such that the following diagram commutes:

\xymatrix{1 \ar[r] & M_{p′} \ar[r] \ar[d]_{ϕ|_{M_{p′}}} & Q̃ \ar[r]^{πQ̃} \ar[d]_{ϕ} & Q \ar[r] \ar[d]_{Id} & 1 \\ 1 \ar[r] & Z \ar[r] & E \ar[r]^{π} & Q \ar[r] & 1}

A standard argument shows that if a universal p′-central extension (Q̃, πQ̃) exists, then it is unique up to isomorphism.

Lemma 6.1. If (Q̃, πQ̃) is a universal p′-central extension of a finite group R, then (Q̃, πQ̃) is p′-representation group of Q.

Proof. Let (Q̌, πQ̌) be an arbitrary p′-representation group of Q. Let V be a finitedimensional k-vector space and θ : Q ⟶ PGL(V) a group homomorphism. Because (Q̌, πQ̌) has the projective lifting property and (Q̃, πQ̃) is universal, there exist group homomorphisms θ̃: Q̌ ⟶ GL(V) and ϕ : Q̃ ⟶ Q̌ such that θ̃∘ ϕ lifts θ. Therefore (Q̃, πQ̃) has the projective lifting property as well.

Now, because (Q̃, πQ̃) is universal, it is easy to see that X(Q̃) = X(Q) = 1. Therefore the Hochschild-Serre 5-term exact sequence associated to (Q̃, πQ̃) is:

1 ⟶ 1 ⟶ 1 ⟶ Hom(M_{p′}, k^{×}) ⟶^{tr} H^{2}(Q, k^{×}) ⟶^{Inf} H^{2}(Q̃, k^{×})

Thus the transgression map tr : Hom(M_{p′}, k^{×}) ⟶ H^{2}(Q, k^{×}) = M_{k}(Q) is injective. But M_{p′} ≅ Hom(M_{p′}, k^{×}), therefore by minimality of (Q̌, πQ̌), we have |M_{p′}| = |M_{k}(Q)| and |Q̃| = |Q̌|, proving that (Q̃, πQ̃) is a p′-representation group of Q. □

Lemma 6.2. Any finite perfect group Q admits a universal p′-central extension.

Proof. Since Q is a perfect group, it is well-known that Q has a representation group relative to ℂ, say (Q̂, πQ̂), which is unique up to isomorphism and that

Ker(πQ̂) =: M ≅ M_{ℂ}(Q) = H^{2}(Q, ℂ^{×}) ,

the Schur multiplier of Q. Moreover, (Q̂, πQ̂) is a universal central extension of Q, in particular perfect, see [Rot95, Theorem 11.11]. Thus, for any central extension

1 ⟶ Z ⟶ E ⟶^{π} Q ⟶ 1

where Z = Ker π, there exists a unique group homomorphism ψ : Q̂ → E such that the following diagram commutes:

\xymatrix{1 \ar[r] & M \ar[r] \ar[d]_{ψ|_{M}} & Q̂ \ar[r]^{πQ̂} \ar[d]_{ψ} & Q \ar[r] \ar[d]_{Id} & 1 \\ 1 \ar[r] & Z \ar[r] & E \ar[r]^{π} & Q \ar[r] & 1}

If Z has order prime to p, then the p-part M_{p} of M lies in the kernel of ψ|_{M}. Passing to the quotient by M_{p}, we define Q̃ := Q̂/M_{p} and denote by ϕ : Q̃ → E the map induced by ψ. Thus we obtain an induced central extension

1 ⟶ M_{p′} ⟶ Q̃ ⟶^{πQ̃} Q ⟶ 1

where M_{p′} := M/M_{p}. This is a universal p′-central extension of Q by construction. □

Given an arbitrary group extension 1 ⟶ A ⟶ G ⟶ Q ⟶ 1 with perfect quotient Q and kernel A of order prime to p, there is an induced p′-central extension:

1 ⟶ A/[G, A] ⟶ G/[G, A] ⟶^{π_{G}} Q ⟶ 1

Moreover, by the above, Q admits a universal p′-central extension, which is in fact a p′-representation group (Q̃, πQ̃) of Q. Therefore, by the universal property, there exists a unique group homomorphism ϕ_{G} : Q̃ → G/[G, A] lifting the identity on Q.

Lemma 6.3. The homomorphism ϕ_{G} : Q̃ → G/[G, A] induces a group homomorphism

ϕ_{G}^{∗} : T (G/[G, A]) ⟶ T (Q̃)

such that ϕ_{G}^{∗} = InfQ̃ ; Im(ϕ_{G}) ∘ Res_{Im(ϕ_{G})}^{G/[G,A]}. Moreover, both InfQ̃ ; Im(ϕ_{G}) and Res_{Im(ϕ_{G})}^{G/[G,A]} preserve indecomposability of endo-trivial modules.

