THE CLASSIFICATION OF TORSION ENDO-TRIVIAL MODULES

JON F. CARLSON^{∗} AND JACQUES THÉVENAZ

1. Introduction

This paper settles a problem raised at the end of the seventies by J.L. Alperin [Al1], E.C. Dade [Da] and J.F. Carlson [Ca1], namely the classification of torsion endo-trivial modules for a finite p-group over a field of characteristic p. Our results also imply, at least when p is odd, the complete classification of torsion endo-permutation modules.

We refer to [CaTh] and [BoTh] for an overview of the problem and its importance in the representation theory of finite groups. Let us only mention that the classification of endo-trivial modules is the crucial step for understanding the more general class of endo-permutation modules, and that endo-permutation modules play an important role in module theory, in particular as source modules, in block theory where they appear in the description of source algebras, and in both derived equivalences and stable equivalence of block algebras, for which many new developments have appeared recently.

Let G be a finite p-group and k be a field of characteristic p. Recall that a (finitely generated) kG-module M is called endo-trivial if End_{k}(M) ≅ k ⊕ F as kG-modules, where F is a free module. Typical examples of endo-trivial modules are the Heller translates Ω^{n}(k) of the trivial module. Any endo-trivial kG-module M is a direct sum M = M_{0} ⊕ L, where M_{0} is an indecomposable endo-trivial kG-module and L is free. Conversely, by adding a free module to an endo-trivial module, we always obtain an endo-trivial module. This defines an equivalence relation among endo-trivial modules and each equivalence class contains exactly one indecomposable module up to isomorphism. The set T (G) of all equivalence classes of endo-trivial kG-modules is a group with multiplication induced by tensor product, called simply the group of endo-trivial kG-modules. Since scalar extension of the coefficient field induces an injective map between the groups of endo-trivial modules, we can replace k by its algebraic closure. So we assume that k is algebraically closed. We refer to [CaTh] for more details about T (G).

Dade [Da] proved that if A is a non-cyclic abelian p-group then T (A) ≅ ℤ, generated by the class of Ω^{1}(k). For any p-group G, Puig [Pu] proved that the abelian group T (G) is finitely generated (but we don’t use this here since it is actually a consequence of our main results). The torsion-free rank of T (G) has been determined recently by Alperin [Al2] and the remaining problem lies in the structure of the torsion subgroup T_{t}(G).

Date: March 3, 2003.

* Partly supported by a grant from NSF.

Let us first recall some important known cases (see [CaTh]). If G = 1 or G = C_{2}, then T (G) = 0. If G = C_{p^{n}} is cyclic of order p^{n}, with n ≥ 1 if p is odd and n ≥ 2 if p = 2, then T (C_{p^{n}}) ≅ ℤ/2ℤ (generated by the class of Ω^{1}(k)). If G = Q_{2^{n}} is a quaternion group of order 2^{n} ≥ 8, then T (Q_{2^{n}}) = T_{t}(Q_{2^{n}}) ≅ ℤ/4ℤ ⊕ ℤ/2ℤ. If G = SD_{2^{n}} is a semi-dihedral group of order 2^{n} ≥ 16, then T (SD_{2^{n}}) ≅ ℤ⊕ℤ/2ℤ and so T_{t}(SD_{2^{n}}) ≅ ℤ/2ℤ. Our first main result asserts that these are the only cases where non-trivial torsion occurs.

Theorem 1.1. Suppose that G is a finite p-group which is not cyclic, quaternion, or semi-dihedral. Then T_{t}(G) = {0}.

As explained in [CaTh], the computation of the torsion subgroup T_{t}(G) is tightly connected to the problem of detecting non-zero elements of T (G) on restriction to a suitable class of subgroups. A detection theorem was proved in [CaTh] and it was conjectured that the detecting family should actually only consist of elementary abelian subgroups of rank at most 2 and, in addition when p = 2, cyclic groups of order 4 and quaternion subgroups Q_{8} of order 8. This conjecture is correct and the largest part of the present paper is concerned with the proof of this conjecture.

It is in fact only for the cases of cyclic, quaternion, and semi-dihedral groups that one needs to include cyclic groups C_{p} or C_{4} and quaternion subgroups Q_{8} in the detecting family. For all the other cases, we are going to prove the following.

Theorem 1.2. Suppose that G is a finite p-group which is not cyclic, quaternion, or semi-dihedral. Then the restriction homomorphism

∏_{E} Res_{E}^{G} : T (G) ⟶ ∏_{E} T (E) ≅ ∏_{E} ℤ

is injective, where E runs through the set of all elementary abelian subgroups of rank 2.

In order to explain the right hand side isomorphism, recall that T (E) ≅ ℤ by Dade’s theorem [Da]. Notice that Theorem 1.1 follows immediately from Theorem 1.2.

In the case of the theorem T (G) is free abelian and the method of Alperin [Al2] describes its rank by restricting drastically the list of elementary abelian subgroups which are actually needed on the right hand side (see also [BoTh] for another approach). However, for a complete classification of all endo-trivial modules, there is still an open problem. Alperin’s method shows that T (G) is a full lattice in a free abelian group A by showing that some explicit subgroup S(G) of the same rank satisfies S(G) ⊆ T (G) ⊆ A. But there is still the problem of describing explicitly the finite group T (G)/S(G) ⊆ A/S(G). However, this additional problem only occurs if G contains maximal elementary subgroups of rank 2 (see [Al2] or [BoTh] for details). In all other cases the rank of T (G) is one and we have the following result.

Corollary 1.3. Suppose that G is a finite p-group for which every maximal elementary abelian subgroup has rank at least 3. Then T (G) ≅ ℤ, generated by the class of the module Ω^{1}(k).

For the proof of Theorem 1.2, we first use the results of [CaTh] which provide a reduction to the case of extraspecial and almost extraspecial p-groups. These are the difficult cases for which we need to prove that the groups can be eliminated from the detecting family. When p is odd, this was already done in [CaTh] for extraspecial p-groups of exponent p^{2} and almost extraspecial p-groups. So we are left with the remaining cases and we have to prove the following theorem, which is in fact the main result we prove in the present paper. Theorem 1.4. Suppose the following:

(a) If p = 2, G is an extraspecial or almost extraspecial 2-group and G is not isomorphic to Q_{8}.

(b) If p is odd, G is an extraspecial p-group of exponent p. Then the restriction homomorphism

∏_{H} Res_{H}^{G} : T (G) ⟶ ∏_{H} T (H)

is injective, where H runs through the set of all maximal subgroups of G.

As mentioned earlier, the classification of endo-trivial modules has immediate consequences for the more general class of endo-permutation modules. The second goal of the present paper is to describe the consequences of the main results for the classification of torsion endo-permutation modules. We prove a detection theorem for the Dade group of all endo-permutation modules and also a detection theorem for the torsion subgroup of the Dade group. For odd p, this yields a complete description of this torsion subgroup, using the results of [BoTh]. Theorem 1.5. If p is odd and G is a finite p-group, the torsion subgroup of the Dade group of all endo-permutation kG-modules is isomorphic to (ℤ/2ℤ)^{s}, where s is the number of conjugacy classes of non-trivial cyclic subgroups of G.

One set of s generators is described in [BoTh]. Since an element of order 2 corresponds to a self-dual module, we obtain in particular the following corollary. Corollary 1.6. If p is odd and G is a finite p-group, then an indecomposable endo-permutation kG-module M with vertex G is self-dual if and only if the class of M in the Dade group is a torsion element of this group.

This is an interesting result in view of the fact that many invariants lying in the Dade group (e.g. sources of simple modules) are either known or expected to lie in the torsion subgroup, while it is not at all clear why the modules should be self-dual.

When p = 2, the situation is more complicated but we obtain that any torsion element of the Dade group has order 2 or 4. Moreover, the detection result is efficient in some cases but examples also show that it is not always sufficient to determine completely this torsion subgroup.

Theorems 1.4 is the result whose proof requires most of the work. The result has to be treated seperately when p = 2 or when p is odd. However, the strategy is similar and many of the same methods are of use for the proof in both cases. After a preliminary Section 2 and two sections about the cohomology of extraspecial groups, the proof of Theorem 1.4 occupies Sections 5–11. We use a large amount of group cohomology, including some very recent results, as well as the theory support varieties of modules. The crucial role of Serre’s theorem on products of Bocksteins appears once again and we actually need a bound for the number of terms in this product that was recently obtained by Yalçin [Ya] for (almost) extraspecial groups. Also, the module-theoretic counterpart of Serre’s theorem described in [Ca2] plays a crucial role. All these results allow us to find an upper bound for the dimension of an indecomposable endo-trivial module which is trivial on restriction to proper subgroups. For the purposes of the present paper, we shall call such a module a critical module. The main goal is to prove that there are no non-trivial critical modules for extraspecial and almost extraspecial 2-groups, except Q_{8}, and also none for extraspecial p-groups of exponent p (with p odd).

The existence of a bound for the dimension of a critical module had been known for more than 20 years and was already used by Puig [Pu] in his proof of the finite generation of T (G). The new aspect is that we are now able to control this bound for (almost) extraspecial groups. One of the differences between the case where p = 2 and the case where p is odd lies in the fact that the cohomology of extraspecial 2-groups is entirely known, so that a reasonable bound can be computed, while for odd p some more estimates are necessary. Another difference is due to the fact that we have three families of groups to consider when p = 2, but only one when p is odd, because the other two were already dealt with in [CaTh].

The other main idea in the proof of Theorem 1.4 is the following. Under the assumption that there exists a non-trivial critical module M, we can construct many others using the action of Out(G) (which is an orthogonal or symplectic group since G is (almost) extraspecial), and then construct a very large critical module by taking tensor products. The dimension of this large module exceeds the upper bound mentioned above and we have a contradiction. It is this part in which the theory of varieties associated to modules plays an essential role. We use it to analyze a suitable quotient module \overline{M} which turns out to be periodic as a module over the elementary abelian group \overline{G} = G/Φ(G).

Once Theorem 1.4 is proved, the proof of Theorem 1.2 requires much less machinery and appears in Section 12. It is very easy if p is odd and, if p = 2, it is essentially an inductive argument using a group-theoretical lemma. Theorem 1.1 also follows easily.

The paper ends with two sections about the Dade group of all endo-permutation modules, where we prove the results mentioned above.

We wish to thank numerous people who have shared ideas and opinions in the course of the writing this paper. Special thanks are due to Cédric Bonnafé, Roger Carter, Ian Leary, Gunter Malle, and Jan Saxl.

2. Preliminaries

Recall that G denotes a finite p-group, and k an algebraically closed field of characteristic p. In this section we write down some of the facts about modules and support varieties that we will need in later developments. All kG-modules are assumed to be finitely generated.

Recall that every projective kG-module is free, because G is a p-group, and that injective and projective modules coincide. Moreover, an indecomposable kG-module M is free if and only if t_{1}^{G} ·M ≠ 0, where t_{1}^{G} = ∑_{g∈G} g (a generator of the socle of kG). More generally, if M is a kG-module and if m_{1}, …, m_{r} ∈ M are such that t_{1}^{G}m_{1}, …, t_{1}^{G}m_{r} are linearly independant, then m_{1}, …, m_{r} generate a free submodule F of M of rank r. Moreover F is a direct summand of M because F is also injective.

Suppose that M is a kG-module. If P ⟶^{θ} M is a projective cover of M then we let Ω(M) denote the kernel of θ. We can iterate the process and define inductively Ω^{n}(M) = Ω(Ω^{n−1}(M)), for n > 1. Suppose that M ⟶^{μ} Q is an injective hull of M. Recall that Q is a projective as well as injective module. Then we let Ω^{−1}(M) be the cokernel of μ. Again we have inductively that Ω^{−n}(M) = Ω^{−1}(Ω^{−n+1}(M)) for n > 1. The modules Ω^{n}(M) are well defined up to isomorphism and they have no non-zero projective submodules. In general we write M = Ω^{0}(M) ⊕ P where P is projective and Ω^{0}(M) has no projective summands.

The basic calculus of the syzygy modules Ω^{n}(M) is expressed in the following.

Lemma 2.1. Suppose that M and N are kG-modules. Then Ω^{m}(M) ⊗ Ω^{n}(N) ≅ Ω^{m+n}(M ⊗ N) ⊕ (free).

Here M ⊗ N is meant to be the tensor product M ⊗_{k} N over k, with the action of the group G defined diagonally, g(m ⊗ n) = gm ⊗ gn. The proof of the lemma is a consequence of the facts that M ⊗_{k} − and − ⊗_{k} N preserve exact sequences and that M ⊗ N is projective whenever either M or N is a projective module.

The cohomology ring H^{*}(G, k) is a finitely generated k-algebra and for any kGmodules M and N, Ext_{kG}^{∗}(M, N) is a finitely generated module over H^{*}(G, k) ≅ Ext_{kG}^{∗}(k, k). We let V_{G}(k) denote the maximal ideal spectrum of H^{*}(G, k). For any kG-module M, let J(M) be the annihilator in H^{*}(G, k) of the cohomology ring Ext_{kG}^{∗}(M, M). Let V_{G}(M) = V_{G}(J(M)) be the closed subset of V_{G}(k) consisting of all maximal ideals that contain J(M). So V_{G}(M) is a homogeneous affine variety. We need some of the properties of support varieties in essential ways in the course of our proofs. See the general references [Be, Ev] for more explanation and details.

Theorem 2.2. Let L, M and N be kG-modules.

(1) V_{G}(M) = {0} if and only if M is projective.

(2) If 0 → L → M → N → 0 is exact then the variety of any one of L, M or N is contained in the union of the varieties of the other two. Moreover, if V_{G}(L) ∩ V_{G}(N) = {0}, then the sequence splits.

(3) V_{G}(M ⊗ N) = V_{G}(M) ∩ V_{G}(N).

(4) V_{G}(Ω^{n}(M)) = V_{G}(M) = V_{G}(M^{∗}) where M^{∗} = Hom_{k}(M, k) is the k-dual of M.

(5) If V_{G}(M) = V_{1} ∪ V_{2} where V_{1} and V_{2} are non-zero closed subsets of V_{G}(k) and V_{1} ∩V_{2} = {0}, then M ≅ M_{1} ⊕M_{2} where V_{G}(M_{1}) = V_{1} and V_{G}(M_{2}) = V_{2}.

(6) A nonprojective module M is periodic (i.e. for some n > 0, Ω^{n}(M) ≅ Ω^{0}(M)) if and only if its variety V_{G}(M) is a union of lines through the origin in V_{G}(k).

(7) Let ζ ∈ Ext_{kG}^{n}(k, k) = H^{n}(G, k) be represented by the (unique) cocycle ζ : Ω^{n}(k) ⟶ k and let L = Ker(ζ), so that there is an exact sequence

0 ⟶ L ⟶ Ω^{n}(k) ⟶^{ζ} k ⟶ 0 .

