THE GROUP OF ENDOTRIVIAL MODULES IN THE NORMAL CASE

NADIA MAZZA

Abstract. The group of endotrivial modules has recently been determined for a finite group having a normal Sylow p-subgroup. In this paper, we give and compare three different presentations of a torsion-free subgroup of maximal rank of the group of endotrivial modules. Finally, we illustrate the constructions in an example. Keywords : Endotrivial modules, finite groups, cohomology of finite groups.

1. Introduction

Endotrivial modules play an important role in the modular representation theory of finite groups, and this may explain why many group theorists have been studying them intensively, since the late seventies. The classification of endotrivial modules of a finite p-group was recently achieved in [12]. Thereafter, in a joint work, Jon Carlson, Daniel Nakano and the author tackled the question of the classification of endotrivial modules for an arbitrary finite group, and they could give an almost complete classification in the case of a finite group of Lie type in its defining characteristic (cf. [9]). The general case is still an open question, in the sense that no presentation by generators and relations of the group of endotrivial modules is yet known. However, the obstacles have been overcome in the case of a finite group having a normal Sylow p-subgroup. The results are presented in [9, Theorem 3.4], where the authors show that in this case, the group of endotrivial modules is generated by the classes of the indecomposable endotrivial modules that are extended from the Sylow p-subgroup. Then, by means of cohomological tools, they construct a minimal set of generators for the group of endotrivial modules.

The primary aim of these notes is to give an alternative construction of a torsion-free subgroup of maximal rank of the group of endotrivial modules of a finite group having a normal Sylow psubgroup, that does not appeal to any cohomological knowledge. The method refers to a theorem proven by Dade, and it is presented in Section 3, after that basic facts about endotrivial modules are recapitulated in Section 2. In Section 4, we review the modules and the techniques used in [8] and [9], and we compare with the approach presented in the previous section. Finally, in Section 5, we work out thoroughly an “odd extraspecial” example.

2. Preliminaries

Throughout these notes, we let k be an algebraically closed field of prime characteristic p. If G is a finite group, we write mod(kG) for the category of finitely generated kG-modules and stmod(kG) for the stable module category. That is, the objects of stmod(kG) are the same as those of mod(kG), and the morphisms are equivalence classes of morphisms. Namely, two morphisms are equivalent if their difference factors through a projective module (cf. [7, § 5]). In addition, we write k for the 1-dimensional trivial kG-module, and if M is a finitely generated kG-module, then End_{k} M = Hom_{k}(M, M) denotes the kG-module that is the k-algebra of k-linear endomorphisms of M. Recall that, for two kG-modules M and N, there is an isomorphism of kG-modules Hom_{k}(M, N) ≅ M^{∗} ⊗ N, where M^{∗} = Hom_{k}(M, k) is the k-linear dual of M, and the tensor product “⊗” is the tensor product over the field k.

Definition 2.1. Let G be a finite group. A finitely generated kG-module M is endotrivial provided that End_{k} M ≅ k in stmod(kG), or equivalently, End_{k} M ≅ k ⊕ (proj) in mod(kG), for some projective kG-module (proj).

We say that two endotrivial kG-modules are equivalent if they are isomorphic in stmod(kG). The set T (G) of isomorphism classes in stmod(kG) of endotrivial kG-modules is an abelian group, called the group of endotrivial modules. The composition law is defined by [M] + [N] = [M ⊗ N].

In particular, in T (G), we have 0 = [k] and −[M] = [M^{∗}].

Endotrivial kG-modules have been defined by E. Dade (cf. [13]), in 1978, for finite p-groups, as a particular case of the capped endo-permutation kG-modules. A capped endo-permutation kGmodule, for a finite p-group G, is a finitely generated kG-module whose endomorphism algebra is a permutation module having a trivial direct summand. Modulo a suitable equivalence relation, they form a finitely generated abelian group D(G), and the group T (G) identifies with a subgroup of D(G).

Also, for a subgroup H of G, the restriction map Res_{H}^{G} : mod(kG) → mod(kH) (also denoted by “·↓_{H}^{G} ”) induces a group homomorphism Res_{H}^{G} : T (G) → T (H).

Non-trivial examples of endotrivial modules are the syzygies of the trivial module, whereas, in the case of a finite p-group, most of the relative syzygies are capped endo-permutation modules (not endotrivial in general, cf. [1]). Let us recall their definitions.

Definition 2.2. If X is a finite G-set, then Ω_{X}^{1}(k) is the relative (to X) syzygy of k, that is, the kernel of the augmentation map kX → k.

If X = G, then we define the syzygy Ω_{G}^{n}(k) of k, for each n ∈ ℤ, as follows. If n ≥ 1, we let Ω_{G}^{n}(k) be the kernel of the (n − 1)-st differential in a minimal projective resolution of k.

\xymatrix{… \ar[r]^{∂_{n+1}} & X_{n} \ar[rr]^{∂_{n}} \ar@{->>}[dr] &  & X_{n−1} \ar[r]^{∂_{n−1}} & … \ar[r]^{∂_{2}} & X_{1} \ar[rr]^{∂_{1}} \ar@{->>}[dr] &  & X_{0} \ar@{->>}[r]^{∂_{0}} & k \\  &  & Ω_{G}^{n}(k) \ar@{^{(}->}[ur] &  &  &  & Ω_{G}^{1}(k) \ar@{^{(}->}[ur]}

If n ≤ −1, we let Ω_{G}^{n}(k) = Hom_{k}(Ω_{G}^{−n}(k), k), and we set Ω_{G}^{0}(k) = k.

Let G be a p-group and suppose that Ω_{X}^{1}(k) is a capped endo-permutation kG-module. Then we let Ω_{X} denote its class in D(G), or T (G) in case it is endotrivial. In particular, for any finite group G, and for any integer n, the syzygies Ω_{G}^{n}(k) are indecomposable endotrivial modules and we have [Ω_{G}^{n}(k)] = nΩ_{G} in T (G). We refer the reader to Section 4 in [7] for more properties of the syzygies, and to Sections 2 to 5 in [4] for those of the relative syzygies.

Elementary abelian p-subgroups play an important role in the analysis of T (G). In particular, we will need the following group theoretical notions for our purposes.

Definition 2.3. Let G be a finite group and p be a prime.

(1) The p-rank of G is the largest integer r such that G has an elementary abelian p-subgroup of rank r.

(2) We write E_{≥2}(G) for the poset of G-conjugacy classes of elementary abelian p-subgroups of G of p-rank at least 2.

Note that the p-rank of a finite group G is the p-rank of a Sylow p-subgroup of G. Moreover, if a Sylow p-subgroup P is normal in G, then E_{≥2}(G) identifies with the quotient of E_{≥2}(P) by the action by conjugation of G.

The number of connected components of E_{≥2}(G) and the torsion-free rank of T (G) are closely related, as shown by the next two theorems. We state them here as they appear in [12] and in [9] respectively. Let us point out the fact that the first result was first obtained by J. Alperin (cf. [1, Theorem 4]).