Proof. The kernel of ϕ_{G} is contained in Ker πQ̃ = M_{p′}, which is a p′-group. Therefore, there

is an induced inflation map InfQ̃ ; Im(ϕ_{G}) : T (Im(ϕ_{G})) → T (Q̃), preserving indecomposability of endo-trivial modules.

Since Im(ϕ_{G}) maps onto Q via π_{G}, the group G/[G, A] is the product of Im(ϕ_{G}) and the central p′-subgroup A/[G, A]. It follows that Im(ϕ_{G}) is a normal subgroup of G/[G, A] of index prime to p. Therefore, by [CMN09, Prop. 3.1], the restriction to Im(ϕ_{G}) of any indecomposable endo-trivial k(G/[G, A])-module remains indecomposable and is endotrivial.

We define ϕ_{G}^{∗} to be the composite of InfQ̃ ; Im(ϕ_{G}) and Res_{Im(ϕ_{G})}^{G/[G,A]} . □

Composing the group homomorphism

ϕ_{G}^{∗} : T (G/[G, A]) ⟶ T (Q̃)

with the inverse of the isomorphism

Inf_{G/[G,A]}^{G} : T (G/[G, A]) ⟶ T (G)

of Lemma 3.2, we obtain a group homomorphism

Φ : T (G) ⟶ T (Q̃) .

We now show that this provides the alternative approach to the map of Theorem 1.1.

Proposition 6.4. Suppose that G satisfies Hypothesis 3.1 and that Q is perfect.

(a) Ker Φ = X(G).

(b) The induced injective group homomorphism

\overline{Φ} : T (G)/X(G) ⟶ T (Q̃) = T (Q̃)/X(Q̃)

coincides with the map Φ_{G,Q̃} of Theorem 1.1.

Proof. Consider the map ϕ_{G}^{∗} : T (G/[G, A]) → T (Q̃) of Lemma 6.3. It is clear that the image of a one-dimensional module is one-dimensional, hence trivial since X(Q̃) = 1 by Lemma 3.3. Therefore X(G) ⊆ Ker Φ. It follows that Φ induces a group homomorphism \overline{Φ} as in the statement.

Our assumption on G implies that, if V is an endo-trivial kG-module, then [G, A] acts trivially on V (Lemma 3.2). Moreover, ρ_{V} : G/[G, A] → GL(V) lifts θ_{V} : Q → PGL(V), as in Section 4. It is then clear that ρ_{V} ϕ_{G} : Q̃ → GL(V) also lifts θ_{V} : Q → PGL(V). Therefore, the definition of Φ_{G,Q̃} (see Proposition 4.2) shows that the class of V is mapped by Φ_{G,Q̃} to the class of the module VQ̃ corresponding to the representation ρ_{V} ϕ_{G}. In other words, [VQ̃] = Φ([V]) and this shows that Φ_{G,Q̃} coincides with \overline{Φ}.

Finally, since Φ_{G,Q̃} is injective and is equal to \overline{Φ}, we have Ker \overline{Φ} = \{0\}. Therefore we obtain Ker Φ = X(G). □

Remark 6.5. The proof we give above shows that Proposition 6.4 remains valid if the assumption that Q is perfect is replaced with the assumption that Q admits a universal p′-central extension. It is possible to prove that this happens if and only if X(Q) = 1, although we could not find a reference to such a statement in the literature. Hence, for simplicity, we restrict ourselves to the perfect case in this article.

7. Examples

In this final section, we provide various examples, in particular illustrating cases where the morphism Φ_{G,Q̃} is not surjective.

Example 7.1. Suppose that Q is simple and take G = Q, hence A = O_{p′}(G) = {1}. Then Φ_{Q,Q̃} is just the inflation map T (Q) → T (Q̃). If Q is a finite simple group listed in the table below, then it is known that its unique p′-representation group Q̃ has indecomposable endo-trivial modules lying in faithful p-blocks, namely not inflated from Q.