Then V_{G}(L) = V_{G}(ζ), the variety of the ideal generated by ζ, consisting of all maximal ideals containing ζ.

We are particularly interested in the case in which the group G is an elementary abelian group. First assume that p = 2 and G = ⟨x_{1}, …, x_{n}⟩ ≅ (C_{2})^{n}. Then H^{*}(G, k) ≅ k[ζ_{1}, …, ζ_{n}] is a polynomial ring in n variables. Here the elements ζ_{1}, …, ζ_{n} are in degree 1 and by proper choice of generators we can assume that res_{G,⟨x_{i}⟩}(ζ_{j}) = δ_{ij} ·γ_{i} where γ_{i} ∈ H^{1}(⟨x_{i}⟩, k) is a generator for the cohomology ring of ⟨x_{i}⟩. Indeed if we assume that the generators are chosen correctly, then for any α = (α_{1}, …, α_{n}) ∈ k^{n}, u_{α} = 1 + ∑_{i=1}^{n} α_{i}(x_{i} − 1) ∈ kG, U = ⟨u_{α}⟩, we have that

res_{G,U}(f(ζ_{1}, …, ζ_{n})) = f(α_{1}, …, α_{n})γ_{α}^{t}

where f is a homogeneous polynomial of degree t and γ_{α} ∈ H^{1}(U, k) is a generator of the cohomology ring of U.

Now suppose that p is an odd prime and let G = ⟨x_{1}, …, x_{n}⟩ ≅ (C_{p})^{n}. Then

H^{*}(G, k) ≅ k[ζ_{1}, …, ζ_{n}] ⊗ Λ(η_{1}, …, η_{n}) ,

where Λ is an exterior algebra generated by the elements η_{1}, …, η_{n} in degree 1 and the polynomial generators ζ_{1}, …, ζ_{n} are in degree 2. We can assume that each ζ_{j} it the Bockstein of the element η_{j} and that the elements can be chosen so that res_{G,⟨x_{i}⟩}(ζ_{j}) = δ_{ij} ·γ_{i} where γ_{i} ∈ H^{2}(⟨x_{i}⟩, k) is a generator for the cohomology ring of ⟨x_{i}⟩. Similarly, assuming that the generators are chosen correctly, for any α = (α_{1}, …, α_{n}) ∈ k^{n}, u_{α} = 1 + ∑_{i=1}^{n} α_{i}(x_{i} − 1) ∈ kG, U = ⟨u_{α}⟩, we have that

res_{G,U}(f(ζ_{1}, …, ζ_{n})) = f(α_{1}^{p}, …, α_{n}^{p})γ_{α}^{t}

where f is a homogeneous polynomial of degree t and γ_{α} ∈ H^{1}(U, k) is a generator of the cohomology ring of U.

Associated to a kG-module M we can define a rank variety

V_{G}^{r}(M) = {α ∈ k^{n} | M↓_{⟨u_{α}⟩} is not a free ⟨u_{α}⟩−module } ∪ {0}

where u_{α} is given as above and where M↓_{⟨u_{α}⟩} denotes the restriction of M to the subalgebra k⟨u_{α}⟩ of kG. Then we have the following result for any p. Theorem 2.3. Let M be any kG-module. If p = 2 then, V_{G}^{r}(M) = V_{G}(M) as subsets of k^{n}. If p > 2 then the map V_{G}(M) ⟶ V_{G}^{r}(M) given by α ↦ α^{p} = (α_{1}^{p}, …, α_{n}^{p}) is an inseparable isogeny (both injective and surjective). In particular, for α ≠ 0, α^{p} ∈ V_{G}(M) (α ∈ V_{G}(M) if p = 2) if and only if M↓_{⟨u_{α}⟩} is not a free k⟨u_{α}⟩-module.

We should emphasize that if v is a unit in kG such that v ≡ u_{α} mod(Rad(kG)^{2}) then M↓_{⟨v⟩} is a free k⟨v⟩-module if and only if α^{p} ∉ V_{G}(M) (α ∉ V_{G}(M) if p = 2). So for example the element x_{1}x_{2}x_{3} fails to act freely on M if and only if (1, 1, 1, 0, …, 0) ∈ V_{G}(M).

3. Extraspecial groups in characteristic 2

In this section and the next, we are interested in the structure and cohomology of extraspecial and almost extraspecial p-groups. These are precisely the p-groups G with the property that G has a unique normal subgroup Z of order p such that G/Z is elementary abelian. Note that the dihedral group D_{8} of order 8 and, more generally, the Sylow p-subgroup of GL(3, p) are extraspecial p-groups. The quaternion group Q_{8} of order 8 and the cyclic group C_{p^{2}} of order p^{2} also have the required property. Indeed, for p = 2 any extraspecial or almost extraspecial group is constructed from copies of D_{8}, Q_{8} and C_{4} by taking central products. In this section we concentrate on the case p = 2 and look more deeply into the structure of the extraspecial and almost extraspecial group and their cohomology.

Suppose that G_{1} and G_{2} are 2-groups with the property that each has a unique normal subgroup of order 2. Let ⟨z_{i}⟩ ∈ G_{i} be the subgroups. Then the central product G_{1} ∗ G_{2} is defined by

G_{1} ∗ G_{2} = (G_{1} × G_{2})/⟨(z_{1}, z_{2})⟩.

It is not difficult to check that D_{8} ∗ D_{8} ≅ Q_{8} ∗ Q_{8} and that D_{8} ∗ C_{4} ≅ Q_{8} ∗ C_{4}. Moreover, C_{4} ∗C_{4} has a central elementary abelian subgroup of order 4 and hence is not of interest to us (it is neither extraspecial nor almost extraspecial). We are left with three types. They are:

\underline{Type 1.} G = D_{8} ∗ D_{8} ∗ ⋯ ∗ D_{8} of order 2^{2n+1} where n is the number of factors in the central product.

\underline{Type 2.} G = D_{8} ∗ ⋯ ∗ D_{8} ∗ Q_{8} of order 2^{2n+1} where n is the number of factors in the central product.

\underline{Type 3.} G = D_{8} ∗ ⋯ ∗ D_{8} ∗ C_{4} of order 2^{2n+2} where n is the number of factors isomorphic to D_{8}.

The groups of type 1 and 2 are the extraspecial groups (see [Go1]) while the groups of type 3 are what we call the almost extraspecial groups.

The groups are also characterized by an associated quadratic form in the following way. Each group is a central extension

0 ⟶ Z ⟶ G ⟶^{μ} E ⟶ 0

where Z = ⟨z⟩ is the unique central normal sugroup of order 2 and E ≅ 𝔽_{2}^{m} is elementary abelian. Recall that a quadratic form on E (as a vector space over 𝔽_{2}) is a map q : E ⟶ 𝔽_{2} with the property that

q(x + y) = q(x) + q(y) + b(x, y)

where b : E × E ⟶ 𝔽_{2} is a symmetric bilinear form. Here the quadratic form q expresses the class of the extension as given in the above sequence. That is, if x̃, ỹ are elements of G and if μ(x̃) = x and μ(ỹ) = y, then

x̃^{2} = z^{q(x)} and [x̃, ỹ] = z^{b(x,y)}.

Notice here that we are writing the operation in G as multiplication. Given the structure of the groups, it is not difficult to write down the associated quadratic forms. With respect to a choice of basis, E can be identified with 𝔽_{2}^{m} and in the sequel we make this identification. Thus we write x = (x_{1}, …, x_{m}) for the elements of E.

Lemma 3.1. Let G be an extraspecial or almost extraspecial group of order 2^{m+1}. Then the quadratic form q associated to G is given on x = (x_{1}, …, x_{m}) ∈ 𝔽_{2}^{m} = E as follows.

For type 1, q(x) = x_{1}x_{2} + ⋯ + x_{2n−1}x_{2n} (m = 2n).

For type 2, q(x) = x_{1}x_{2} +⋯+x_{2n−3}x_{2n−2} +x_{2n−1}^{2} +x_{2n−1}x_{2n} +x_{2n}^{2} (m = 2n). For type 3, q(x) = x_{1}x_{2} + ⋯ + x_{2n−1}x_{2n} + x_{2n+1}^{2} (m = 2n + 1).

Now on the k-vector space V = k^{m} of dimension m, let q, b denote the same forms but with the field of coefficients expanded from 𝔽_{2} to k. Let F : k → k be the Frobenius homomorphism, F (a) = a^{2}. If ν = (x_{1}, …, x_{m}) ∈ V, let F act on ν by F (ν) = (x_{1}^{2}, x_{2}^{2}, …, x_{m}^{2}). Recall that a subspace W ⊆ V is isotropic if q(w) = 0 for all w ∈ W. The following is not difficult:

Lemma 3.2. Let h be the codimension in V of a maximal isotropic subspace of V. The values of h for the quadratic forms associated to the above groups are:

h = n for G of type 1 (m = 2n),

h = n + 1 for G of type 2 (m = 2n) or type 3 (m = 2n + 1). Moreover 2^{h} is the index in G of a maximal elementary abelian subgroup.

We are now prepared to state the theorem of Quillen on the cohomology. See [BeCa] for one treatment.

Theorem 3.3. [Qu] Let G be an extraspecial or almost extraspecial group of order 2^{m+1}. If we write ν = (x_{1}, …, x_{m}), then we have

H^{*}(G, k) = k[x_{1}, …, x_{m}]/(q(ν), b(ν, F (ν)), …, b(ν, F^{h−1}(ν))) ⊗ k[δ]

where δ is an element of degree 2^{h} that restricts to a non-zero element of Z. Moreover the elements q(ν), b(ν, F (ν)), …, b(ν, F^{h−1}(ν)) form a regular sequence in k[x_{1}, …, x_{m}] and H^{*}(G, k) is a Cohen-Macaulay ring.

The following will be vital for the proof of our main results.

Theorem 3.4. Let G be an extraspecial or almost extraspecial 2-group. Define t = t_{G} to be the natural number given as follows. If G is of type 1 of order 2^{2n+1}, let

t_{G} = \begin{cases}2^{n−1} + 1 & for n ≤ 4 , \\ 2^{n−1} + 2^{n−4} & for n ≥ 4 . \end{cases}

If G is of type 2 of order 2^{2n+1} or of type 3 of order 2^{2n+2}, then let

t_{G} = \begin{cases}3 & for n = 1 , \\ 2^{n} + 2^{n−2} & for n ≥ 2 . \end{cases}

Then there exists non-zero elements ζ_{1}, …, ζ_{t} ∈ H^{1}(G, 𝔽_{2}) such that ζ_{1} ⋯ ζ_{t} = 0. Moreover, in the isomorphism H^{1}(G, 𝔽_{2}) ≅ Hom(G, 𝔽_{2}), each ζ_{i} corresponds to a homomorphism whose kernel is a maximal subgroup of G and is the centralizer of a noncentral involution in G.

Proof. The proof is contained in the paper [Ya]. For the groups of type 1, t_{G} is actually equal to the cohomological length, that is, the least number of non-zero elements in H^{1}(G, 𝔽_{2}) such that the product of those elements is zero (see [Ya], Theorem 1.3).

So suppose that G has type 2 or 3. Then t_{G} in our theorem is equal to the cardinality s(G) of a representing set in G (see [Ya], Propositions 6.2 and 6.3). A representing set for G is a collection of elements of G that contains at least one noncentral element from each elementary abelian subgroup of G. But now Proposition 1.1 of [Ya] shows that the cohomological length is at most s(G).

The point of the last statement is that the centralizer of any maximal elementary abelian subgroup of G is contained in the centralizers of some elements in a representing set. Because the cohomology ring H^{∗}(G, 𝔽_{2}) is Cohen-Macaulay (see Theorem 3.3), any element whose restriction to the centralizer of every maximal elementary abelian subgroup of G vanishes, is the zero element (see Theorem 3.4 in [Ya]). Hence if we choose the elements ζ_{i} to correspond to the centralizers of the elements in a representing set as in the last statement, then their product is zero as desired. □

The next theorem will be very important to the proof of the general case. It is part of the effort to get an explicit upper bound on the dimensions of critical modules. Theorem 3.5. Let G be an extraspecial or almost extraspecial group of order 2^{m+3} and let H be the centralizer of a noncentral involution in G. Then H ≅ C_{2} × U where U is an extraspecial or almost extraspecial group of order 2^{m+1} of the same type as G. Assume that m ≥ 2 and, if m = 2, that U ≅ D̸_{8}. Then for 2 ≤ r ≤ t_{G} we have that

Dim H^{r}(H, k) ≤ \binom{m + r}{r} − \binom{m + r − 2}{r − 2}.

Proof. The structure of the centralizer H can be verified directly from what we know of G. For one thing it can be checked that all noncentral involutions in G are conjugate by an element in the automorphism group of G and hence their centralizers are all isomorphic.

Throughout the proof we use the notation in Theorem 3.3, for the cohomology of the group U, so that H^{∗}(U, k) is generated by x_{1}, …, x_{m} and δ, with deg(δ) = 2^{h} (where h is the value associated to the group U as in Lemma 3.2). We know that

H^{∗}(H, k) ≅ H^{∗}(U, k) ⊗ H^{∗}(C_{2}, k)

and moreover we know that H^{∗}(C_{2}, k) ≅ k[y] is a polynomial ring in one variable y in degree 1. We want to focus on the polynomial ring S generated by x_{1}, …, x_{m}, y. We have a homomorphism from S to H^{∗}(H, k) whose kernel contains the elements q(ν) and β(ν, F (ν)) where ν = (x_{1}, …, x_{m}). Let Q denote the image of S in H^{∗}(H, k). For this argument, let S^{#} = S/(q(ν)) and let S^{##} = S/(q(ν), β(ν, F (ν))). If R denotes any of these graded rings, we let R_{r} denote the homogeneous part of R in degree exactly r. Note that R_{r} = 0 if r < 0.

First notice that Dim S_{r} = \binom{m+r}{m} = \binom{m+r}{r}. Because q(ν) and β(ν, F (ν)) are two terms of a regular sequence of elements in S we must have that

Dim S_{r}^{#} = Dim S_{r} − Dim S_{r−2}

and

Dim S_{r}^{##} = Dim S_{r}^{#} − Dim S_{r−3}^{#}

for all r ≥ 2. Moreover Dim S_{r} ≥ Dim S_{r}^{##} ≥ Dim Q_{r} for all values of r.