Let P be a finite p-group. Assume that P has p-rank at least 2 and is not semi-dihedral. Let F_{1}, …, F_{m} denote a set of representatives of the components D_{1}, …, D_{m} of the poset E_{≥2}(P). By [12], we know that we can make such a choice satisfying that all F_{i}’s have p-rank 2, all but possibly F_{m} are maximal, and F_{m} is normal in P. For each 1 ≤ i ≤ m let S_{i} be a non-central subgroup of P of order p that is contained in F_{i}. By [12, Lemma 2.2], if F_{i} is maximal then C_{P} (S_{i}) = S_{i} × L_{i}, where L_{i} has normal p-rank one. That is, L_{i} is cyclic if p is odd, and can be either cyclic or generalized quaternion in case p = 2.

Theorem 2.4 ([12], Theorem 3.1). The group T (P) of endotrivial kP -modules is torsion-free of rank m and it is generated by Ω_{P} and the classes of the modules N_{i} for i = 1, …, (m − 1), where N_{i} is the unique indecomposable summand with vertex P of

M_{i}^{⊗2} if C_{P} (S_{i})/S_{i} is cyclic of order ≥ 3,
M_{i} if p = 2 and |C_{P} (S_{i})/S_{i}| = 2,
M_{i}^{⊗4} if p = 2 and C_{P} (S_{i})/S_{i} is generalized quaternion,

and where M_{i} = Ω_{P}^{−1}(k) ⊗ Ω_{P/S_{i}}^{1}(k). Moreover,

Res_{F_{j}}^{P} N_{i} ≅ \{ \begin{matrix}Ω_{F_{i}}^{a_{i}}(k) ⊕ (proj) & if i = j \\ k ⊕ (proj) & otherwise \end{matrix}

where

a_{i} = −2p if C_{P} (S_{i})/S_{i} is cyclic of order ≥ 3,
a_{i} = −2 if p = 2 and |C_{P} (S_{i})/S_{i}| = 2,
a_{i} = −8 if p = 2 and C_{P} (S_{i})/S_{i} is generalized quaternion.

In particular, if T (P) is cyclic, then T (P) = ⟨Ω_{P} ⟩.

We refer the reader to [10], [12] and [13] for a detailed description of the group of endotrivial modules in the case that m = 1, since we will not consider that situation later.

Let G be a finite group having a normal Sylow p-subgroup P, and let F_{1}, …, F_{m} be as above. Choose representatives E_{1}, …, E_{n} of the components C_{1}, …, C_{n} of the poset E_{≥2}(G).

Remark 2.5. If P has p-rank at least 3 and m ≥ 2, then, for any index 1 ≤ i ≤ m − 1, the subgroups F_{i} and F_{m} are not G-conjugate. Indeed, since P is normal in G, then P contains all p-subgroups of G, and since the p-rank is at least 3, then F_{m} is contained in an elementary abelian p-subgroup F of rank 3. Thus, if ^{g}F_{i} = F_{m} < F, for some g ∈ G, then F_{i} < F^{g} < P. But this means that F_{i} is not maximal in P, which implies i = m. In other words, there is only one connected component of elementary abelian p-subgroups of rank at least 3, which is thus invariant by any automorphism of P, in particular here by G-conjugation.

This observation allows us, without loss of generality and only for the convenience of the notation, to choose the indexes of the F_{i}’s so that E_{i} = F_{i}, for 1 ≤ i ≤ n − 1 and E_{n} = F_{m}. The structure of T (G) is as follows (cf. [9]).

Theorem 2.6. The group T (G) is finitely generated, and hence, it splits as a direct sum T T (G) ⊕ T F (G), where T T (G) denotes the torsion subgroup and T F (G) is a torsion-free subgroup of maximal rank (as ℤ-module).

Moreover, T (G) is generated by the classes of the indecomposable endotrivial kG-modules whose restriction to P is an indecomposable module, and T F (G) ≅ ℤ^{n}.

In particular, if T T (P) is trivial, then T T (G) is generated by the classes of the one-dimensional kG-modules. Also, if n = 1, then T F (G) = ⟨Ω_{G}⟩.

Besides the purpose of the classification of endotrivial kG-modules, the fact that the syzygies of k are indecomposable endotrivial modules has the following consequence.

Corollary 2.7. Let P be a normal Sylow p-subgroup of a finite group G, and let (X., ∂.) be a kG-projective resolution of the trivial module k. Then, (X., ∂.) is minimal ⟺ (X.↓_{P}^{G} , ∂.↓_{P}^{G} ) is a minimal kP -projective resolution of k.

Proof. “⟸” is obvious.

“⟹” A projective kG-resolution of k is minimal if and only if it is the splice of short exact sequences, one for each integer n ≥ 0,

\xymatrix{0 \ar[r] & Ω_{G}^{n+1}(k) \ar[r]^{i_{n}} & X_{n} \ar[r]^{∂_{n}} & Ω_{G}^{n}(k) \ar[r] & 0}

By restriction to P, we get then

\xymatrix{0 \ar[r] & Ω_{P}^{n+1}(k) ⊕ (proj) \ar[r]^{i_{n}↓_{P}^{G}} & X_{n} ↓_{P}^{G} \ar[r]^{∂_{n}↓_{P}^{G}} & Ω_{P}^{n} (k) ⊕ (proj) \ar[r] & 0}

But the kG-modules Ω_{G}^{n}(k) are indecomposable endotrivial and P ◁ G, and so, by Theorem 2.6, Ω_{G}^{n}(k)↓_{P}^{G} is indecomposable. Therefore the projective factors “(proj)” are all zero. In other words, for each integer n ≥ 0, the above short exact sequence is in fact

\xymatrix{0 \ar[r] & Ω_{P}^{n+1}(k) \ar[r]^{i_{n}↓_{P}^{G}} & X_{n} ↓_{P}^{G} \ar[r]^{∂_{n}↓_{P}^{G}} & Ω_{P}^{n} (k) \ar[r] & 0}

Hence, (X.↓_{P}^{G} , ∂.↓_{P}^{G} ) is a minimal kP -projective resolution of k.

□

3. Dade’s result and its consequences

Throughout this section, we assume that G is a finite group and that P is a normal Sylow p-subgroup of G. We continue with the notation used in the previous section.

The only cases in which generators for the group T F (G) are not given by Theorem 2.6 are for T F (G) not cyclic, i.e. for n ≥ 2, according to our notation. Hence, we will assume that n ≥ 2. In this case, we also have m ≥ 2, and so T T (P) = 0, by Theorem 2.4. Moreover, this also implies that the center of P is cyclic non trivial. Thus, it has a unique cyclic subgroup Z of order p, which is contained in any maximal elementary abelian p-subgroup. Furthermore, Z ◁ G, since Z is characteristic in P.