The results concerning the sporadic groups can be found in [LM15b, Table 3], and those about the alternating group 𝔄_{6} in [LM15c, Theorem A & Theorem B] together with [CMN09, Theorem A & Theorem B].

\begin{matrix}Q & p & Q̃ & T (Q) & T (Q̃) \\ 𝔄_{6} & 3 & 2.𝔄_{6} & ℤ ⊕ ℤ/4 & ℤ ⊕ ℤ/8 \\ 𝔄_{6} & 2 & 3.𝔄_{6} & ℤ^{2} & ℤ^{2} ⊕ ℤ/3 \\ M_{22} & 3 & 4.M_{22} & ℤ ⊕ (ℤ/2)^{2} & ℤ ⊕ ℤ/2 ⊕ ℤ/4 \\ J_{3} & 2 & 3.J_{3} & ℤ & ℤ ⊕ ℤ/3 \\ Ru & 3 & 2.Ru & ℤ ⊕ ℤ/2 & ℤ ⊕ ℤ/4 \\ F i_{22} & 5 & 6.F i_{22} & ℤ ⊕ (ℤ/2)^{2} & ℤ ⊕ ℤ/6 ⊕ ℤ/2 \end{matrix}

Further examples are given by the exceptional covering group 2.F_{4}(2) of the exceptional group of Lie type F_{4}(2), which possesses simple torsion endo-trivial modules lying in faithful blocks in characteristics 5 and 7 (see [LM15a, Proposition 5.5]), although the full structure of the group of endo-trivial modules has not been determined in these cases.

Example 7.2. Assume p > 2, let n ≥ max{2p, p + 4} be an integer and denote by 𝔖̃_{n} and 𝔖̂_{n} the two isoclinic p′-representation groups of the symmetric group 𝔖_{n}. Corollary 4.3 yields

T (𝔖̃_{n})/X(𝔖̃_{n}) ≅ T (𝔖̂_{n})/X(𝔖̂_{n}) .

However, [LM15c, Thm. B(1),(2)] proves a stronger result, namely T (𝔖̃_{n}) = Inf𝔖̃_{n} ; 𝔖_{n}(T (𝔖_{n}))

and T (𝔖̂_{n}) = Inf𝔖̂_{n} ; 𝔖_{n}(T (𝔖_{n})).

Consequently, given any finite group G such that G/O_{p′}(G) is isomorphic to one of 𝔖_{n}, 𝔖̃_{n} or 𝔖̂_{n} (with n ≥ max{2p, p + 4}), by Theorem 1.1 there exist injective group homomorphisms

T (𝔖_{n})/X(𝔖_{n}) ⟶ T (G)/X(G) ⟶^{Φ_{G,𝔖̂n}} T (𝔖̂_{n})/X(𝔖̂_{n}) ⟶^{∼} T (𝔖_{n})/X(𝔖_{n}) ,

where the first map is induced by inflation. Hence we have T (G)/X(G) ≅ T (𝔖_{n})/X(𝔖_{n}). We recall that the structure of T (𝔖_{n}) is known by [CMN09].

Example 7.3. In this final example, we outline a method which allows us to show that the maps InfQ̃ ; Q is an isomorphism on the torsion-free part of the groups of endo-trivial modules of Q and Q̃ in some concrete cases.

Specifically, we may use the fact that endo-trivial modules are liftable to characteristic zero, and afford characters taking root-of-unity values at p-singular conjugacy classes, see [LMS13, Theorem 1.3 and Corollary 2.3]. Therefore, if for every faithful p-block B of kQ̃ (of full defect) no elements of ℤ Irr_{ℂ}(B) take root-of-unity values at p-singular conjugacy classes of Q̃, then any endo-trivial kQ̃-module is inflated from Q, hence

InfQ̃ ; Q : T F (Q) ⟶ T F (Q̃)

is an isomorphism.

This was used in [LM15c, Theorem B] in the case that Q = 𝔖_{n}, n ≥ max{2p, p + 4} (as mentioned in Example 7.2 above), as well as in [LM15b, Lemma 4.3 and Lemma 6.2] for a large number of sporadic simple groups Q. More precisely, in characteristic p = 2 for Q = M_{12}, M_{22}, J_{2}, HS, McL, Ru, Suz, ON, F i_{22}, Co_{1}, F i′_{24}, or B, in characteristic p = 3 for Q = M_{12}, J_{2}, HS, Suz, F i_{22}, Co_{1}, or B, in characteristic p = 5 for Q = J_{2}, HS, Ru, Suz, Co_{1}, F i′_{24}, or B, and in characteristic p = 7 for Q = Co_{1}, F i′_{24}, or B.

Acknowledgements. The authors are indebted to Shigeo Koshitani for several useful discussions and for providing references.

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

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

Jacques Thévenaz, EPFL, Section de Mathématiques, Station 8, CH-1015 Lausanne, Switzerland.

  E-mail address: jacques.thevenaz@epfl.ch