By Theorem 3.4, t_{G} ≤ 2t_{U} (with equality in most cases) and moreover, by Lemma 3.2, we see that t_{U} < 2^{h} in all cases. The choice that r ≤ t_{G} now means that

r ≤ t_{G} ≤ 2t_{U} < 2 · 2^{h} = 2 · deg(δ)

and this implies that we must have either Dim H^{r}(H, k) = Dim Q_{r} , if r < deg(δ), or Dim H^{r}(H, k) = Dim Q_{r} + Dim(δ · Q_{r−deg(δ)}), if deg(δ) ≤ r < 2deg(δ). Notice also that deg(δ) = 2^{h} ≥ 4 in all cases because we assumed that m ≥ 2 and U ≅ D̸_{8} (if U ≅ D_{8}, then h = 1 and deg(δ) = 2). Hence we have that

Dim H^{r}(H, k) ≤ Dim Q_{r} + Dim Q_{r−deg(ζ)}

≤ Dim S_{r}^{#} − Dim S_{r−3}^{#} + Dim S_{r−deg(ζ)}^{#} − Dim S_{r−deg(ζ)−3}^{#}

≤ Dim S_{r}^{#} − Dim S_{r−3}^{#} + Dim S_{r−deg(ζ)}^{#}

≤ Dim S_{r}^{#} = \binom{m + r}{r} − \binom{m + r − 2}{r − 2} .

The last inequality follows from the facts that r−deg(δ) ≤ r−3 and that Dim S_{s}^{#} is an increasing function of s. □

Corollary 3.6. Suppose that G and H are as in the theorem. If 2 ≤ r ≤ t_{G}, then

∑_{i=0}^{r} Dim Ω^{i}(k_{H})↑_{H}^{G} ≤ \binom{m + r − 1}{m}|G| + 2.

Proof. For any i we have an exact sequence

0 ⟶ Ω^{i+1}(k_{H}) ⟶ P_{i} ⟶ Ω^{i}(k_{H}) ⟶ 0

where P_{i} is the degree i term in a minimal kH-projective resolution of the trivial kH-module k_{H}. Recall that Dim P_{i} = Dim H^{i}(H, k) · |H|. Then by the Theorem, for r = 2s + 1, we have that

∑_{i=0}^{r} Dim Ω^{i}(k_{H}) = ∑_{j=0}^{s} ( Dim Ω^{2j+1}(k_{H}) + Dim Ω^{2j}(k_{H})) = ∑_{j=0}^{s} Dim P_{2j}

≤ Dim P_{0} + ∑_{j=1}^{s} [\binom{m + 2j}{2j} − \binom{m + 2j − 2}{2j − 2}]|H|

= |H| + [\binom{m + 2s}{2s} − \binom{m}{0}]|H|

= \binom{m + r − 1}{r − 1}|H| = \binom{m + r − 1}{m}|H| .

On the other hand if r = 2s is even, then we use the fact that Dim P_{1} = \binom{m+1}{1}|G| and we obtain similarly

∑_{i=0}^{r} Dim Ω^{i}(k_{H}) = Dim k + Dim P_{1} + ∑_{j=2}^{s} Dim P_{2j−1}

≤ 1 + \binom{m + 2s − 1}{2s − 1}|H| = 1 + \binom{m + r − 1}{m}|H|.

In both cases, inducing from H to G, the dimension of Ω^{i}(k_{H})↑_{H}^{G} is doubled and the result follows. □

4. Extraspecial groups in odd characteristic

Our aim in this section is to get results similar to those of the last section for extraspecial p-groups in the case that the prime p is not 2. As in the characteristic 2 case, for any positive integer n there are two isomorphism types of extraspecial groups of order p^{2n+1} and one isomorphism type of almost extraspecial group of order p^{2n+2}. For each n, one of the two nonisomorphic groups of order p^{2n+1} has exponent p^{2} and the other one has exponent p. In the earlier paper [CaTh] we showed that Theorem 1.4 holds for extraspecial groups of exponent p^{2} and almost extraspecial groups (i.e. for these groups there are no non-trivial critical modules). As a consequence, the only groups of interest to us are the extraspecial groups of order p^{2n+1} and exponent p.

Up to isomorphism, there is exactly one extraspecial group G_{1} of order p^{3} and exponent p. It is generated by elements x, y and z, which satisfy the relations that z is in the center of G_{1}, z^{p} = x^{p} = y^{p} = 1 and [x, y] = z. It is isomorphic to the Sylow p-subgroup of the general linear group GL(3, p). For n > 1, the extraspecial group of order p^{2n+1} is a central product

G_{n} = G_{1} ∗ G_{1} ∗ … ∗ G_{1}

of n copies of G_{1} as in the last section. That is, G_{n} is the quotient group obtained by taking the direct product of n copies of G_{1} and then identifying the centers (see [Go1]). The center of G_{n} is a cyclic subgroup Z = ⟨z⟩ of order p and G_{n}/Z is an elementary abelian p-group of order p^{2n}.

We need an analogue to Theorem 3.4 for our case. Theorem 4.1. For G = G_{1}, let t_{G} = 2(p + 1), while for G = G_{n}, n > 1, let t_{G} = (p^{2} + p − 1)p^{n−2}. Then there exists non-zero elements η_{1}, …, η_{t} ∈ H^{1}(G, 𝔽_{p}) such that β(η_{1}) ⋯ β(η_{t}) = 0 where t = t_{G}. Moreover, in the isomorphism H^{1}(G, 𝔽_{p}) ≅ Hom(G, 𝔽_{p}), each η_{i} corresponds to a homomorphism whose kernel is a maximal subgroup of G and is the centralizer of a noncentral element of order p in G.

Proof. The proof of the theorem is contained in the paper by Yalçin as Theorem 1.2 of [Ya]. In this case the dimension of H^{1}(G, 𝔽_{p},) is the same as that of Hom(G, 𝔽_{p}) which is 2n. □

As in the last section we are going to need estimates on the dimensions of the cohomology groups H^{r}(G_{n}, k) where k is a field of characteristic p. We begin with the case of the extraspecial group G = G_{1} of order p^{3}. Ian Leary [Le1] has given a complete description of the cohomology ring H^{*}(G, k) except that he did not fully compute the Poincaré series, which is something that we need. The calculation is, of course, implicit in his work, and he did calculate it in the special case that p = 3 [Le2]. Note that our results agree with his in that situation. Theorem 4.2. The Poincaré series for the cohomology ring of the group G = G_{1} is given by the rational function

∑_{n=0}^{∞} Dim H^{n}(G, k) t^{n} = \frac{1 + t + 2t^{2} + 2t^{3} + t^{4} + t^{5} + … + t^{2p−1}}{(1 − t)(1 − t^{2p})} .

Proof. We will not repeat the long list of relations given by Leary (Theorem 6 of [Le1]). However we will use exactly the notation of that paper and the interested reader can follow the computation. The strategy is first to ignore the contribution of the regular element z in degree 2p. This element is a nondivisor of zero as it restricts non-trivially to the center of G. We also know that it is regular from the given relation and from the fact that it is represented on the E_{2} of the spectral sequence, by an element in E_{2}^{0,2p} which survives to the E_{∞} page of the spectral sequence. Consequently, the Poincaré series f(t) of H^{*}(G, k) is obtained by multiplying 1/(1−t^{2p}) times the Poincaré series of the subalgebra A generated by all of the given generators other than z.

Next we consider the subalgebra A as a module over the subring R generated by x and x′. Note that x and x′ are in degree 2 and satisfy the relation x^{p}x′−xx′^{p} = 0 in degree 2p + 2. So the Poincaré series for R is f_{1} = (1 − t^{2p+2})/(1 − t^{2})^{2}. This is also the series for the R-submodule M_{1} generated by the element 1 in degree 0. The first thing that needs to be established from the relations is that the Rgenerators are the elements of the sequence

S = [1, y, y′, Y, Y′, X, X′, yY′, XY′, XX′, d_{4}, c_{4}, d_{5}, …, c_{p−1}, d_{p}]

of length 2p + 3. Let M_{i} be the R-submodule generated by the first i elements of the sequence, and let f_{i} be the Poincaré series for M_{i}/M_{i−1}. Then the desired Poincaré series for A is f_{1} + f_{2} + … + f_{2p+3}. Note that f_{1} has been calculated.

• For f_{2}, we note that xy′ = x′y and x^{p}y′ = x′^{p}y. So x′(x^{p−1} − x′^{p−1})y = 0. Therefore f_{2} = t(1 − t^{2p})/(1 − t^{2})^{2}.

• Since xy′ = x′y ∈ M_{2}, we have that f_{3} = t/(1 − t^{2}).

• Similarly to the calculation for f_{2}, we have that f_{4} = t^{2}(1 − t^{2p})/(1 − t^{2})^{2} and f_{6} = t^{3}(1 − t^{2p})/(1 − t^{2})^{2}.

• For f_{5}, note that x^{2}Y′ = xx′Y ∈ M_{4} and xx′Y′ ∈ M_{4}. Therefore f_{5} = t^{3} + t^{2}/(1 − t^{2}).

• The calculation for f_{7} is similar to that of f_{3} and we get that f_{7} = t^{3}/(1 − t^{2}).

• For i := 8, …, 2p + 3, it should be checked that xS_{i}, x′S_{i} ∈ M_{i−1} where S_{i} is the i^{th} element of the sequence S. Consequently, f_{i} = t^{j_{i}}, where j_{i} is the degree of S_{i}. Note that j_{8} = 3 while j_{i} = i − 4 for i ≥ 9.

Finally it is necessary to verify that

f_{1} + f_{2} + … + f_{2p+3} = (1 + t + 2t^{2} + 2t^{3} + t^{4} + … + t^{2p−1})/(1 − t)

by routine but tedious calculation. □

We need to derive two facts from the above theorem. The first is an upper bound which is not optimal but will be sufficient for our purposes. Corollary 4.3. For G = G_{1}, we have that

Dim H^{r}(G, k) ≤ 2(r + 1) = 2\binom{r + 1}{1}.

Moreover, Dim H^{r}(G, k) = 2r if 1 ≤ r ≤ 3 and Dim H^{r}(G, k) = r + 3 if 4 ≤ r ≤ 2p − 1.

Proof. Consider the series expansion

g(t) = \frac{1 + t + 2t^{2} + 2t^{3} + t^{4} + … + t^{2p−1}}{1 − t} = ∑_{r=0}^{∞} a_{r}t^{r} .

A routine computation yields the value of the coefficients a_{0} = 1, a_{r} = 2r if 1 ≤ r ≤ 3, a_{r} = r + 3 if 4 ≤ r ≤ 2p − 1, and a_{r} = 2p + 2 if r ≥ 2p − 1. The Poincaré series for the cohomology ring of G_{1} is obtained by multiplying g(t) with \frac{1}{1−t^{2p}} = ∑_{i=0}^{∞} t^{2ip}. Therefore Dim H^{r}(G, k) = a_{r} for r ≤ 2p−1 and this proves the second statement of the lemma. Moreover, for arbitrary r, writing r = j + q(2p) with 0 ≤ j < 2p, we have that

Dim H^{r}(G, k) = a_{j} + qa_{2p} ≤ (j + 3) + q(2p + 2) ≤ 2(r + 1) .
□

Corollary 4.4. For G = G_{1} we have that Dim Ω^{2p}(k) = p^{3}(p + 1) + 1.

Proof. If P_{j} is the j-th term of a minimal projective resolution of k, we have Dim(P_{j}) = Dim H^{j}(G, k) |G| and so Dim Ω^{j+1}(k) = Dim H^{j}(G, k)|G|−Dim Ω^{j}(k). Using this relation and the dimensions given in the previous corollary, we obtain Dim Ω^{2}(k) = p^{3} + 1 and then by induction Dim Ω^{2j−1}(k) = (j + 1)p^{3} − 1 and Dim Ω^{2j}(k) = (j + 1)p^{3} + 1 for 2 ≤ j ≤ p. □

In the rest of the section, we require the following well known combinatorial identity. Lemma 4.5. For any integers c, i, j ≥ 0 we have that

∑_{a+b=c} \binom{a + i}{i}\binom{b + j}{j} = \binom{c + i + j + 1}{i + j + 1}.

Proof. Recall that if P is a polynomial ring in n variables, then the number of monomials of degree r is \binom{r+n−1}{n−1}. Now the tensor product of a polynomial ring in i + 1 variables with a polynomial ring in j + 1 variables yields a polynomial ring in i + j + 2 variables. The identity follows by counting the number of monomials of degree c. □

We also need to know the dimension of the cohomology groups of elementary abelian groups. Lemma 4.6. Let p be an odd prime and let E be an elementary abelian p-group of rank m. Then Dim H^{r}(E, k) = \binom{r+m−1}{m−1}.

Proof. Recall that H^{∗}(E, k) ≅ k[ζ_{1}, …, ζ_{m}] ⊗ Λ(η_{1}, …, η_{m}) where ζ_{1}, …, ζ_{m} are in degree 2 and η_{1}, …, η_{m} are in degree 1. A basis of H^{r}(E, k) consists of the elements ζ_{1}^{a_{1}} …, ζ_{m}^{a_{m}}η_{1}^{e_{1}}, …, η_{m}^{e_{m}} where 0 ≤ a_{i} ≤ r/2, 0 ≤ e_{i} ≤ 1 and ∑_{i=1}^{m}(2a_{i} + e_{i}) = r. This basis is in bijection with the set of monomials of degree r in k[x_{1}, …, x_{m}] by mapping the above basis element to x_{1}^{2a_{1}+e_{1}} … x_{m}^{2a_{m}+e_{m}}. Now the number of monomials of degree r is \binom{r+m−1}{m−1}. □

Our main result in this section gives estimates for the dimensions of the cohomology of the centralizers of p-elements. Theorem 4.7. Let G = G_{n} be an extraspecial of order p^{2n+1} and exponent p. Let H be the centralizer of a noncentral element of order p in G. Then H ≅ C_{p}×G_{n−1}. Moreover,

Dim H^{m}(H, k) ≤ 2\binom{m + 2n − 2}{2n − 2}.

Proof. As with the characteristic 2 case, the structure of the centralizer H can be verified directly from what we know of G. All noncentral elements of order p in G are conjugate by an element in the automorphism group of G and hence their centralizers are isomorphic.

Next we need to approximate the dimensions of the cohomology groups of the group G_{n−1} for n ≥ 1. The estimate in Corollary 4.3 will serve in the case that n = 2. Let N be a normal subgroup of G_{n−1} such that N ≅ G_{1}. We can take N to be the first factor in the central product that expresses G_{n−1}. Then G_{n−1}/N ≅ C_{p}^{2(n−2)}, an elementary abelian group of order p^{2(n−2)}. The Lyndon-Hochschild-Serre spectral sequence of the extension of G_{n−1}/N by N has E_{2} term

E_{2}^{r,s} = H^{r}(G_{n−1}/N, H^{s}(N, k)) ⇒ H^{r+s}(G_{n−1}, k).