A key tool to determine which endotrivial kP -modules extend to G is provided by the following theorem, proven by Dade and never published (as far as we are aware of).

Theorem 3.1 (Theorem 7.1, [14]). Let G be a finite group having a normal Sylow p-subgroup P, let k be an algebraically closed field of characteristic p, and let M be an endo-permutation kP -module. Then M extends to a kG-module if and only if M is G-stable.

By definition, if H is a normal subgroup of G and M is a kH-module, then we define the conjugate kH-module^{g}M of M by setting \underbrace{h ·^{g}m}_{in ^{g}M} = ^{g}( \underbrace{h^{g} · m}_{in M} ) , ∀ h ∈ H, ∀ m ∈ M, ∀ g ∈ G. We

say then that M is G-stable if we have an isomorphism of kH-modules M ≅ ^{g}M, ∀ g ∈ G. Equivalently, a kH-module M is G-stable if and only if ^{g}M ≅ M, ∀ g ∈ G/H.

Since an endotrivial module is an endo-permutation module, and since a module is endotrivial if and only if its restriction to a Sylow p-subgroup is, we have the following.

Corollary 3.2. The map Res_{P}^{G} : T (G) → T (P) induces an isomorphism of abelian groups T F (G) ≅ T F (P)^{G/P}, where T F (P)^{G/P} is the subset of the G/P -fixed points. That is, T F (P)^{G/P} is the subgroup of T F (P) generated by the classes of the G-stable endotrivial kP -modules.

Proof. By Theorem 2.6, we have ker(Res_{P}^{G}) = T T (G), and the image of Res_{P}^{G} is necessarily a subgroup of T (P)^{G/P}. Hence, Res_{P}^{G} induces an injection T F (G) → T F (P)^{G/P}. Now, by Theorem 3.1, an endotrivial kP -module M extends to a kG-module (necessarily endotrivial) if and only if M is G-stable. That is, the above map is also surjective.

□

At this stage, determining a set of generators for T F (G) comes down to a question of linear

algebra. Indeed, if x ∈ T (P) then x is uniquely expressible as a ℤ-linear combination x = ∑_{i=1}^{m} x_{i}e_{i},

where e_{i} = [N_{i}], for the module N_{i} defined in Theorem 2.4, for 1 ≤ i ≤ m − 1, and e_{m} = Ω_{P}.

Lemma 3.3. For all g ∈ G, and for all 1 ≤ i ≤ m, we have ^{g}e_{i} = e_{σ_{g}(i)}, for the permutation σ_{g}, acting on the set {1, …, m}, defined by ^{g}S_{j} =_{P} S_{σ_{g}(j)}, ∀ 1 ≤ j ≤ m − 1 and σ_{g}(m) = m.

Here, the notation “ =_{P} ” means “is P -conjugate to”.

Proof. Fix an index 1 ≤ i ≤ m − 1, and write S = S_{i} and N = N_{i}. Let g ∈ G and consider the map φ : ^{g}k[P/S] ⟶ k[P/^{g}S], defined by φ(^{g}(uS)) = ^{g}u^{g}S on a k-basis {^{g}(uS) | uS ∈ P/S} of the permutation kP -module ^{g}k[P/S]. Let us verify that φ is an isomorphism of kP -modules. It is immediate that φ is an isomorphism of k-vector spaces, and so we only need to check that φ commutes with the action of P:

v · φ(^{g}(uS)) = v ·^{g}u^{g}S = ^{g}(v^{g} u)^{g}S = φ(^{g}(v^{g} uS)) = φ(v ·^{g}(uS)) , ∀ u ∈ P, ∀ uS ∈ P/S, ∀ g ∈ G.

It follows that we have exact sequences forming the commutative diagram:

\xymatrix{0 \ar[r] & ^{g} Ω_{P/S}^{1}(k) \ar[r] \ar[d]^{φ↓_{ker(^{g}ε)}} & ^{g}k[P/S] \ar[r]^{^{g}ε} \ar[d]^{φ} & ^{g}k \ar[r] \ar[d]^{Id} & 0 \\ 0 \ar[r] & Ω_{P/^{g}S}^{1}(k) \ar[r] & k[P/^{g}S] \ar[r]^{ε′} & k \ar[r] & 0}

Since φ is an isomorphism, the left hand kP -modules ^{g}Ω_{P/S}^{1}(k) and Ω_{P/^{g}S}^{1}(k) are isomorphic. Similarly, we have ^{g}Ω_{P}^{1} (k) ≅ Ω_{P}^{1} (k) and the result follows.

□

This lemma has an immediate consequence.

Proposition 3.4. x = ∑_{i=1}^{m} x_{i}e_{i} ∈ T (P)^{G} ⟺ x_{i} = x_{j} whenever S_{i} =_{G} S_{j}, ∀ 1 ≤ i ≤ m − 1.

We will need a group theoretical result, complementary to Lemma 2.2 in [12]. For this, let us fix an index 1 ≤ i ≤ n − 1 and write S = S_{i}, E = E_{i} and N = N_{i}. Recall that our choice of the E_{i}’s forces E_{i} = F_{j} for some j.

Lemma 3.5. The stabilizer of the P -conjugacy class of S is the subgroup P N_{G}(E) of G.

Proof. By [12, Lemma 2.2], we know that C_{P} (S) = N_{P} (S) = S × L, where L has normal p-rank 1 and L contains the unique central subgroup Z of P of order p. Moreover, the index of N_{P} (S) in N_{P} (E) is p. In particular, we deduce that E = S × Z, and that N_{P} (E)/N_{P} (S) acts transitively (by conjugation) on the p non central subgroups of order p of E.

Since Z ◁ G, we have N_{G}(S) ≤ N_{G}(E), and so P N_{G}(S) ≤ P N_{G}(E).

Conversely, for g ∈ N_{G}(E), we have ^{g}S ≤ ^{g}E = E. Thus, there exists v ∈ P (actually, we can take v ∈ N_{P} (E) − N_{P} (S)) such that ^{g}S = ^{v}S, and so, v^{−1}g ∈ N_{G}(S). It follows then that g ∈ P N_{G}(S), as was to be shown.

□

Since P N_{G}(E) is the stabilizer of the P -conjugacy class of S, Theorem 3.1 implies that N extends to a k[P N_{G}(E)]-module Ñ, which is necessarily endotrivial.

Let C be a set of representatives of the left cosets G/P N_{G}(E). Then the correspondence c ↦ ^{c}N is a bijection from C to the G-conjugacy class of N.

Definition 3.6. Let H be a subgroup of G and let M be a kH-module. We define the tensor induced module Ten_{H}^{G} M as follows (cf. [15, § 5.1]). It is the k-vector space

⨂_{s∈[G/H]} (s ⊗ M) , where [G/H] is a set of representatives of the left cosets G/H

endowed with the structure of kG-module given by

g · ⊗_{s}(s ⊗ m_{s}) = ⊗_{s}(τ_{g}(s) ⊗ h_{s}m_{s}) = ⊗_{s}(s ⊗ h_{τ_{g}^{−1}(s)}m_{τ_{g}^{−1}(s)}) ,

where gs = τ_{g}(s)h_{s}, with h_{s} ∈ H and a permutation τ_{g} of [G/H].