As k-vector spaces, it is true that E_{2}^{r,s} ≅ H^{r}(G_{n−1}/N, k) ⊗ H^{s}(N, k) because N commutes with the other factors of the central product. So we have that

Dim H^{m}(G_{n−1}, k) ≤ ∑_{r+s=m} Dim(E_{2}^{r,s})

= ∑_{r+s=m} Dim H^{r}(G_{n−1}/N, k) Dim H^{s}(N, k)

≤ ∑_{r+s=m} \binom{r + 2(n − 2) − 1}{2(n − 2) − 1}2\binom{s + 1}{1} = 2\binom{m + 2n − 3}{2n − 3},

using Lemma 4.6, Corollary 4.3 and the combinatorial identity of Lemma 4.5. Now H^{m}(H, k) ≅ ⨁_{r+s=m} H^{r}(G_{n−1}, k) ⊗ H^{s}(C_{p}, k). Therefore,

Dim H^{m}(H, k) = ∑_{r+s=m} Dim H^{r}(G_{n−1}, k) · Dim H^{s}(C_{p}, k)

≤ ∑_{r+s=m} 2\binom{r + 2n − 3}{2n − 3}\binom{s}{0} = 2\binom{m + 2n − 2}{2n − 2},

using again Corollary 4.3 and Lemma 4.5. □

Corollary 4.8. Suppose that G and H are as in the theorem. If r ≥ 1, then

∑_{i=0}^{r} Dim Ω^{i}(k_{H})↑_{H}^{G} ≤ 2p^{2n+1}\binom{r + 2n − 2}{2n − 1}.

Proof. Suppose that … → P_{1} → P_{0} → k → 0 is a minimal kH-projective resolution of the trivial module k. Then we know that Dim Ω^{0}(k) + Dim Ω^{1}(k) = Dim P_{0}. For j ≥ 2, Ω^{j}(k_{H}) is a submodule of P_{j−1}. The dimension of P_{j} is precisely |H| Dim H^{j}(H, k) and the dimension of Ω^{j}(k_{H})↑_{H}^{G} is p times the dimension of Ω^{j}(k_{H}). So from the theorem we have that

∑_{i=0}^{r} Dim Ω^{i}(k_{H})↑_{H}^{G} ≤ p|H| ∑_{i=0}^{r−1} Dim H^{i}(H, k)

≤ p^{2n+1} ∑_{i=0}^{r−1} 2\binom{i + 2n − 2}{2n − 2}\binom{r − 1 − i}{0}

= 2p^{2n+1}\binom{r + 2n − 2}{2n − 1},

by the identity 4.5. □

5. New endo-trivial modules from old endo-trivial modules

Here we start the proof of Theorem 1.4. Suppose that G is an extraspecial or almost extraspecial p-group and that G ≅ Q̸_{8}. Let Z = ⟨z⟩ be the Frattini subgroup of G, of order p, with elementary abelian quotient \overline{G} = G/Z of rank m. Let x_{1}, …, x_{m} ∈ G such that \overline{G} = ⟨\overline{x}_{1}, …, \overline{x}_{m}⟩. Recall that Z is the unique normal subgroup of order p. Moreover every maximal subgroup of G contains Z and G is not elementary abelian. Some of the results in this section hold more generally if G has a Frattini subgroup Z of order p, but we leave this generalization to the reader.

Let M be an endo-trivial kG-module whose class in T (G) lies in the kernel of the restriction to proper subgroups. This means that M↓_{H}^{G} ≅ k ⊕ (free) for every maximal subgroup H of G. For the purpose of the proof of Theorem 1.4 (Sections 5–11), we make the following definition: Definition 5.1. We say that a kG-module M is critical if it is an indecomposable endo-trivial module such that M↓_{H}^{G} ≅ k ⊕ (free) for every maximal subgroup H of G.

Actually, the last condition implies that the module M is endo-trivial because its restriction to every elementary abelian subgroup is isomorphic to k ⊕ (free), hence endo-trivial (see Lemma 2.9 of [CaTh]). In fact M is a torsion endo-trivial module by a theorem of Puig [Pu], but we do not need this fact in our arguments. By factoring out all free summands of an endo-trivial module M, one can always assume that M is indecomposable and this is why we do so. We shall often omit to mention this indecomposability condition, to the effect that we shall usually only prove that a module satisfies the condition on restriction to maximal subgroups in order to deduce that it is critical. Since our aim is to prove that the kernel above is trivial, we have to show that any critical kG-module M is isomorphic to k as a kG-module. We will often assume, by contradiction, the existence of a non-trivial critical kG-module.

In this section, we prove several results concerning the structure of a critical module M and the construction of new modules with the same property. For some of the results, we only need to assume that M↓_{H}^{G} ≅ k ⊕ (free) for a single subgroup H of G.

For any critical kG-module M, and more generally for any kG-module M such that M↓_{Z}^{G} ≅ k ⊕ (free), we let M′ = {m ∈ M | (z − 1)^{p−1}m = 0} and we set

\overline{M} = M/M′ .

We let ^{−} : M ⟶ \overline{M} be the quotient map. Since (z − 1)\overline{M} = 0, the module \overline{M} can be viewed as a k\overline{G}-module. A large part of this paper is devoted to an analysis of the properties of the module \overline{M}. Lemma 5.2. Let M be a kG-module. Suppose that M↓_{Z}^{G} ≅ k ⊕ (free).

(a) The module M has two filtrations

K_{1} ⊂ K_{2} ⊂ … ⊂ K_{p−1} ⊂ K_{p} = M
∪ ∪ ∪
{0} ⊂ I_{p−1} ⊂ I_{p−2} ⊂ … ⊂ I_{1}

where K_{i} = {m ∈ M | (z − 1)^{i}m = 0} is the kernel of multiplication by (z − 1)^{i} (in particular K_{p−1} = M′) and I_{i} = (z − 1)^{i}M is the image of multiplication by (z − 1)^{i}.

(b) K_{i}/I_{p−i} ≅ k for any i = 1, …, p−1. Moreover K_{p−1}/I_{p−1} ≅ k ⊕ (I_{1}/I_{p−1}).

(c) The module I_{1} = (z − 1)M is free as a module over the ring kZ/(z − 1)^{p−1}. Moreover, I_{i}/I_{i+1} ≅ \overline{M} for any i = 1, …, p−1.

(d) The module M/K_{1} is isomorphic to I_{1}. In particular it is free as a module over the ring kZ/(z − 1)^{p−1} and K_{i+1}/K_{i} ≅ \overline{M} for any i = 1, …, p−1.

(e) Dim(M) = p Dim(\overline{M}) + 1.

Proof. (a) Note that K_{i} and I_{i} are submodules because z is central in kG. We have I_{p−i} ⊂ K_{i} because (z − 1)^{p} = 0. The filtrations are clear.

(b) In order to prove (b), it suffices to restrict to the subgroup Z. But we have M↓_{Z}^{G} = k ⊕ F for some free kZ-module F, and therefore

K_{i} = k ⊕ (z − 1)^{p−i}F , I_{p−i} = (z − 1)^{p−i}F .

Moreover it is clear that K_{p−1}/I_{p−1} = K_{1}/I_{p−1} ⊕ (I_{1}/I_{p−1}) ≅ k ⊕ (I_{1}/I_{p−1}). (c) Multiplication by (z − 1)^{i} induces a map

M ⟶ (z − 1)^{i}M/(z − 1)^{i+1}M = I_{i}/I_{i+1}

and we claim that its kernel is M′. Again, in order to prove this, it suffices to restrict to the subgroup Z and consider the decomposition M↓_{Z}^{G} = k ⊕ F as above. Then the kernel is k ⊕ (z − 1)F = M′. It is also clear that

(z − 1)M = (z − 1)F ≅ F/(z − 1)^{p−1}F

and this is free over the ring kZ/(z − 1)^{p−1}.

(d) Multiplication by (z − 1) induces an isomorphism M/K_{1} ≅ I_{1}.

(e) Since M↓_{Z}^{G} = k⊕F, we have that Dim(\overline{M}) = Dim(F/(z−1)F) = Dim(F)/p and Dim(M) = p Dim(\overline{M}) + 1. □

Lemma 5.3. Let M be a kG-module. Suppose that there is a maximal subgroup H of G such that M↓_{H}^{G} ≅ k ⊕ (free).

(a) M ≅ k ⊕(free) as a kG-module if and only if \overline{M} is a free k\overline{G}-module. More precisely, M has a free summand with r generators as a kG-module if and only if \overline{M} has a free summand with r generators as a k\overline{G}-module. In particular, if M is indecomposable, then \overline{M} has no projective summands.

(b) M ≅ k̸ ⊕ (free) as a kG-module if and only if \overline{M} is a periodic k\overline{G}-module.

Proof. (a) It is easy to see that if M has a free summand L ≅ (kG)^{r} as a kGmodule then \overline{M} has a free summand L/(z − 1)L ≅ (k\overline{G})^{r} as a k\overline{G}-module.

The converse is essentially contained in Lemma 3.3 of [CaTh] and we recall the argument. Assume that M = N ⊕L where L is free and N has no free summands. Then t_{1}^{G} · N = 0 where

t_{1}^{G} = ∑_{g∈G} g = (z − 1)^{p−1} ∏_{i=1}^{m}(x_{i} − 1)^{p−1} ,

x_{i} being a lift in G of the generator \overline{x}_{i} of \overline{G}. Let X = ∏_{i=1}^{m}(x_{i} − 1)^{p−1}. If \overline{N} has

a free submodule then X · \overline{N} ≠ 0, since \overline{X} = t\overline{G} ; 1. But if X · \overline{N} ≠ 0 then, via the isomorphism \overline{N} ≅ (z − 1)^{p−1}N of Lemma 5.2, we would obtain (z − 1)^{p−1}X · N = t_{1}^{G} · N ≠ 0, which is a contradiction.

(b) The hypothesis on M↓_{H}^{G} implies that \overline{M} is free on restriction to H/Z. But \overline{H} = H/Z is a maximal subgroup of \overline{G} = G/Z, so \overline{G}/\overline{H} is a cyclic group of order p. Tensoring with \overline{M} the exact sequence

0 ⟶ k ⟶ k[\overline{G}/\overline{H}] ⟶ k[\overline{G}/\overline{H}] ⟶ k ⟶ 0 ,

we obtain an exact sequence with \overline{M} at both ends and free k\overline{G}-modules in the middle, because k[\overline{G}/\overline{H}] ⊗ \overline{M} ≅ \overline{M}↓\overline{G} ; \overline{H}↑\overline{G} ; \overline{H}. If now M ≅ k̸ ⊕ (free), then \overline{M} is not zero and is not free as a k\overline{G}-module, by part (a), so \overline{M} is periodic. If conversely \overline{M} is periodic, then \overline{M} is not free and M ≅ k̸ ⊕ (free) by part (a). □

Lemma 5.4. Suppose that p = 2 and that M is a non-trivial critical kG-module. Then the number of generators of M is the same as the number of generators of \overline{M} and is equal to 4 Dim(\overline{M})/|G|. Moreover Dim(Ω(M)) = Dim(Ω^{−1}(M)) = Dim(M) − 2.

Proof. Let H be a maximal subgroup of G. Since M↓_{H}^{G} ≅ k⊕(free), we know that \overline{M} is free as a module over k\overline{H}. So the number of generators of \overline{M} as a k\overline{H}-module is Dim(\overline{M})/|\overline{H}|. Our first claim is that G acts trivially on \overline{M}/ Rad(k\overline{H})\overline{M}. So the number of generators of \overline{M} as a k\overline{G}-module is also Dim(\overline{M}/ Rad(k\overline{H})\overline{M}) = Dim(\overline{M})/|\overline{H}|. In order to prove the claim, we note that the group G/H acts on \overline{M}/ Rad(k\overline{H})\overline{M}. If there was a free summand generated by the class of an element \overline{m}, then \overline{m} would generate a free summand of \overline{M} as a module over k\overline{G}, contrary to part (c) of the previous lemma. Since the group G/H has order 2, the only possibility is that G/H acts trivially on \overline{M}/ Rad(k\overline{H})\overline{M}.

Now our second claim is that, given a set of generators of \overline{M}, some lifts of those generators in M will generate M. Assuming this, it follows that the number of generators of M is Dim(\overline{M})/|\overline{H}| = 4 Dim(\overline{M})/|G|. If r = 4 Dim(\overline{M})/|G|, then the projective cover of M is the free module (kG)^{r}. Using Lemma 5.2 we obtain

Dim(Ω(M)) = Dim((kG)^{r})−Dim(M) = 4 Dim(\overline{M})−2 Dim(\overline{M})−1 = Dim(M)−2

as desired. Finally, since the dual module M^{∗} also satisfies the assumptions of the lemma, we have that

Dim(Ω^{−1}(M)) = Dim(Ω^{−1}(M)^{∗}) = Dim(Ω(M^{∗})) = Dim(M^{∗}) − 2 = Dim(M) − 2

and this completes the proof.

We are left with the proof of the second claim. Let L be the submodule of M generated by some lifts in M of the generators of \overline{M}. Assume by contradiction that L ≠ M. Since M↓_{H}^{G} = k ⊕ F for some free kH-module F, we have \overline{M}↓\overline{G} ; \overline{H} = F/(z−1)F and so we can choose the lifts of the generators of \overline{M} so that L↓_{H}^{G} = F. Now for any other maximal subgroup H′ of G, we have M↓_{H′}^{G} = k ⊕ F′ for some free kH′-module F′. The subgroup H ∩ H′ is non-trivial because it contains Z and there are two decompositions

M↓_{H∩H′}^{G} = T ↓_{H∩H′}^{H} ⊕ F ↓_{H∩H′}^{H} = T′↓_{H∩H′}^{H′} ⊕ F′↓_{H∩H′}^{H′}

where T, respectively T′, denotes a trivial one-dimensional module for kH, respectively kH′. By comparing the fixed points M^{H∩H′} and the relative traces t_{1}^{H∩H′} · M in both decompositions, we see that T′↓_{H∩H′}^{H′} cannot be contained in F ↓_{H∩H′}^{H} and therefore

M↓_{H∩H′}^{G} = T′↓_{H∩H′}^{H′} ⊕ F ↓_{H∩H′}^{H}

(see Lemma 8.2 in [CaTh] for details). Since F is the restriction of a kGsubmodule, this is a decomposition of M as a kH′-module, namely

M↓_{H′}^{G} = T′ ⊕ L↓_{H′}^{G} .

By the Krull-Schmidt theorem, we deduce that L↓_{H′}^{G} is free. Since this holds for any maximal subgroup H′ and since G is not elementary abelian, Chouinard’s theorem (see [Be] or [Ev]) implies that L is free as a kG-module and so M ≅ k⊕L. But M is indecomposable and non-trivial by assumption. This contradiction completes the proof of the claim. □

For our next theorem, we first need a technical lemma. Lemma 5.5. Let W be a kG-module satisfying the following two conditions:

(a) W/(z − 1)W = U_{1} ⊕ U_{2} where U_{1} and U_{2} are k\overline{G}-submodules such that the varieties satisfy V\overline{G}(U_{1}) ∩ V\overline{G}(U_{2}) = \{0\}.