Proposition 3.7. The kG-module Ten_{P N_{G}(E)}^{G} Ñ is endotrivial.

Proof. It is enough to check that Res_{P}^{G} Ten_{P N_{G}(E)}^{G} Ñ is endotrivial. By the tensor version of Mackey formula (cf. [15, Prop. 5.2.1]), and since P ◁ G, we have isomorphisms of kP -modules

Res_{P}^{G} Ten_{P N_{G}(E)}^{G} Ñ ≅ ⨂_{c∈C} Ten_{P ∩ (^{c}P N_{G}(E))}^{P} Res_{P ∩ (^{c}P N_{G}(E))}^{(^{c}P N_{G}(E))} Ñ^{c} =

= ⨂_{c∈C} Res_{P}^{(^{c}P N_{G}(E))} Ñ^{c} ≅ ⨂_{c∈C}^{c(} Res_{P}^{P N_{G}(E)} Ñ) ≅ ⨂_{c∈C}^{c}N

which is a tensor product of endotrivial kP -modules and hence is endotrivial.

□

For any elementary abelian p-group E of rank at least 2, we may identify the group T (E) = ⟨Ω_{E}⟩ with ℤ, via nΩ_{E} ↦ n.

Definition 3.8. Let H be a finite group and let E_{1}, …, E_{l} be representatives of the connected components of E_{≥2}(H). The product of all restriction maps Res_{E_{i}}^{H} : T (H) ⟶ T (E_{i}), composed with the isomorphism T (E_{i}) ≅ ℤ, 1 ≤ i ≤ l, yields a well-defined homomorphism

\xymatrix{res_{E(H)} : T (H) \ar[r] & ∏_{i=1}^{l}T (E_{i}) \ar[r] & ℤ^{l}}

For any endotrivial kH-module M, the element res_{E(H)}([M]) ∈ ℤ^{l} is called the type of M.

Throughout these notes, δ_{i,j} denotes the Kronecker symbol.

Proposition 3.9. We have ker(res_{E(G)}) = T T (G). Moreover,

res_{E(G)}([Ten_{P N_{G}(E_{j})}^{G} Ñ_{j}]) = (δ_{i,j}a_{i})_{i=1}^{n} ,

where a_{i} is the integer defined in Theorem 2.4.

Proof. The first statement has been proved in [9, Proposition 2.3], so we only need to show the second one.

By transitivity of the restriction and Proposition 3.7, we have

Res_{E_{i}}^{G} Ten_{P N_{G}(E_{j})}^{G} Ñ_{j} ≅ ⨂_{c∈C_{j}} Res_{E_{i}}^{P} ^{c}N_{j}

where C_{j} is a set of representatives of the left cosets G/P N_{G}(E_{j}). We may assume that 1 ∈ C_{j} for all j. Now, for any c ∈ C_{j}, the kP -module ^{c}N_{j} is endotrivial and defined in Theorem 2.4 as the “unique” indecomposable direct summand with vertex P of a r-fold tensor product of the module

c(Ω_{P}^{−1}(k) ⊗ Ω_{P/S_{j}}^{1}(k)) ≅ Ω_{P}^{−1}(k) ⊗ Ω_{P/^{c}S_{j}}^{1}(k)

where the integer r is either 1, 2 or 4. By “unique”, we mean unique up to isomorphism.

Even though it is not an endotrivial module, Res_{E_{i}}^{P} Ω_{P/^{c}S_{j}}^{1}(k) has a “unique” indecomposable summand V with vertex E_{i}, and V is isomorphic to Ω_{(P/^{c}S_{j})↓_{E_{i}}^{P}}^{1} (k). By Mackey formula (for

E_{i}-sets) we have that

(P/^{c}S_{j}) ↓_{E_{i}}^{P} ≅ ∐_{x∈[E_{i}\P/^{c}S_{j}]} E_{i}/(E_{i} ∩^{xc}S_{j})

If i ≠ j then E_{j} ≠_{G} E_{i} and no G-conjugate of S_{j} is contained in E_{i}, since E_{i} = S_{i} × Z and Z ◁ G. It follows that

(P/^{c}S_{j})↓_{E_{i}}^{P} ≅ ∐_{x∈[E_{i}\P/^{c}S_{j}]} E_{i} . Thus, V ≅ Ω_{E_{i}}^{1}(k) if i ≠ j, by [4, Lemma 3.2.7],

and therefore ^{c}N_{j} is trivial.

Otherwise, we have i = j. If ^{xc}S_{i} < E_{i}, for x ∈ P, then there is y ∈ N_{P} (E_{i}) − C_{P} (S_{i}) such that xcS_{i} = ^{y}S_{i}, since N_{P} (E_{i})/C_{P} (S_{i}) acts transitively on the p non central subgroups of order p of E_{i}. It follows that y^{−1}xc ∈ N_{G}(S_{i}), and so c ∈ P N_{G}(S_{i}) = P N_{G}(E_{i}). By the choice of representatives in C_{i}, this implies c = 1. Then, Theorem 2.4 shows that Res_{E_{i}}^{P} N_{i} ≅ Ω_{E_{i}}^{a_{i}} ⊕ (proj).

Finally, if i = j and c ∉ P N_{G}(E_{i}), then the previous argument shows that ^{xc}S_{i} is not contained in E_{i}, for any x ∈ P. Therefore, Res_{E_{i}}^{P} ^{c}N_{i} ≅ k ⊕ (proj), as in the case i ≠ j.

In conclusion, by tensoring all the pieces together, we deduce that

Res_{E_{i}}^{G} Ten_{P N_{G}(E_{j})}^{G} Ñ_{j} ≅ ⨂_{c∈C_{j}} Res_{E_{i}}^{P} ^{c}N_{j} ≅ \{ \begin{matrix}Ω_{E_{i}}^{a_{i}}(k) ⊕ (proj) & if i = j \\ k ⊕ (proj) & if i ≠ j \end{matrix}
□

Theorem 3.10. Let G be a finite group having a normal Sylow p-subgroup P. Consider the same notation as above, and write the group T (G) of endotrivial kG-modules as a direct sum T (G) = T T (G) ⊕ T F (G), where T T (G) is the torsion subgroup and T F (G) is torsion-free. Then, we may choose the set

{Ω_{G}, x_{i} = [Ten_{P N_{G}(E_{i})}^{G} Ñ_{i}] | 1 ≤ i ≤ n − 1} as a basis for the free ℤ-module T F (G).

Moreover, we have res_{E(G)}(Ω_{G}) = (1)_{i=1}^{n}, and res_{E(G)}(x_{j}) = (δ_{i,j}a_{i})_{i=1}^{n}.

Proof. The kG-modules Ten_{P N_{G}(E_{i})}^{G} Ñ_{i} are endotrivial and their classes are linearly independent, by Theorem 2.4 and the proof of Proposition 3.7. Hence they generate a torsion-free submodule of T (G) of rank equal to the rank of T F (G).

Moreover, the proof of Proposition 3.7 shows that in T (P) we have

Res_{P}^{G}[Ten_{P N_{G}(E_{i})}^{G} Ñ_{i}] = ∑_{c∈C_{i}}[^{c}N_{i}]

where C_{i} denotes a set of the representatives of the left cosets G/P N_{G}(E_{i}). Since {^{c}N_{i} | c ∈ C_{i}} is a sequence without repetition that contains all the G-conjugates of N_{i}, it also makes the element ∑_{c∈C_{i}}[^{c}N_{i}] of T (P) satisfy the minimal necessary condition for being G-stable, by Proposition 3.4.

This proves the first statement. The second claim has been proved in Proposition 3.9.

□

4. Alternative constructions

Throughout this section, we let G be a finite group having a normal Sylow p-subgroup P. We assume n ≥ 2, and we choose the representatives E_{1}, …, E_{n} and F_{1}, …, F_{m} of the connected components of E_{≥2}(G) and E_{≥2}(P), respectively, such that E_{i} = F_{i}, ∀ 1 ≤ i ≤ n − 1 and E_{n} = F_{m}, as before. Write also Z = ⟨z⟩ for the unique central subgroup of P of order p. Finally, let a_{i} be the integer defined in Theorem 2.4.

The purpose of this section is to review the two constructions of endotrivial modules providing free sets of generators for T F (G), as given in [9]. Then, we compare the modules obtained for each construction. We assume that the reader is familiar with the notions of cohomology of finite groups and (cohomological) varieties of modules given in [7] or [15]. The notation we use in this section is as follows. For an inclusion of finite groups H → K, we denote the induced restriction map in cohomology H^{∗}(K, k) → H^{∗}(H, k) either Res_{H}^{K} or ↓_{H}^{K} indifferently. Meanwhile, we denote by res_{K,H}^{∗} : V_{H}(k) → V_{K}(k) the induced map on the varieties.

Let us now recall the presentation of T F (G) given in [9].

Theorem 4.1 ([9, Theorem 3.4]). For each 1 ≤ i ≤ n − 1 there exists an endotrivial kG-module U_{i} such that

U_{i} ↓_{E_{i}}^{G} ≅ Ω_{E_{i}}^{a_{i}} ⊕ (proj) and U_{i} ↓_{E_{j}}^{G} ≅ k ⊕ (proj) , ∀ j ≠ i ,

where the a_{i}’s are the integers defined in Theorem 2.4.

The classes Ω_{G}, [U_{i}], 1 ≤ i ≤ n − 1 form a free set of generators for T F (G).

The structure of the modules U_{i} is described in the proof, which uses the construction originating from [11] and quoted by the authors as the “deconstruction method”. Briefly, it consists in analysing the variety V = V_{G/Z}(\overline{Ω_{G}^{a_{i}}(k)}), where \overline{Ω_{G}^{a_{i}}(k)} is the kG/Z-module that is the quotient of Ω_{G}^{a_{i}}(k) by the submodule (z − 1)^{p−1}Ω_{G}^{a_{i}}(k). The arguments of [8, Theorem 7.2] show that for each index j, the variety V decomposes as a union of varieties V_{j} ∪ V′_{j} such that

V_{j} ∩ V′_{j} = {0} , V_{j} = res_{G/Z,E_{j}/Z}^{∗}(V_{E_{j}/Z}(k)) and res_{G/Z,E_{l}/Z}^{∗}(V_{E_{l}/Z}(k)) ⊆ V′_{j} , ∀ l ≠ j

Then, by [8, Corollary 4.3], there exists an endotrivial module of the desired type, whose construction appeals to previous results (and ultimately from a theorem of [2], applied to the Lyndon-Hochschild-Serre spectral sequence of the group extension 1 → Z → G → G/Z → 1). Indeed, it turns out that each V_{j} is the variety of the kG/Z-module that is the quotient U_{j}/(z − 1)^{p−1}U_{j} of an indecomposable endotrivial kG-module U_{j}.

We now turn to the third construction. Note that it does not require P ◁ G, and that, instead, it necessites an additional assumption on the cohomology group H^{∗}(G, k).

Proposition 4.2 ([8, Corollary 4.6]). Suppose that Z is a subgroup of order p in the center of a Sylow p-subgroup P of G. Suppose that for d > 0 the group H^{2d}(G, k) has an element ζ such that Res_{Z}^{G}(ζ) ≠ 0. Then G has an endotrivial module of type (2dδ_{i,j})_{i=1}^{n} for each 1 ≤ j ≤ n.

Let us recall how these endotrivial modules are obtained, starting from a generalization of the definition of Carlson’s L_{ζ}’s modules (cf. [7, § 6]).

Definition 4.3. Let ζ ∈ H^{s}(G, S) be non zero, where S is a 1-dimensional kG-module and s ≥ 1. Let ζ̃ ∈ Hom_{G}(Ω_{G}^{s}(k), S) represent ζ. We define the kG-module L_{ζ,S} = ker ζ̃.

Among many properties of these modules, all that we need to know for our concern is that the modules L_{ζ,S} are defined up to isomorphism by the class ζ ∈ H^{s}(G, S). In particular, for s = 2d, as in Proposition 4.2, we have a short exact sequence of kG-modules

\xymatrix{0 \ar[r] & L_{ζ,S} \ar[r] & Ω_{G}^{2d}(k) \ar[r]^{ζ̃} & S \ar[r] & 0}

which does not depend on the choice of ζ̃, and which does not involve any projective summand. Since Ω_{Z}^{2d}(k) = k, we have that Ω_{G}^{2d}(k)↓_{Z}^{G} ≅ k⊕(proj). Thus, the assumption that ζ ↓_{Z}^{G} ≠ 0 implies that L_{ζ,S} ↓_{Z}^{G} is projective. By Quillen Dimension Theorem (cf. [15, Corollary 8.3.3]), it follows that the variety V_{G}(L_{ζ,S}) of L_{ζ,S} is disconnected, since the number n of connected components of E_{≥2}(G) is at least 2. More precisely, V_{G}(L_{ζ,S}) is a union of n subspaces V_{1}, …, V_{n}, each V_{i} being the variety of a kG-submodule L_{i} of L_{ζ,S}, such that L_{ζ,S} decomposes as direct sum L_{1} ⊕ ⋯ ⊕ L_{n}. Note that V_{i} is a line for each 1 ≤ i ≤ n − 1. The similar reasoning applies to the group P instead of G, and since Ω_{G}(k) ↓_{P}^{G} ≅ Ω_{P} (k), we conclude that L_{ζ,S} ↓_{P}^{G} ≅ L_{ζ↓_{P}^{G}} splits as a direct sum X_{1} ⊕ ⋯ ⊕ X_{m}, where m ≥ 2 is the number of connected components of E_{≥2}(P). Moreover, if V_{P} (L_{ζ,S} ↓_{P}^{G} ) = W_{1} ∪ ⋯ ∪ W_{m}, with W_{i} = V_{P} (X_{i}), then G acts by permutation on the W_{i}’s and hence on the X_{i}’s. It follows that L_{i} ↓_{P}^{G} ≅ ⊕_{g∈[G/G_{i}]} g · X_{i}, for the stabilizer G_{i} = P N_{G}(F_{i}) of the P -conjugacy class of F_{i}, as in the previous section.