(b) For some r ≤ p, we have (z − 1)^{r}W = 0 and W is free as a module over the ring kZ/(z − 1)^{r}.

Then W = W_{1} ⊕ W_{2} where W_{1} and W_{2} are kG-submodules of W such that W_{i}/(z − 1)W_{i} ≅ U_{i} for i = 1, 2.

Proof. We use induction on r. There is nothing to prove if r = 1 so we assume r ≥ 2. By induction, W/(z − 1)^{r−1}W = V_{1} ⊕ V_{2} where V_{1} and V_{2} are kGsubmodules of W/(z − 1)^{r−1}W such that V_{i}/(z − 1)V_{i} ≅ U_{i} for i = 1, 2. Now, since W is free as a module over kZ/(z − 1)^{r}, multiplication by (z − 1) induces an isomorphism W/(z − 1)^{r−1}W ≅ (z − 1)W and we write L_{i} for the image of V_{i}. So (z − 1)W = L_{1} ⊕ L_{2}.

Let π : W → W/(z − 1)W = U_{1} ⊕ U_{2} be the canonical surjection. Passing to the quotient by L_{1}, we obtain a short exact sequence

0 ⟶ L_{2} ⟶ W/L_{1} \underrightarrow{π̃} U_{1} ⊕ U_{2} ⟶ 0

where π̃ is induced by π. Let K = {x ∈ W/L_{1} | (z − 1)x = 0}. We claim that π̃(K) = U_{1}. Let x ∈ K and let w ∈ W be a lift of x. Then (z − 1)w ∈ L_{1}. Since multiplication by (z −1) induces an isomorphism W/(z −1)^{r−1}W ≅ (z −1)W, the class of w in W/(z − 1)^{r−1}W is in V_{1}. It follows that π(w) ∈ U_{1}, hence π̃(x) ∈ U_{1}, proving the claim.

Therefore we obtain a short exact sequence

0 ⟶ (z − 1)^{r−2}L_{2} ⟶ K \underrightarrow{π̃} U_{1} ⟶ 0

because L_{2} ∩ Ker(z − 1) = (z − 1)^{r−2}L_{2}. This is a sequence of k\overline{G}-modules since (z − 1)K = 0 by construction. Now multiplication by (z − 1)^{r−1} induces an isomorphism W/(z − 1)W ≅ (z − 1)^{r−1}W mapping U_{2} onto (z − 1)^{r−2}L_{2}. By applying our assumption on the varieties of U_{1} and U_{2} we deduce that the sequence splits (see Theorem 2.2). Let σ be a section of π̃ : K → U_{1} and let W_{1} be the inverse image of σ(U_{1}) in W, so that W_{1}/L_{1} = σ(U_{1}). We have obtained a short exact sequence

0 ⟶ L_{1} ⟶ W_{1} ⟶^{π} U_{1} ⟶ 0 .

We can construct similarly a submodule W_{2} and a short exact sequence

0 ⟶ L_{2} ⟶ W_{2} ⟶^{π} U_{2} ⟶ 0 .

Then π(W_{1} ∩ W_{2}) = 0, so W_{1} ∩ W_{2} ⊆ Ker(π) = L_{1} ⊕ L_{2}. But since W_{i} ∩ Ker(π) = L_{i}, we obtain W_{1} ∩ W_{2} = 0. For reasons of dimensions (or by a direct argument), the direct sum W_{1} ⊕ W_{2} must be the whole of W. □

Theorem 5.6. Let M be a critical kG-module and suppose that \overline{M} = \overline{M}_{1} ⊕ \overline{M}_{2} where \overline{M}_{1} and \overline{M}_{2} are k\overline{G}-submodules. Suppose that the varieties satisfy

V\overline{G}(\overline{M}_{1}) ∩ V\overline{G}(\overline{M}_{2}) = \{0\}.

Then there exist critical kG-modules N_{1} and N_{2} such that \overline{N}_{i} ≅ \overline{M}_{i} for 1 ≤ i ≤ 2.

Proof. As before, let M′ = {m ∈ M | (z − 1)^{p−1}m = 0}. Let M_{1} ⊆ M be the inverse image of \overline{M}_{1} under the quotient map M ⟶ M/M′ = \overline{M}. Let M_{2} be the inverse image of \overline{M}_{2}. Then M′ = M_{1} ∩ M_{2} and M_{1}/M′ ≅ \overline{M}_{1}, M_{2}/M′ = \overline{M}_{2}.

By Lemma 5.2, (z − 1)M is free over kZ/(z − 1)^{p−1} and

(z − 1)M/(z − 1)^{2}M ≅ M/M′ = \overline{M} = \overline{M}_{1} ⊕ \overline{M}_{2} .

Therefore Lemma 5.5 applies and we have (z − 1)M = W_{1} ⊕ W_{2} such that W_{i}/(z − 1)W_{i} ≅ \overline{M}_{i} for i = 1, 2. Now define N_{1} = M_{1}/W_{2} and N_{2} = M_{2}/W_{1}. If r_{i} = Dim(\overline{M}_{i}), then Dim(\overline{M}) = r_{1} + r_{2} and by Lemma 5.2 we obtain Dim(M) = pr_{1} + pr_{2} + 1 and Dim((z − 1)M) = (p − 1)r_{1} + (p − 1)r_{2}. Therefore we have Dim(M_{1}) = pr_{1} + (p − 1)r_{2} + 1 and Dim(M_{2}) = (p − 1)r_{1} + pr_{2} + 1. Also Dim(W_{i}) = (p − 1)r_{i}, hence Dim(N_{i}) = pr_{i} + 1 for i = 1, 2.

We claim that N_{1}↓_{H}^{G} ≅ k ⊕ (free) for every maximal subgroup H of G (and similarly for N_{2}). Let H = ⟨z, y_{1}, …, y_{m−1}⟩ where \overline{y}_{1}, …, \overline{y}_{m−1} are generators of \overline{H} = H/Z. The assumption on M↓_{H}^{G} implies that \overline{M} is free as k\overline{H}-module.

Therefore \overline{M}_{1} and \overline{M}_{2} must be free as k\overline{H}-modules. Let Y = ∏_{i=1}^{m−1}(y_{i} − 1)^{p−1} so

that \overline{Y} = t\overline{H} ; 1 and Y (z − 1)^{p−1} = t_{1}^{H}. Then we get

Dim(\overline{M}_{1}) = |\overline{H}| · Dim(\overline{Y} · \overline{M}_{1}) .

Now (z − 1)^{p−1}N_{1} ≅ (z − 1)^{p−1}M_{1} because N_{1} = M_{1}/W_{2} and (z − 1)^{p−1}W_{2} = 0. Therefore

t_{1}^{H} · N_{1} = Y (z − 1)^{p−1}N_{1} ≅ Y (z − 1)^{p−1}M_{1} ≅ Y · \overline{M}_{1} = \overline{Y} · \overline{M}_{1} .

It follows that

|H| Dim(t_{1}^{H} · N_{1}) = p · |\overline{H}| · Dim(\overline{Y} · \overline{M}_{1}) = p · Dim(\overline{M}_{1}) = pr_{1} = Dim(N_{1}) − 1 .

Therefore N_{1}↓_{H}^{G} has a free submodule of dimension Dim(N_{1}) − 1. The only way this can happen is if N_{1}↓_{H}^{G} ≅ k ⊕ (free).

Now we prove that \overline{N}_{1} ≅ \overline{M}_{1} (and similarly for N_{2}). We have to compute the submodule N′_{1} = {x ∈ N_{1} | (z − 1)^{p−1}x = 0}. But N_{1} = M_{1}/W_{2} and we have W_{2} ⊆ M′ ⊆ M_{1} and (z − 1)^{p−1}M′ = 0. Therefore M′/W_{2} ⊆ N′_{1} and \overline{N}_{1} = N_{1}/N′_{1} is a quotient of N_{1}/(M′/W_{2}) ≅ M_{1}/M′ = \overline{M}_{1}. In order to prove that this is not a proper quotient, it suffices to prove that \overline{N}_{1} and \overline{M}_{1} have the same dimension. But by the previous part of the proof, we know that N_{1}↓_{H}^{G} ≅ k ⊕ (free) for every maximal subgroup H. By Lemma 5.2 this implies

Dim(\overline{N}_{1}) = \frac{Dim(N_{1}) − 1}{p} = r_{1} = Dim(\overline{M}_{1}) ,

as was to be shown.

Finally we conclude that N_{1} is critical. Indeed, since \overline{M} has no free summand as a k\overline{G}-module, \overline{N}_{1} cannot have a free summand and therefore N_{1} has no free summand as a kG-module by Lemma 5.3. This implies that N_{1} is critical since we know that N_{1}↓_{H}^{G} ≅ k ⊕ (free) for every maximal subgroup H. □

Theorem 5.7. Let M_{1} and M_{2} be critical kG-modules and suppose that the varieties satisfy

V\overline{G}(\overline{M}_{1}) ∩ V\overline{G}(\overline{M}_{2}) = \{0\}.

Then M_{1} ⊗ M_{2} ≅ M ⊕ (free) where M is a critical kG-module such that \overline{M} ≅ \overline{M}_{1} ⊕ \overline{M}_{2}.

Proof. Let r_{j} = Dim(\overline{M}_{j}) for j = 1, 2. Thus Dim(M_{j}) = pr_{j} + 1. Consider the filtration of M_{1} as in Lemma 5.2

{0} ⊂ (z − 1)^{p−1}M_{1} ⊂ K_{1} ⊂ … ⊂ K_{p−1} ⊂ K_{p} = M_{1} ,

where K_{i} = {m ∈ M_{1} | (z − 1)^{i}m = 0}. This induces a filtration on M_{1} ⊗ M_{2}

{0} ⊂ (z − 1)^{p−1}M_{1} ⊗ M_{2} ⊂ K_{1} ⊗ M_{2} ⊂ … ⊂ K_{p−1} ⊗ M_{2} ⊂ M_{1} ⊗ M_{2} ,

with all quotients but one isomorphic to \overline{M}_{1}⊗M_{2}. We need to prove the following.

Lemma 5.8. \overline{M}_{1} ⊗ M_{2} = F ⊕ L where L ≅ \overline{M}_{1} and F is a free kG-module of dimension pr_{1}r_{2} such that (z − 1)^{p−1}F = \overline{M}_{1} ⊗ (z − 1)^{p−1}M_{2}.

Proof. By hypothesis V\overline{G}(\overline{M}_{1}) ∩ V\overline{G}(\overline{M}_{2}) = \{0\} and hence \overline{M}_{1} ⊗ \overline{M}_{2} is projective as a k\overline{G}-module. Choose elements m_{1}, …, m_{r} ∈ \overline{M}_{1} ⊗ M_{2} such that \overline{m}_{1}, …, \overline{m}_{r} is a free k\overline{G}-basis for \overline{M}_{1} ⊗ \overline{M_{2}}. Here \overline{m}_{i} = m_{i} + (\overline{M}_{1} ⊗ M′_{2}) denotes the class of m_{i} in \overline{M}_{1} ⊗ \overline{M_{2}} = (\overline{M}_{1} ⊗ M_{2})/(\overline{M}_{1} ⊗ M′_{2}).

As before, let X = ∏_{i=1}^{m}(x_{i} − 1)^{p−1} so that \overline{X} = t\overline{G} ; 1 and X(z − 1)^{p−1} = t_{1}^{G}. Then

\overline{X}\overline{m}_{1}, …, \overline{X}\overline{m}_{r} are linearly independent in \overline{M}_{1}⊗\overline{M}_{2}. Since z acts trivially on \overline{M}_{1}, multiplication by (z−1)^{p−1} induces an isomorphism \overline{M}_{1}⊗\overline{M}_{2} ≅ \overline{M}_{1}⊗(z−1)^{p−1}M_{2} and it follows that X(z − 1)^{p−1}m_{1}, …, X(z − 1)^{p−1}m_{r} are linearly independent in \overline{M}_{1} ⊗ (z − 1)^{p−1}M_{2}. Therefore t_{1}^{G}m_{1}, …, t_{1}^{G}m_{r} are linearly independent in \overline{M}_{1} ⊗ M_{2}. So the kG-submodule F of \overline{M}_{1} ⊗ M_{2} generated by m_{1}, …, m_{r} is a free kG-module. Moreover we have (z − 1)^{p−1}F = \overline{M}_{1} ⊗ (z − 1)^{p−1}M_{2}.

Consider now the exact sequence of k\overline{G}-modules

0 ⟶ \overline{M}_{1} ⊗ (z − 1)^{p−1}M_{2} ⟶ \overline{M}_{1} ⊗ (M_{2})^{Z} ⟶ \overline{M}_{1} ⊗ k ⟶ 0 ,

where (M_{2})^{Z} = \{x ∈ M_{2} | (z − 1)x = 0\}. Since the kernel is free over k\overline{G}, the sequence splits and we have \overline{M}_{1}⊗(M_{2})^{Z} ≅ (z−1)^{p−1}F ⊕L where L is a submodule isomorphic to \overline{M}_{1}. If we had F ∩ L ≠ 0, then we would have Soc(F) ∩ L ≠ 0, hence (z − 1)^{p−1}F ∩ L ≠ 0, a contradiction. Therefore F ∩ L = 0 and \overline{M}_{1} ⊗ M_{2} contains a submodule F ⊕ L.

We now show that F ⊕ L = \overline{M}_{1} ⊗ M_{2} by proving that both modules have the same dimension. We have Dim(F) = p · Dim(\overline{M}_{1} ⊗ \overline{M}_{2}) = pr_{1}r_{2} and therefore

Dim(F ⊕ L) = pr_{1}r_{2} + r_{1} = r_{1}(pr_{2} + 1) = Dim(\overline{M}_{1}) Dim(M_{2}) ,

as was to be shown. □

Now, continuing with the proof of the theorem, we note that each quotient (K_{i+1}⊗M_{2})/(K_{i}⊗M_{2}) is isomorphic to \overline{M}_{1}⊗M_{2}, hence contains a free submodule \overline{F_{i}} of dimension pr_{1}r_{2} by the lemma. Now remember that projective modules are also injective and, as a result, if a projective module is a direct summand of a section of a module V, then it is a direct summand of V. Thus we can lift the free module \overline{F_{i}} and obtain a free submodule F_{i} of M_{1} ⊗ M_{2} mapping isomorphically onto \overline{F_{i}} under the quotient map M_{1} ⊗ M_{2} → (M_{1} ⊗ M_{2})/(K_{i} ⊗ M_{2}). Similarly, (z − 1)^{p−1}M_{1} ⊗ M_{2} is isomorphic to \overline{M}_{1} ⊗ M_{2}, hence contains a free submodule F_{0} of dimension pr_{1}r_{2} by the lemma. Therefore we have

M_{1} ⊗ M_{2} = M ⊕ F

where F = F_{0} ⊕ … ⊕ F_{p−1} is free of dimension p^{2}r_{1}r_{2} and M is a submodule of dimension (pr_{1} + 1)(pr_{2} + 1) − p^{2}r_{1}r_{2} = p(r_{1} + r_{2}) + 1.

Since, for any maximal subgroup H of G, we have M_{j}↓_{H}^{G} ≅ k ⊕ (free) for j = 1, 2, the same holds for M_{1} ⊗ M_{2} and hence M↓_{H}^{G} ≅ k ⊕ (free). We are going to prove that \overline{M} ≅ \overline{M}_{1} ⊕ \overline{M}_{2}. This will imply that M is critical. Indeed \overline{M}_{j} has no k\overline{G}-free summand, because M_{j} is critical (j = 1, 2), so \overline{M}_{1} ⊕ \overline{M}_{2} has no k\overline{G}-free summand and therefore M has no kG-free summand by Lemma 5.2. This forces the endo-trivial module M to be indecomposable.

Instead of working with \overline{M}, we consider the isomorphic module (z − 1)^{p−1}M and our goal now is to prove that (z − 1)^{p−1}M ≅ \overline{M}_{1} ⊕ \overline{M}_{2}. We work with the submodule K_{1} ⊗ M_{2} of our filtration and we first analyze its submodule K_{1} ⊗ K′_{1}, where K′_{1} = {m ∈ M_{2} | (z − 1)m = 0} is the analog of K_{1} for M_{2}. Notice that K_{1} ⊗ K′_{1} is a k\overline{G}-module with a filtration

(z−1)^{p−1}M_{1}⊗(z−1)^{p−1}M_{2} ⊂ ((z−1)^{p−1}M_{1}⊗K′_{1})+(K_{1}⊗(z−1)^{p−1}M_{2}) ⊂ K_{1}⊗K′_{1} .

In the filtration, the bottom submodule is free over k\overline{G} and is equal to (z−1)^{p−1}F_{0} by the lemma. The middle quotient of this filtration is the direct sum of

((z−1)^{p−1}M_{1} ⊗ K′_{1})/((z−1)^{p−1}M_{1} ⊗ (z−1)^{p−1}M_{2}) ≅ (z−1)^{p−1}M_{1} ⊗ k ≅ \overline{M}_{1}

and

(K_{1} ⊗ (z−1)^{p−1}M_{2})/((z−1)^{p−1}M_{1} ⊗ (z−1)^{p−1}M_{2}) ≅ k ⊗ (z−1)^{p−1}M_{2} ≅ \overline{M}_{2} .

This direct sum can be lifted in K_{1} ⊗ K′_{1}, because the submodule (z − 1)^{p−1}F_{0} = (z−1)^{p−1}M_{1} ⊗ (z−1)^{p−1}M_{2} is k\overline{G}-free and so the sequence

0 ⟶ (z − 1)^{p−1}F_{0} ⟶ K_{1} ⊗ K′_{1} ⟶ (K_{1} ⊗ K′_{1})/(z − 1)^{p−1}F_{0} ⟶ 0

splits. Therefore K_{1} ⊗ K′_{1} contains a submodule V_{1} ⊕ V_{2} with V_{j} ≅ \overline{M}_{j} and (V_{1} ⊕ V_{2}) ∩ (z − 1)^{p−1}F_{0} = 0. It follows that (V_{1} ⊕ V_{2}) ∩ Soc(F_{0}) = 0 and so (V_{1} ⊕ V_{2}) ∩ F_{0} = 0.

We now have V_{1} ⊕ V_{2} ⊕ F_{0} ⊂ K_{1} ⊗ M_{2} and therefore V_{1} ⊕ V_{2} ⊕ F_{0} intersects trivially F_{1} ⊕ … ⊕ F_{p−1} because this free module has been lifted from quotients of (M_{1} ⊗ M_{2})/(K_{1} ⊗ M_{2}). This shows that M_{1} ⊗ M_{2} contains the submodule V_{1} ⊕ V_{2} ⊕ F_{0} ⊕ F_{1} ⊕ … ⊕ F_{p−1} = V_{1} ⊕ V_{2} ⊕ F. Therefore V_{1} ⊕ V_{2} is isomorphic to a submodule of M.

Now we show that V_{1} ⊕ V_{2} ⊂ (z − 1)^{p−1}(M_{1} ⊗ M_{2}). Since z acts trivially on K′_{1}, we have (z − 1)^{p−1}(M_{1} ⊗ K′_{1}) = (z − 1)^{p−1}M_{1} ⊗ K′_{1} and this contains V_{1} by construction of V_{1}. Similarly V_{2} ⊂ K_{1} ⊗ (z − 1)^{p−1}M_{2} = (z − 1)^{p−1}(K_{1} ⊗ M_{2}). Passing to the quotient by F, we deduce that V_{1}⊕V_{2} is isomorphic to a submodule of

(z − 1)^{p−1}((M_{1} ⊗ M_{2})/F ) = (z − 1)^{p−1}((M ⊕ F)/F ) ≅ (z − 1)^{p−1}M .

In order to prove that this submodule is the whole of (z − 1)^{p−1}M, it suffices to prove that they have the same dimension. But V_{j} ≅ \overline{M}_{j} has dimension r_{j} (for j = 1, 2) and we know that Dim(M) = p(r_{1}+r_{2})+1. Therefore Dim(\overline{M}) = r_{1}+r_{2} and we are done. This shows that (z − 1)^{p−1}M ≅ V_{1} ⊕ V_{2} and this completes the proof of the theorem. □

Theorem 5.6 and Theorem 5.7 provide the basic tools for constructing a large critical module from any given finite set of such modules, as follows. Theorem 5.9. For every i = 1, …, t, let M_{i} be a non-trivial critical kG-module. Let ℓ_{i} be a line in the variety of the periodic k\overline{G}-module \overline{M}_{i} and assume that ℓ_{i} ≠ ℓ_{j} for i ≠ j. Then there exists a non-trivial critical kG-module M such that V\overline{G}(\overline{M}) = ⋃_{i=1}^{t} ℓ_{i}. Moreover, Dim(M) ≥ t|G|/2 + 1 if p = 2 and Dim(M) ≥ t|G| + 1 if p is odd.

Proof. Recall that \overline{M}_{i} is periodic by Lemma 5.3, and hence V\overline{G}(\overline{M}_{i}) is a union of lines. By Theorem 2.2, \overline{M}_{i} = L_{i} ⊕ N_{i} such that V\overline{G}(L_{i}) = ℓ_{i} and the variety of N_{i} is the union of the other lines (if any, otherwise set simply L_{i} = \overline{M}_{i}). By Theorem 5.6, there exists a critical kG-module U_{i} such that \overline{U}_{i} = L_{i}.

Now by Theorem 5.7 and the assumption that the lines ℓ_{i} are distinct, we obtain a critical kG-module M such that

U_{1} ⊗ U_{2} ⊗ … ⊗ U_{t} = M ⊕ (free)

and \overline{M} = \overline{U}_{1} ⊕ \overline{U}_{2} ⊕ … ⊕ \overline{U}_{t} so that V\overline{G}(\overline{M}) = ⋃_{i=1}^{t} ℓ_{i}.

Since U_{i}↓_{H}^{G} ≅ k ⊕ (free) where H is a maximal subgroup of G, Dim(U_{i}) − 1 is a multiple of |H| and therefore Dim(\overline{U}_{i}) is a multiple of |H|/p = |G|/p^{2}. It follows that Dim(\overline{M}) ≥ t|G|/p^{2} and Dim(M) ≥ t|G|/p + 1. We can do better if p is odd because U_{i} is an endo-trivial module and so Dim(U_{i}) ≡ ±1 (mod |G|) by Lemma 2.10 in [CaTh]. A plus sign is forced here and therefore Dim(U_{i}) − 1 is a multiple of |G|. The same argument then yields Dim(M) ≥ t|G| + 1. □

Remark. By a theorem of Puig [Pu], the torsion subgroup T_{t}(G) is finite. Therefore, there are actually finitely many possible choices for the modules M_{i} in the last theorem. It then follows from the theorem that one can construct an indecomposable torsion endo-trivial module M such that V\overline{G}(\overline{M}) contains V\overline{G}(\overline{N}) for any torsion endo-trivial module N. Moreover, Dim(M) ≥ t|G|/2+1, respectively t|G| + 1, where t is the number of components of V\overline{G}(\overline{M}). However, in view of the main theorem of this paper, it will turn out that T_{t}(G) = 0 and so M ≅ k.

6. Lower bounds on dimensions of critical modules

In this section we prove a theorem that is essential to the general cases of our main result. Basically it says that, if an extraspecial group or an almost extraspecial group has a non-trivial critical module, then it has one of large dimension. For the proof, we need a few lemmas. Lemma 6.1. Suppose that M is a non-trivial critical kG-module and let ℓ be a line in V\overline{G}(\overline{M}). Then ℓ is not contained in any 𝔽_{p}-rational subspace of V\overline{G}(k).

Proof. Note that V\overline{G}(k) = k^{m} where |G| = p^{m+1}. An 𝔽_{p}-rational subspace (i.e. a subspace defined by a linear equation with 𝔽_{p}-coefficients) corresponds to a maximal subgroup \overline{H} ⊆ \overline{G}. That is, the 𝔽_{p}-rational subspaces of V\overline{G}(k) are precisely the subspaces of the form res_{\overline{G},\overline{H}}^{∗}(V\overline{H}(k)). If ℓ were in res_{\overline{G},\overline{H}}^{∗}(V\overline{H}(k))

then it would have to be the case that V\overline{H}(\overline{M}↓\overline{G} ; \overline{H}) ≠ \{0\} and hence \overline{M}↓\overline{G} ; \overline{H} would not be free as a k\overline{H}-module. By Lemma 5.3, this would contradict the hypothesis that M↓_{H}^{G} ≅ k ⊕ (free) where H is the inverse image of \overline{H} in G. □

Recall that a p′-group is a group of order prime to p. Lemma 6.2. Suppose that ℓ is a line through the origin in V\overline{G}(k) = k^{m} and suppose that ℓ is not contained in any 𝔽_{p}-rational subspace of k^{m}. Then the stabilizer S of ℓ for the action of GL_{m}(𝔽_{p}) on k^{m} is a cyclic p′-subgroup.

Proof. Suppose that y ∈ GL_{m}(𝔽_{p}) stabilizes ℓ and that v is a point on ℓ. Then v is an eigenvector of y with eigenvalue λ. That is, simply, y · v = λv. So the line ℓ is a kS-submodule for the action of S on k^{m}, corresponding to a homomorphism ρ : S ⟶ GL(ℓ) ≅ k^{∗} mapping y ∈ S to the eigenvalue λ.

We claim that ρ is injective on the stabilizer S. For suppose that ρ(y) = λ = 1. Then, viewing y as a matrix, we have that (y − I_{m})v = 0. If y is not the identity then some row (a_{1}, a_{2}, …, a_{m}) of y −I_{m} is not zero. But then v is in the subspace defined by the equation a_{1}x_{1} + a_{2}x_{2} + … + a_{m}x_{m} = 0. Because the coefficients of y are in 𝔽_{p} we have a contradiction.

Now S is isomorphic to a finite subgroup of k^{∗} and therefore it must consist of roots of unity. Thus it is a cyclic p′-group and we are done. □

Recall that for any automorphism α of G, the conjugate module N^{α} is defined to be the k-vector space N with the action of G given by g · n = α(g)n for g ∈ G and n ∈ N. If α is an inner automorphism of G, then N^{α} ≅ N and it follows that the group Out(G) of outer automorphisms of G acts on the set of isomorphism classes of kG-modules. We shall also write N^{y} for a conjugate module defined by an outer automorphism y ∈ Out(G).

Since G is extraspecial or almost extraspecial, we control Out(G) in the following sense. Recall that if p = 2, there is an associated quadratic form on the 𝔽_{2}-vector space G/Z(G) (see Lemma 3.1). If p is odd, there is a symplectic form b on the 𝔽_{p}-vector space G/Z(G) defined by [x̃, ỹ] = z^{b(x,y)}, where x, y ∈ G/Z(G), x̃, ỹ ∈ G are elements of G that lift x and y, and z is a generator of Z(G). Lemma 6.3. Let G be an extraspecial or almost extraspecial p-group. Let Out_{0}(G) be the subgroup of Out(G) consisting of outer automorphisms fixing the center Z(G) pointwise.

(a) If p is odd and G is extraspecial of exponent p, then Out_{0}(G) is isomorphic to the symplectic group O_{G} associated to the symplectic form corresponding to G.

(b) If p = 2, Out_{0}(G) is isomorphic to the orthogonal group O_{G} associated to the quadratic form corresponding to G.

Proof. When G is extraspecial, this is one of the main results in Winter’s paper [Wi]. If G is almost extraspecial, the arguments given in Sections 3F and 4 of [Wi] extend and yield the same result. Alternatively, this appears explicitly in Exercise 5 of Chapter 8 of [As]. □

Theorem 6.4. Suppose that there exists a non-trivial critical kG-module N. (a) If p is odd, there exists a critical kG-module M such that

Dim(M) > |G| · \frac{|O_{G}|}{|C|}

where O_{G} is the symplectic group associated to G and C is a cyclic p′-subgroup of O_{G} of maximal order.

(b) If p = 2, there exists a critical kG-module M such that

Dim(M) > \frac{|G|}{2} · \frac{|O_{G}|}{|C|}

where O_{G} is the orthogonal group associated to G and C is an odd order cyclic subgroup of O_{G} of maximal order.

Proof. Let ℓ be a line in V\overline{G}(\overline{N}). Notice that if y ∈ O_{G} then N^{y} is also a nontrivial kG-module such that N^{y}↓_{H}^{G} ≅ k ⊕ (free). But then y(ℓ) is in the variety V\overline{G}(\overline{N}^{y}). If B denotes the stabilizer of ℓ in O_{G}, we obtain a family of modules N^{y} indexed by the set of cosets O_{G}/B. So by Theorem 5.9, there exists a critical

kG-module M such that V\overline{G}(\overline{M}) = ⋃_{y∈O_{G}/B} y(ℓ). Moreover, Dim M > \frac{|O_{G}|}{|B|} · \frac{|G|}{2}

if p = 2 and Dim M > \frac{|O_{G}|}{|B|} · |G| if p is odd.