For each index 1 ≤ i ≤ n, set L′_{i} = ⊕_{j∈J_{i}}L_{j}, where J_{i} = {j | 1 ≤ j ≤ n, i ≠ j}.

Proposition 4.4. Each L_{i} gives rise to an indecomposable endotrivial kG-module M_{i}, of type (2dδ_{i,j})_{j=1}^{n}, built as the push-out in the following commutative diagram.

[figure: 0; 0; L′_{i}; L′_{i}; 0; L_{ζ,S}; Ω_{G}^{2d}(k); ζ̃; S; 0; 0; L_{i}; M_{i}; S; 0; 0; 0]

Proof. The proof is a paraphrase of an argument used in the demonstration of [9, Theorem 3.1], and goes as follows.

Let 1 ≤ i, j ≤ n. We need to show that

M_{i} ↓_{E_{j}}^{G} ≅ \{ \begin{matrix}Ω_{E_{j}}^{2d}(k) ⊕ (proj) & if i = j \\ k ⊕ (proj) & otherwise \end{matrix}

We have L_{i} ↓_{P}^{G} = ⊕_{g∈[G/G_{i}]} g · X_{i}, with X_{i} ↓_{E_{j}}^{P} projective (and hence injective) if and only if j ≠ i, i.e. E_{j} ≠ F_{i}, by our choice of the representatives of the connected components of E_{≥2}(P) and E_{≥2}(G). This implies that exactly one of the restrictions to E_{j} of the exact sequences consisting in the second column or the bottom row is split, depending on whether X_{j} is a summand in L_{i} ↓_{P}^{G} or not. Namely, the restriction to E_{j} of the second column splits if and only if X_{j} | L_{i} ↓_{P}^{G} , that is, if and only if X_{j} = g · X_{i}, for some g ∈ G. This happens if and only if i = j, and in this case we have

M_{j} ↓_{E_{j}}^{G} ⊕(proj) ≅ Ω_{P}^{2d}(k)↓_{E_{j}}^{P} ≅ Ω_{E_{j}}^{2d}(k) ⊕ (proj).

Hence M_{j} ↓_{E_{j}}^{G} ≅ Ω_{E_{j}}^{2d}(k) ⊕ (proj).

Otherwise, i ≠ j and the restriction to E_{j} of the bottom row splits, since the left term is injective. In this case, we have M_{i} ↓_{E_{j}}^{G} ≅ k ⊕ (proj).

Thus M_{i} ↓_{E_{j}}^{G} is endotrivial for all 1 ≤ i, j ≤ n, and the type of M_{i} is (2dδ_{i,j})_{i=1}^{n}, as asserted. □

We discuss the possible additional assumptions needed for this construction to be carried out. That is, we want to give a sufficient and necessary condition on the cohomology group H^{2d}(P, k), that would detect when there exists a one-dimensional kG-module S such that H^{2d}(G, S) contains an element that restrict non trivially to Z.

Let H be a normal subgroup of G. Then Hom_{H}(M ↓_{H}^{G} , k) is a kG-module, for any kG-module M, and since the restriction map commutes with the differentials in complexes of kG- and kH-modules, it induces a map of kG-modules Res_{H}^{G} : H^{s}(G, k) ⟶ H^{s}(H, k) in cohomology (cf. [15, § 4.1]). In fact, H acts trivially on Hom_{H}(M ↓_{H}^{G} , k), and so, we can consider Res_{H}^{G} as a map of kG/H-modules. Similarly, for G, P, Z and 2d as in Proposition 4.2, the map Res_{Z}^{P} : H^{2d}(P, k) ⟶ H^{2d}(Z, k) is a map of kG/P -modules. Since p does not divide the order of G/P, the map Res_{Z}^{P} splits. Now, the fact that H^{2d}(Z, k) is one dimensional implies that, if Res_{Z}^{P} is non zero, then any non zero element β ∈ H^{2d}(Z, k) lifts to an element ζ ∈ H^{2d}(P, k) that restricts non trivially to Z and such that (g · ζ)↓_{Z}^{P} = g · ζ ↓_{Z}^{P} = μ_{g}β, for some |G : P |-root of unity μ_{g} ∈ k, for any g ∈ G.

On the other hand, we also have that the action of G on the 1-dimensional k-vector space H^{s}(Z, k), for any integer s ≥ 0, defines a representation ρ : G → Aut(H^{s}(Z, k)) of G. Set H for the kernel of ρ, that is, H = {g ∈ G | g · ζ = ζ , ∀ ζ ∈ H^{s}(Z, k)} is the stabilizer of H^{s}(Z, k) in G. In particular, H is a normal subgroup of G containing P, and the quotient group G/H is cyclic of order prime to p, since it is isomorphic to a finite subgroup of the group of units in k. Hence, H^{s}(Z, k) is a kG/H-module.

Lemma 4.5. Let s ≥ 0 be an integer and assume there exists ζ ∈ H^{s}(P, k), such that ζ ↓_{Z}^{P} ≠ 0. Then Tr_{P}^{H} ζ ∈ H^{s}(H, k) and we have (Tr_{P}^{H} ζ)↓_{Z}^{H} ≠ 0, where Tr_{P}^{H} denotes the transfer in cohomology ( cf. [15, § 4.2]).

Proof. Let ζ ∈ H^{s}(P, k), for some s ≥ 0, and assume that ζ ↓_{Z}^{P} ≠ 0. By definition of H, we have

Res_{Z}^{H} Tr_{P}^{H} ζ = ∑_{g∈[H/P]} g · ζ ↓_{Z}^{P} = |H : P | · ζ ↓_{Z}^{P} .

Since |H : P | is invertible in k, we have that Res_{Z}^{H} Tr_{P}^{H} ζ ≠ 0 if ζ ↓_{Z}^{P} ≠ 0.

□

This observation leads us to a sufficient and necessary condition for determining when the last construction of endotrivial modules can be applied, provided that it applies for T (P).

Proposition 4.6. There exist a 1-dimensional kG-module S and η ∈ H^{s}(G, S) such that η ↓_{Z}^{G} ≠ 0 if and only if there exists ζ ∈ H^{s}(P, k) such that ζ ↓_{Z}^{P} ≠ 0.

In particular, for s = 2d, there exists then an indecomposable endotrivial module of type (2dδ_{i,j})_{i=1}^{n}, for each index 1 ≤ j ≤ n.

Proof. The “if ” part is clear.

Assume there exists ζ ∈ H^{s}(P, k) such that ζ ↓_{Z}^{P} ≠ 0, and let H be the stabilizer of H^{s}(Z, k) in G, as above. By Lemma 4.5, we have that Tr_{P}^{H} ζ ∈ H^{s}(H, k) and that (Tr_{P}^{H} ζ) ↓_{Z}^{H} ≠ 0. By the Eckmann-Shapiro Lemma (cf. [15, Proposition 4.1.3]), we have H^{s}(H, k) ≅ H^{s}(G, k↑_{H}^{G}). Thus, the image of Tr_{P}^{H} ζ under this isomorphism is an element of H^{s}(G, k↑_{H}^{G}) that restricts non trivially to the subgroup Z. Now, k↑_{H}^{G} is a direct sum of 1-dimensional kG-modules, since G/H is a cyclic group of order prime to p, as noted above, and since k is algebraically closed. Therefore, there is a 1-dimensional direct summand S of k↑_{H}^{G} and η ∈ H^{s}(G, S) such that η ↓_{Z}^{G} ≠ 0, as was to be shown.

The last assertion is then the application of Proposition 4.2.

□

We end this section with the comparison of the modules obtained in each construction, and, for convenience, we use the same notation. First, observe that a module Ten_{G_{i}}^{G} Ñ_{i} of Section 3 is not indecomposable and it has a large dimension, in general. Moreover the corresponding kP -module N_{i} is the cap of a (larger, in general) endo-permutation module. That is, N_{i} is the unique direct summand with vertex P of a capped endo-permutation kP -module. Consequently, if one wishes to compute T (G) with a computer support, the generators for T F (G) provided by the Ten_{G_{i}}^{G} Ñ_{i} are likely to involve time-consuming algorithms. Instead, the modules of Section 4 are indecomposable and their construction is likely to be more handy by an algebra software.

In order to compare the three presentations of T F (G), we use the injectivity of the restriction map res_{E(G)} : T F (G) → ∏_{i=1}^{n} T (E_{i}). Then, it is the simple matter of comparing the computations in Theorems 3.10, 4.1, and Proposition 4.2.

Remark 4.7. In stmod(kG), we have isomorphisms Ten_{G_{i}}^{G} Ñ_{i} ≅ U_{i}^{∗} ≅ M_{i}^{∗} , ∀ 1 ≤ i ≤ n. In particular, since U_{i} and M_{i} are indecomposable, we have U_{i} ≅ M_{i} in mod(kG), for all 1 ≤ i ≤ n.

It is remarkable, and remains yet unexplained, that the modules U_{i} and M_{i} are isomorphic, since their construction is apparently different. Moreover, the modules M_{i} cannot be built in general, whereas the modules U_{i} always exist.

5. Odd extraspecial example

Let p be an odd prime and let P be a Sylow p-subgroup of PSL_{3}(p). That is, P is an extraspecial p-group of order p^{3} and exponent p. Let G be its normalizer in PSL_{3}(p).

There are p+1 conjugacy classes in P of maximal elementary abelian p-subgroups of rank 2, each consisting of a single maximal normal subgroup of P. Let us call these subgroups F_{1}, …, F_{p+1}. Choose also non central subgroups S_{i} < F_{i}, 1 ≤ i ≤ (p + 1) of order p.

By Theorem 2.4, the group T (P) of endotrivial modules is free abelian of rank p + 1 and admits a presentation T (P) = ⟨Ω_{P} , [N_{i}], 1 ≤ i ≤ p⟩, where [N_{i}] denotes the class of the indecomposable endotrivial summand of the kP -module (Ω_{P}^{−1}(k) ⊗ Ω_{P/S_{i}}^{1}(k))^{⊗2}, for each index i. The elements [N_{i}], for 1 ≤ i ≤ p + 1, verify a non trivial relation in T (P) (cf. also [5, § 11])

∑_{i=1}^{p+1}[N_{i}] = 2p Ω_{P} , since both have the same type, namely (2p, …, 2p) ∈ ℤ^{p+1}.

Let us also point out that, for each index i, the module (Ω_{P}^{−1}(k) ⊗ Ω_{P/S_{i}}^{1}(k))^{⊗2} is not endotrivial (and hence it is decomposable). Indeed, it has dimension (p^{3} −1)^{2}(p^{2} −1)^{2}, which is not congruent to ±1( mod p^{3}), ∀ p > 2.

Let us now consider T (G). By Theorem 2.6, the torsion subgroup T T (G) is generated by the one-dimensional kG-modules, since T T (P) is trivial. That is, T T (G) is isomorphic to the character group of the abelian p′-group G/P ≅ C_{p−1} × C_{p−1}. On the other hand, for T F (G), we note that there are exactly two elementary abelian p-subgroups which are normal in G, and the p − 1 others form a unique G-conjugacy class. Say F_{1}, F_{2} ◁ G, and F_{3} =_{G} ⋯ =_{G} F_{p+1}. Then, we may choose E_{i} = F_{i}, 1 ≤ i ≤ 3, and so we get G_{1} = G_{2} = G and G_{3} has index p − 1 in G, where G_{i} = P N_{G}(E_{i}) = N_{G}(E_{i}) for 1 ≤ i ≤ 3. Indeed, by the Class Formula, the cardinality of the G-orbit of E_{3} is p − 1, which is hence the index in G of the stabilizer P N_{G}(E_{3}) of the G-conjugacy class of E_{3}.

From Proposition 3.4 and Lemma 3.5, we get T (P)^{G/P} = ⟨Ω_{P} , [N_{1}], [N_{2}]⟩ ≅ ℤ^{3}. Note also

that ∑_{i=3}^{p+1}[N_{i}] = −[N_{1}] − [N_{2}] ∈ T (P)^{G/P}.

Theorem 3.1 implies that the modules N_{1} and N_{2} extend to G, into indecomposable endotrivial modules Ñ_{1} and Ñ_{2}, respectively. Therefore, T F (G) = ⟨Ω_{P} , [Ñ_{1}], [Ñ_{2}]⟩, by Theorem 3.10.

By Proposition 3.7, the module N_{3} extends to an indecomposable endotrivial kG_{3}-module Ñ_{3}, yielding then an endotrivial kG-module Ten_{G_{3}}^{G} Ñ_{3}. Moreover, [Ten_{G_{3}}^{G} Ñ_{3}] + [Ñ_{1}] + [Ñ_{2}] ∈ T T (G).