By Lemma 6.1, the line ℓ is not contained in any 𝔽_{p}-rational subspace of V\overline{G}(k) = k^{m}. Thus by Lemma 6.2, the group B = S ∩ O_{G} is cyclic of order prime to p. If C is of maximal order among cyclic p′-subgroups of O_{G}, we deduce the lower bound of the statement. □

For use in the following sections, we need to have some estimates of the orders of the orthogonal and symplectic groups and their cyclic p′-subgroups. Proposition 6.5. Let G be an extraspecial or almost extraspecial p-group. Let O_{G} be the orthogonal or symplectic group associated to G.

1. If p is odd and G is extraspecial of exponent p and order p^{2n+1}, then O_{G} = Sp(2n, 𝔽_{p}) and

|O_{G}| = p^{n^{2}} ∏_{i=1}^{n}(p^{2i} − 1) .

2. If p = 2 and G ≅ D_{8} ∗ ⋯ ∗ D_{8} is extraspecial of order 2^{2n+1} (type 1), then

O_{G} = O^{+}(2n, 𝔽_{2}) and

|O_{G}| = 2 · 2^{n(n−1)}(2^{n} − 1) ∏_{i=1}^{n−1}(2^{2i} − 1) .

3. If p = 2 and G ≅ D_{8} ∗ ⋯ ∗ D_{8} ∗ Q_{8} is extraspecial of order 2^{2n+1} (type 2),

then O_{G} = O^{−}(2n, 𝔽_{2}) and

|O_{G}| = 2 · 2^{n(n−1)}(2^{n} + 1) ∏_{i=1}^{n−1}(2^{2i} − 1) .

4. If p = 2 and G ≅ D_{8} ∗ ⋯ ∗ D_{8} ∗ C_{4} is almost extraspecial of order 2^{2n+2}

(type 3), then O_{G} = Sp(2n, 𝔽_{2}) and

|O_{G}| = 2^{n^{2}} ∏_{i=1}^{n}(2^{2i} − 1) .

Moreover, if C is any cyclic p′-subgroup of O_{G}, then |C| ≤ (p + 1)^{n}.

Proof. In the first three cases, we have O_{G} = Sp(2n, 𝔽_{p}), respectively O_{G} = O^{±}(2n, 𝔽_{2}), essentially by definition (see also [Wi]). In the third case, we obtain O_{G} = O(2n + 1, 𝔽_{2}) ≅ Sp(2n, 𝔽_{2}) (see Theorem 11.9 of Taylor’s book [Ta]). The orders of the four groups appear on pages 70 and 141 of Taylor’s book [Ta]. The list can also be found in any of a number of text books on Chevalley groups or finite simple groups (e.g. Gorenstein’s book [Go2]). The types of the groups of Lie type in the four cases listed are C_{n}(p), D_{n}(2), ^{2}D_{n}(2) and C_{n}(2), respectively. In the first case the corresponding simple group is O_{G}/{±1}. In the next two cases the corresponding simple group has index 2 in O_{G}, while the group is simple in the fourth case.

For the statement about the cyclic p′-subgroups, note first that elements of order prime to p are semi-simple, hence contained in a maximal torus. Now, for a Chevalley group of rank n over the field 𝔽_{q}, the order of a maximal torus is equal to

| det(w^{−1}F − 1)| = | det(F − w)|

where F (x) = x^{q} is the Frobenius morphism, w is an element of the corresponding Weyl group, and where F and w act on the cocharacter group of a fixed maximal torus of the corresponding algebraic group (see Proposition 3.3.5 in Carter’s book [Cart]). Since w has finite order, we obtain a product ∏_{i=1}^{n}(q − ζ_{i}) for suitable roots of unity ζ_{i} (the eigenvalues of w). In our case, q = p and the rank n is the same as the integer n of the statement. Therefore if C is any cyclic p′-subgroup of O_{G}, we get

|C| ≤ | det(F − w)| = | ∏_{i=1}^{n}(p − ζ_{i})| ≤ ∏_{i=1}^{n}(p + 1) = (p + 1)^{n} ,

as was to be shown. □

7. Upper bounds on dimensions of critical modules

Throughout the section we assume that G is a p-group and that k is an algebraically closed field of characteristic p.

We will need the following results. Recall that a non-zero element ζ of H^{1}(G, 𝔽_{p}) corresponds to a maximal subgroup of G in the sense that there is a unique maximal subgroup H of G such that res_{G,H}(ζ) = 0. When p is odd, we also need the Bockstein map β : H^{1}(G, 𝔽_{p}) ⟶ H^{2}(G, 𝔽_{p}) (see [Be] or [Ev]). Theorem 7.1. Suppose that G is a p-group which is not elementary abelian. Suppose that η_{1}, …, η_{t} are non-zero elements in H^{1}(G, 𝔽_{p}) and have the property that

η_{1} ⋯ η_{t} = 0 if p = 2,
β(η_{1}) ⋯ β(η_{t}) = 0 if p is odd.

(a) Assume that p = 2. For each i, let H_{i} be the maximal subgroup of G corresponding to η_{i}. Then there is a projective module P such that k⊕Ω^{1−t}(k)⊕P has a filtration

{0} = L_{0} ⊆ L_{1} ⊆ ⋯ ⊆ L_{t} ≅ k ⊕ Ω^{1−t}(k) ⊕ P

where L_{i}/L_{i−1} ≅ (Ω^{1−i}(k))↑_{H_{i}}^{G} for each i = 1, …, t.

(b) Assume that p is odd. For each i, let K_{i} be the maximal subgroup of G corresponding to η_{i} and set H_{2i} = H_{2i−1} = K_{i}. Then there is a projective module P such that k ⊕ Ω^{1−2t}(k) ⊕ P has a filtration

{0} = L_{0} ⊆ L_{1} ⊆ ⋯ ⊆ L_{2t} ≅ k ⊕ Ω^{1−2t}(k) ⊕ P

where L_{i}/L_{i−1} ≅ (Ω^{1−i}(k))↑_{H_{i}}^{G} for each i = 1, …, 2t.

Proof. This is the essence of Lemma 3.10 of [Ca2]. That lemma is stated for ℤG-modules but that does not really matter since we can tensor the whole thing with k. Because the emphasis of our theorem is different from that of the results of [Ca2] we give a brief sketch of the proof here. However, all of the ideas as well as the details are given in the paper [Ca2].

(a) We first give the proof when p = 2 and then indicate how to modify the arguments for odd p. Each of the cohomology elements η_{i} corresponds to an exact sequence

0 ⟶ 𝔽_{2} ⟶ 𝔽_{2}↑_{H_{i}}^{G} ⟶ 𝔽_{2} ⟶ 0 .

Now we splice all of these together and tensor with k to get a sequence of the form

0 ⟶ k ⟶ k↑_{H_{t}}^{G} ⟶ … ⟶ k↑_{H_{2}}^{G} ⟶ k↑_{H_{1}}^{G} ⟶ k ⟶ 0 ,

which represents the element η_{1} ⋯ η_{t} = 0 in H^{t}(G, k). Note that we are using the same notation η_{i} for the element of H^{1}(G, 𝔽_{2}) and its image under the change of rings in H^{1}(G, k). Now we consider the complex C obtained by truncating the ends off of the sequence. That is, C_{i} = k↑_{H_{i+1}}^{G} for i = 0, …, t − 1 and C_{i} = 0 otherwise. We see that the homology of C is a result of the truncations. That is, H_{i}(C) = k if either i = 0 or i = t − 1 and H_{i}(C) = 0 otherwise.

The next step is to collapse the complex C into a single module. This is accomplished exactly as in the paragraphs preceding Proposition 3.7 of [Ca2]. That is, we tensor, over k, the complex C with a projective resolution of the trivial module k. This gives us a projective resolution of the complex C and it has the same homology as C. So in degrees above t, it is exact and is the projective resolution of a module U, which we can take to be the image of the t^{th} boundary map of the total complex. The only problem with U is that it is in the wrong degree. So we take W = Ω^{−t}(U). This is the module that we want.

There are now two things to note about W. First because the terms of the complex C are induced from the maximal subgroup H_{1}, …, H_{t}, the module W has a filtration by the modules k↑_{H_{i}}^{G} suitably translated by Ω, exactly as described in the statement of the theorem. That is, the projective resolution of the complex C as constructed above is filtered by the projective resolutions of the terms of the complex, suitably translated. See the proof of Proposition 3.8 of [Ca2] for this part.

Next we note that the module W is isomorphic to k⊕Ω^{1−t}(k)⊕P for some projective module P. This is because the original sequence that represented η_{1} ⋯ η_{t} splits and hence the projective resolution of the complex is, in high degrees, a projective resolution of the homology groups of the complex, suitably translated. See Proposition 3.8 of [Ca2] for this part. This proves the theorem if p = 2.

(b) If p is odd, the cohomology element η_{i} has to be replaced by its Bockstein β(η_{i}) which corresponds to an exact sequence

0 ⟶ 𝔽_{p} ⟶ 𝔽_{p}↑_{K_{i}}^{G} ⟶ 𝔽_{p}↑_{K_{i}}^{G} ⟶ 𝔽_{p} ⟶ 0 .

Again we splice all of these together and tensor with k. Using our numbering of the subgroups H_{i}, we obtain a sequence of the form

0 ⟶ k ⟶ k↑_{H_{2t}}^{G} ⟶ … ⟶ k↑_{H_{2}}^{G} ⟶ k↑_{H_{1}}^{G} ⟶ k ⟶ 0 ,

which represents the element β(η_{1}) ⋯ β(η_{t}) = 0 in H^{2t}(G, k). The complex C is obtained by truncating the ends off of the sequence and the rest of the argument is the same, except that the integer t has to be replaced by 2t throughout. □

The upper bounds that we want for the dimensions of our critical modules is contained in the following. Theorem 7.2. Suppose that G is a p-group which is not elementary abelian. Suppose that η_{1}, …, η_{t} ∈ H^{1}(G, 𝔽_{p}) are non-zero and have the property that

η_{1} ⋯ η_{t} = 0 if p = 2,
β(η_{1}) ⋯ β(η_{t}) = 0 if p is odd.

Let r = t if p = 2 and r = 2t if p is odd. Let H_{1}, …, H_{r} be the maximal subgroups of G as in the previous theorem. Suppose that M is an indecomposable kG-module with the property that M↓_{H_{i}}^{G} ≅ k ⊕ (free) for every i. Then for any s,

Dim Ω^{s}(M) + Dim Ω^{s−r+1}(M) ≤ ∑_{i=1}^{r} Dim (Ω^{s+1−i}(k)↑_{H_{i}}^{G}) .

Proof. Let P be a projective module such that W = k ⊕ Ω^{1−r}(k) ⊕ P has a filtration as in the last theorem. Then tensoring W and all of the factors in the filtration with Ω^{s}(M) we get that

{0} = L_{0} ⊗ Ω^{s}(M) ⊆ L_{1} ⊗ Ω^{s}(M) ⊆ ⋯ ⊆ L_{r} ⊗ Ω^{s}(M)
≅ W ⊗ Ω^{s}(M) ≅ Ω^{s}(M) ⊕ Ω^{s+1−r}(M) ⊕ (free).

Then we have

(L_{i} ⊗ Ω^{s}(M))/(L_{i−1} ⊗ Ω^{s}(M)) ≅ (L_{i}/(L_{i−1}) ⊗ Ω^{s}(M)
≅ Ω^{1−i}(k)↑_{H_{i}}^{G} ⊗ Ω^{s}(M)
≅ Ω^{1−i}(Ω^{s}(M↓_{H_{i}}^{G}))↑_{H_{i}}^{G} ⊕ Q
≅ Ω^{s+1−i}(k)↑_{H_{i}}^{G} ⊕ Q′

for some projective modules Q and Q′. Now the important thing to remember is that kG is a self injective algebra and hence projective modules are also injective. As a result, if a projective module is a direct summand of a section of a module V, then it is a direct summand of V. The consequence of this is that (after stripping away the unnecessary projective modules Q′) we can get that, for some projective module R, the module Ω^{s}(M) ⊕ Ω^{s+1−r}(M) ⊕ R has a filtration

{0} = X_{0} ⊆ X_{1} ⊆ ⋯ ⊆ X_{r} ≅ Ω^{s}(M) ⊕ Ω^{s+1−r}(M) ⊕ R

where X_{i}/X_{i−1} ≅ Ω^{s+1−i}(k)↑_{H_{i}}^{G}. The statement about dimensions follows immediately. □

8. Special cases of 2-groups of small order

In this section we consider some special cases of 2-groups that we need to treat separately, for they are not covered by the general argument of Section 10. For each of the groups we show directly Theorem 1.4. We discuss the groups of order 8, the almost extraspecial group D_{8} ∗ C_{4} of order 16 and the extraspecial group D_{8} ∗ D_{8} of order 32 (type 1).

Let us start with the groups of order 8. First Q_{8} is excluded by assumption (and there is actually a non-trivial critical kQ_{8}-module of dimension 5, see [CaTh]). For G = D_{8} the structure of T (D_{8}) is known (see [CaTh]) and every non-trivial endo-trivial kD_{8}-module is non-trivial on restriction to one of the two elementary abelian 2-subgroups of D_{8}. Thus the only critical module is the trivial one. Alternatively, we can also prove the result in the following way. Proposition 8.1. Let G = D_{8}. Then there exists no non-trivial critical kGmodule.

Proof. Let M be a critical kG-module. By Theorem 3.4, the number of cohomology classes whose product vanishes is equal to t_{G} = 2. Applying now Theorem 7.2 with s = 1, we get

Dim Ω^{1}(M) + Dim M ≤ Dim Ω^{1}(k_{H_{1}})↑_{H_{1}}^{G} + Dim k↑_{H_{2}}^{G}

for some maximal subgroups H_{1} and H_{2}. Since H_{i} has order 4, Ω^{1}(k_{H_{i}}) has dimension 3 and we obtain

Dim Ω^{1}(M) + Dim M ≤ 6 + 2 = 8 .

By Lemma 5.4, Dim Ω^{1}(M) = Dim M − 2. So Dim M ≤ 5. This part of the argument is essentially the same as the one appearing in Theorem 5.3 of [CaTh]).

If we assume now that there exists a non-trivial critical kG-module, then by Theorem 6.4, there exists a non-trivial critical kG-module M of dimension

Dim M > \frac{|G|}{2} · \frac{|O_{G}|}{|C|} = 4 · 2 = 8 ,

since |O_{G}| = 2 by Proposition 6.5. This contradicts the previous upper bound. □

We turn now to the group G = D_{8} ∗ D_{8} of order 32. Proposition 8.2. Let G = D_{8} ∗ D_{8}. Then there exists no non-trivial critical kG-module.