Lastly, we remark that in the presentation of T F (G) above, the underlying modules are indecomposable, which is not the case in general. But it need to be pointed out that the modules N_{i} are proper direct summands of (endo-permutation) modules of dimension (p^{3} − 1)^{2}(p^{2} − 1)^{2}, and that at this point we do not know what Dim(N_{i}) is. The second half of this section will give us the answer.

We turn now to the presentation of T (P) and T F (G) described in Section 4, and for convenience, we adopt the same notation. In particular, since p is odd, we have a_{i} = 2p, for the integer a_{i} of Theorem 2.4. Hence, we can apply Proposition 4.2 to find generators for T (P). Indeed, by [3, Theorem 10.1] (or [16, Theorems 6 and 7]), there exists ζ ∈ H^{2p}(P, k) such that ζ ↓_{Z}^{P} ≠ 0. In fact, ζ ∈ H^{2p}(P, 𝔽_{p}) and ζ is obtained as the norm from any F ∈ E_{≥2}(P) to P of the inflation of a non zero element η ∈ H^{2}(Z, 𝔽_{p}), and where Z is identified with a quotient of F.

Let ζ̃: Ω_{P}^{2p}(k) → k represent ζ and set L_{ζ} = ker ζ̃. Then L_{ζ} is the direct sum L_{1} ⊕ ⋯ ⊕ L_{p+1} of indecomposable kP -modules. Moreover, all the modules L_{i} have same dimension, since the automorphism group Aut(P) of P permutes transitively the subgroups F_{1}, …, F_{p+1} and hence Aut(P) also acts by permuting transitively the L_{i}’s.

Now, the endotrivial modules M_{i}, for 1 ≤ i ≤ p + 1, are obtained as the push-out

\xymatrix{L_{ζ} \ar[r] \ar[d] & Ω_{P}^{2p}(k) \ar@{-->}[d] \\ L_{i} \ar@{-->}[r] & M_{i} .}

In addition, by [11, Corollary 4.4], we have Dim(Ω_{P}^{2p}(k)) = p^{3}(p + 1) + 1. Therefore, we get Dim(L_{i}) = p^{3}, and so Dim(M_{i}) = p^{3} + 1. Note that this is the minimal possible dimension that we could expect for M_{i}. Indeed, it has to be greater than 1 and congruent to 1 ( mod |P |).

Last but not least, we handle T F (G). Note that since H^{2p}(P, k) contains an element that restricts non trivially to Z, we can apply any of the two constructions presented in Section 4. We choose the “deconstruction method” (cf. Theorem 4.1).

We consider the kG/Z-module

U = Ω_{G}^{2p}(k)/(z − 1)^{p−1}Ω_{G}^{2p}(k) .

Note that the action of (z−1)^{p−1} = ∑_{i=0}^{p−1} z^{i} on a kZ-module M annihilates any non-free summand of M, and sends a free module onto its socle. By a dimension argument (cf. [11, Corollary 4.4]), we deduce that Ω_{G}^{2p}(k)↓_{Z}^{G} ≅ k ⊕ (kZ)^{p^{2}(p+1)}, and so (z − 1)^{p−1}Ω_{G}^{2p}(k) has dimension p^{2}(p + 1). It follows that Dim(U) = p^{2}(p + 1)(p − 1) + 1.

In addition, in the notation of the discussion after Theorem 4.1, the variety V = V_{G/Z}(U) decomposes as the union V_{1} ∪ V_{2} ∪ V_{3} ∪ W, where W = V′_{1} ∩ V′_{2} ∩ V′_{3}. This is because E_{≥2}(G) consists in three isolated vertices E_{1}, E_{2}, E_{3}. Then, each V_{i} is the variety of the kG/Z-module Ū_{i} = U_{i}/(z − 1)^{p−1}U_{i}, for an indecomposable endotrivial kG-module U_{i} satisfying

U_{i} ↓_{E_{j}}^{G} = \{ \begin{matrix}Ω_{E_{i}}^{2p}(k) ⊕ (proj) & if i = j \\ k ⊕ (proj) & if i ≠ j \end{matrix}

Since the group Aut(G) of automorphisms of G permutes transitively E_{1}, E_{2} and E_{3}, the induced action on V permutes transitively V_{1}, V_{2} and V_{3}. Consequently, the modules Ū_{1}, Ū_{2} and Ū_{3} have same dimension. Since the U_{i}’s are endotrivial, we have U_{i} ↓_{Z}^{G} = k ⊕ (proj). It follows that Dim(U_{i}) = 1 + p Dim(Ū_{i}), for 1 ≤ i ≤ 3, and thus the modules U_{i} have same dimension.

Now, E_{i} = F_{i}, for 1 ≤ i ≤ 3, and E_{3} =_{G} F_{i} , ∀ 3 ≤ i ≤ p + 1. That is, the G-conjugacy classes of F_{1} and F_{2} coincide with their respective P -conjugacy classes. Therefore, U_{1} ↓_{P}^{G} and U_{2} ↓_{P}^{G} are isomorphic to the indecomposable endotrivial modules obtained using the techniques of Theorem 4.1 applied to P instead of G, and corresponding to F_{1} and F_{2} respectively. By Remark 4.7, we have U_{i} ↓_{P}^{G} ≅ M_{i} for i = 1, 2. Thus, Dim(U_{i}) = p^{3} + 1 and Dim(Ū_{i}) = p^{2}, for i = 1, 2 and 3. Note also that the isomorphism U_{3} ↓_{P}^{G} ⊕ (proj) ≅ ⊗_{i=3}^{p+1}M_{i} tells us that Dim((proj)) = (p^{3} + 1)^{p−1} − (p^{3} + 1). On the other hand, the variety W is the variety of a kG/Z-module of dimension Dim(U) − 3 Dim(Ū_{1}) = p^{2}((p + 1)(p − 1) − 3) + 1.

We can now answer the question concerning the dimension of the modules Ñ_{1}, Ñ_{2} and Ten_{G_{3}}^{G} Ñ_{3}. Indeed, by duality and indecomposability, we have Dim(Ñ_{i}) = p^{3} + 1 for i = 1, 2, whereas Dim(Ten_{G_{3}}^{G} Ñ_{3}) = (p^{3} + 1)^{p−1}.

In particular, Ten_{G_{3}}^{G} Ñ_{3} is not indecomposable, since

Dim(M_{3}) = (p − 1)p^{3} + 1 < (p^{3} + 1)^{p−1} = Dim(Ten_{G_{3}}^{G} Ñ_{3}), ∀ p > 2 .

More precisely, we have Ten_{G_{3}}^{G} Ñ_{3} ≅ M_{3}^{∗} ⊕ (proj), where

Dim((proj)) = Dim(Ten_{G_{3}}^{G} Ñ_{3}) − Dim(M_{3}) = ∑_{i=2}^{p−1} \binom{p − 1}{i}p^{3}i.

Acknowledgments

The author would like to thank in a special way Jon Carlson for sharing his deep knowledge of the topic. In particular, she is grateful to him for the many enlightening discussions and suggestions that helped to generalize several results.

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