Proof. Let M be a critical kG-module. By Theorem 3.4, the number of cohomology classes whose product vanishes is equal to t_{G} = 3. Applying now Theorem 7.2 with s = 1, we get

Dim Ω^{1}(M) + Dim Ω^{−1}(M) ≤ Dim Ω^{1}(k)↑_{H_{1}}^{G} + Dim k↑_{H_{2}}^{G} + Dim Ω^{−1}(k)↑_{H_{3}}^{G}

for some maximal subgroups H_{1}, H_{2}, and H_{3}. Since H_{i} has order 16, Ω^{±1}(k_{H_{i}}) has dimension 15 and we obtain

Dim Ω^{1}(M) + Dim Ω^{−1}(M) ≤ 30 + 2 + 30 = 62 ,

so that Dim Ω^{1}(M) ≤ 62.

If we assume now that there exists a non-trivial critical kG-module, then by Theorem 6.4, there exists a non-trivial critical kG-module M of dimension

Dim M > \frac{|G|}{2} · \frac{|O_{G}|}{|C|} ≥ 16 · \frac{72}{9} = 128 ,

using Proposition 6.5. So Dim Ω^{1}(M) > 126 by Lemma 5.4, a contradiction. □

In the last case, G is the almost extraspecial group D_{8} ∗ C_{4} of order 16. The method of the previous cases does not work because the orthogonal group O_{G} is too small. Instead of using the action of O_{G}, we shall give an argument using the action of a Galois group. Lemma 8.3. Let G = D_{8} ∗ C_{4}. If M is a critical kG-module, then Dim M ≤ 17.

Proof. By Theorem 3.4, the number of cohomology classes whose product vanishes is equal to t_{G} = 3. Applying now Theorem 7.2 with s = 1, we get

Dim Ω^{1}(M) + Dim Ω^{−1}(M) ≤ Dim Ω^{1}(k)↑_{H_{1}}^{G} + Dim k↑_{H_{2}}^{G} + Dim Ω^{−1}(k)↑_{H_{3}}^{G}

for some maximal subgroups H_{1}, H_{2}, and H_{3}. Since H_{i} has order 8, Ω^{±1}(k) has dimension 7 and we obtain

Dim Ω^{1}(M) + Dim Ω^{−1}(M) ≤ 14 + 2 + 14 = 30 .

By Lemma 5.4, Dim Ω^{1}(M) = Dim Ω^{−1}(M) = Dim M −2. So Dim Ω^{1}(M) ≤ 15 and Dim M ≤ 17. □

Proposition 8.4. Let G = D_{8} ∗ C_{4}. Then there exists no non-trivial critical kG-module.

Proof. Suppose that there is such a module N. We need to look at V\overline{G}(\overline{N}) ⊆ V\overline{G}(k) ≅ k^{3}. Suppose that p = (α, β, γ) is a point in V\overline{G}(\overline{N}). By dividing by α we may assume that α = 1, so that p = (1, β, γ) ∈ V\overline{G}(\overline{N}). Notice that p ∉ res_{G,H}^{∗}(V\overline{H}(\overline{N}_{H})) for any maximal subgroup H since \overline{N}_{H} is a free k\overline{H}-module. Therefore p is not in any 𝔽_{2}-rational subspace of k^{3}, and hence β and γ can not both be in the field with four elements (otherwise 1, β, γ would be linearly independent over 𝔽_{2}). It follows that if F : k^{3} ⟶ k^{3} is the Frobenius map, F (a, b, c) = (a^{2}, b^{2}, c^{2}), then p, F (p) and F^{2}(p) lie on different lines in V_{G}(k).

Next we need to notice that using the Frobenius homomorphism we can create a new module from N, by letting it act on the coefficients of the action of the elements of G on N. That is, if the module N is defined by a representation G ⟶ GL(N), and if we consider the homomorphism F : GL(N) ⟶ GL(N) that takes a matrix (a_{ij}) to (a_{ij}^{2}), we let N^{F} be the module defined by the composition. It is not difficult to see that N^{F} is also critical. Moreover, F (p) is a point in V\overline{G}(\overline{N^{F}}). It follows that the lines through p, F (p), and F^{2}(p) are all lines in the variety of the quotient module \overline{L} for some non-trivial critical module L. Thus by Theorem 5.9, kG has a non-trivial critical module of dimension at least 25. This contradicts Lemma 8.3. □

9. The groups of order p^{3} for odd p

When the prime p is odd, there is one special case in the proof of Theorem 1.4 that must be handled with extra care. This involves the groups of order p^{3}. The problem is that the general estimates of the dimensions of critical modules that we use later are not sufficient to handle this case. The result that we want is the following.

Proposition 9.1. Let G = G_{1}, an extraspecial group of order p^{3} and exponent p, for p an odd prime. Then there exists no non-trivial critical kG-module.

The proof proceeds in several steps. Throughout assume that a non-trivial critical kG-module exists and use Theorem 6.4 to obtain one of large dimension, as follows. Lemma 9.2. If a non-trivial critical kG-module exists, then there exists a critical kG-module M whose dimension is at least equal to (p − 1)p^{4} + 1. Moreover Dim Ω(M) ≥ (p − 1)p^{4} − 1 and Dim Ω^{−1}(M) ≥ (p − 1)p^{4} − 1.

Proof. By Theorem 6.4 there exists a critical module M whose dimension is at least |G| |Sp(2, 𝔽_{p})|/|C| where C is a cyclic p′-subgroup of the symplectic group Sp(2, 𝔽_{p}) of maximal order. Now Sp(2, 𝔽_{p}) = SL(2, 𝔽_{p}) has order p(p^{2} − 1) and its cyclic p′-subgroup of maximal order has order p + 1. So the dimension of M must be greater than (p − 1)p^{4} and must be congruent to 1 modulo p.

Now to compute the dimension of Ω(M), we notice from the proof of Theorem 6.4 that the variety of the module \overline{M} is the union of at least p(p − 1) distinct lines in V\overline{G}(k) = k^{2}. So \overline{M} = \overline{U}_{1}⊕…⊕\overline{U}_{t} where, for each i, V\overline{G}(\overline{U}_{i}) is a single line and t > p(p − 1). Now, as in the proof of Theorem 5.9, Dim \overline{U}_{i} = r_{i}p^{2} for some r_{i} (we use here the fact that U_{i} is endo-trivial and p is odd). Because \overline{U}_{i} is not a free k\overline{G}-module (and, in fact, has no free submodules) and because a projective cover of \overline{U}_{i} has dimension p^{2} Dim \overline{U}_{i}/Rad(\overline{U}_{i}), we must have Dim \overline{U}_{i}/Rad(\overline{U}_{i}) > r_{i}. Therefore \overline{U}_{i} is minimally generated by at least r_{i} + 1 generators and the number of generators of \overline{M} is at least

m = ∑_{i=1}^{t}(r_{i} + 1) = (∑_{i=1}^{t} r_{i}) + t .

Now \overline{M}/Rad(\overline{M}) is a quotient of M/Rad(M), so the minimal number of generators of M is at least m. As a result, the number of copies of kG appearing in the projective cover of M must be at least m. Now the dimension of M is p^{3}(∑_{i=1}^{t} r_{i}) + 1 and so the dimension of Ω(M) is at least

p^{3}m − Dim(M) = p^{3}((∑_{i=1}^{t} r_{i}) + t) − p^{3}(∑_{i=1}^{t} r_{i}) − 1 = tp^{3} − 1 ≥ (p − 1)p^{4} − 1 .

By applying the same argument to the dual module M^{∗} (which also satisfies the properties we need), we obtain

Dim Ω^{−1}(M) = Dim Ω^{−1}(M)^{∗} = Dim Ω(M^{∗}) ≥ (p − 1)p^{4} − 1 .

This proves the Lemma. □

Lemma 9.3. We have that

Dim Ω^{2p}(M) + Dim Ω^{−1}(M) ≤ p^{3}(p^{2} + p + 1).

Proof. ¿From any one of the papers [Le1, Ya, BeCa] we have that there exist η_{1}, … η_{p+1} ∈ H^{1}(G, k) such that β(η_{1}) ⋯ β(η_{p+1}) = 0. In Leary [Le1] the relation is given as x^{p}x′ − xx′^{p} = 0. Now applying Theorem 7.2 with t = p + 1 (hence r = 2t = 2(p + 1)) and choosing s = 2p in that Theorem, we get

Dim Ω^{2p}(M) + Dim Ω^{−1}(M) ≤ ∑_{i=1}^{2p+2} Dim(Ω^{2p+1−i}(k)↑_{H_{i}}^{G}) ,

where H_{i} is a maximal subgroup of G corresponding to the appropriate η_{j}. In our case, every H_{i} is an elementary abelian group of order p^{2}, and hence the dimensions on the right hand side of the inequality are independent of the particular η_{j}. Because Dim H^{j}(H_{i}, k) = j + 1 (see Lemma 4.6), we have that (for H_{i} = H)

Dim Ω^{2j−1}(k_{H}) + Dim Ω^{2j}(k_{H}) = p^{2} Dim H^{2j−1}(H, k) = p^{2}(2j) .

Induction to G multiplies the dimensions by p. Consequently the right hand side of the above inequality has the form

∑_{i=1}^{2p+2} Dim(Ω^{2p+1−i}(k)↑_{H_{i}}^{G})

= p Dim Ω^{−1}(k_{H}) + p Dim k + p ∑_{j=1}^{p} ( Dim Ω^{2j−1}(k_{H}) + Dim Ω^{2j}(k_{H}))

= p(p^{2} − 1 + 1 + ∑_{j=1}^{p} 2p^{2}j )

= p^{3} + 2p^{3}(p)(p + 1)/2 = p^{3}(1 + p^{2} + p)

as desired. □

At this point we should notice that the two lemmas above are not sufficient to give us the contradiction that we want. We need some further analysis of the dimension of Ω^{2p}(M). For this purpose we need to recall that there exists an element ζ ∈ H^{2p}(G, k) which has the property that its restriction res_{G,Z}(ζ) is not zero where Z = ⟨z⟩ is the center of G. In Leary’s paper [Le1], the element that he calls z will do. The element ζ can also be obtained by applying the Evens norm map to an element in the degree 2 cohomology of a maximal elementary abelian subgroup whose restriction to Z is not trivial.

The element ζ can be represented by a unique cocycle ζ : Ω^{2p}(k) ⟶ k. Hence we have an exact sequence

0 ⟶ L ⟶ Ω^{2p}(k) ⟶^{ζ} k ⟶ 0

where L is the kernel of ζ. Now by Theorem 2.2, V_{G}(L) = V_{G}(ζ), the variety of the ideal generated by ζ. In particular, the restriction L↓_{Z}^{G} is free as a kZ-module. This fact can also be derived from the observation that the above sequence is split as a sequence of kZ-modules because the restriction of ζ to Z is not zero and Ω^{2p}(k)↓_{Z}^{G} ≅ k ⊕ (free).

Let \overline{L} = L/(z − 1)L ≅ (z − 1)^{p−1}L. Then \overline{L} is a k\overline{G}-module where \overline{G} = G/Z.

Lemma 9.4. The k\overline{G}-module \overline{L} has no projective submodules, and moreover,

V\overline{G}(\overline{L}) ⊆ ⋃ res_{\overline{G},\overline{E}}^{∗}V\overline{E}(k)

where the union is over the set of all subgroups \overline{E} = E/Z where E is a maximal subgroup of G.

Notice that every maximal subgroup of G is elementary abelian and the union in the lemma is over all subgroups of order p in \overline{G}. Thus the right hand side of the containment is the union of all of the 𝔽_{p}- rational lines in V\overline{G}(k) ≅ k^{2}. It can be proved that the two sides are actually equal, but we do not need to know this.

Proof. If \overline{L} had a k\overline{G}-projective submodule then L and hence also Ω^{2p}(k) would have projective kG-submodules. That is, if t\overline{G} ; 1\overline{L} ≠ 0 then also t_{1}^{G}L ≠ 0. But clearly this is impossible.

Now suppose that ℓ ⊆ V\overline{G}(k) is a line that is not 𝔽_{p}-rational. Let N be a k\overline{G}-module such that V\overline{G}(N) = ℓ (e.g. take N = k\overline{G}/(σ − 1) where ⟨σ⟩ is a cyclic shifted subgroup corresponding to the line ℓ). Then the restriction N↓\overline{G} ; \overline{E} is a free k\overline{E}-module for any maximal subgroup E of G. So, viewing N as a kG-module by inflation, we have that V_{E}(N↓_{E}^{G}) is the line determined by the center Z, because Z acts trivially on N↓_{E}^{G}. Therefore N is periodic as a kGmodule and we must have that V_{G}(N) = res_{G,Z}^{∗}(V_{Z}(k)), the line determined by the center Z. Because L is free on restriction to Z we know that V_{G}(L)∩V_{G}(N) = {0} and hence L ⊗ N is a free kG-module. Now Z acts trivially on N and hence (z − 1)(L ⊗ N) = ((z − 1)L) ⊗ N. Hence \overline{L ⊗ N} ≅ \overline{L} ⊗ N is a free k\overline{G}-module. It follows from Theorem 2.2 that V\overline{G}(\overline{L}) ∩ V\overline{G}(N) = \{0\}. Hence the line ℓ is not in V\overline{G}(\overline{L}) and this holds for all lines in V\overline{G}(k) which are not 𝔽_{p}-rational. Thus the variety V\overline{G}(\overline{L}) must be contained in the union of the 𝔽_{p}-rational lines. □

Lemma 9.5. If M is a critical kG-module, V\overline{G}(\overline{M}) ∩ V\overline{G}(\overline{L}) = \{0\} and M ⊗ L ≅ L ⊕ (free).

Proof. We first show that \overline{M} ⊗ L is a free kG-module. That is, L is free as a kZ-module and \overline{\overline{M} ⊗ L} ≅ \overline{M} ⊗ \overline{L}. But from Lemmas 9.4 and 6.1 we have that V\overline{G}(\overline{M}) ∩ V\overline{G}(\overline{L}) = \{0\}. Hence \overline{M} ⊗ \overline{L} is free as a k\overline{G}-module. Thus \overline{M} ⊗ L is free as a kG-module.

It follows that M ⊗ L has a filtration

0 ⊆ ((z − 1)^{p−1}M) ⊗ L ⊆ … ⊆ ((z − 1)M) ⊗ L ⊆ M′ ⊗ L ⊆ M ⊗ L

where M′ = {m ∈ M|(z − 1)^{p−1}m = 0}. All of the factors are isomorphic to \overline{M} ⊗ L and hence are projective, except for the factor

(M′ ⊗ L)/((z − 1)M ⊗ L) ≅ (M′/(z − 1)M) ⊗ L ≅ k ⊗ L ≅ L .

The lemma follows from the fact that free modules are also injective and hence any free composition factor is a direct summand. □

Now tensoring the sequence given above with M we get an exact sequence

0 ⟶ M ⊗ L ⟶ M ⊗ Ω^{2p}(k) ⟶^{1⊗ζ} M ⟶ 0 .

Any projective submodule of M ⊗ L is also a direct summand of the middle term and can be factored out. So we have an exact sequence of the form

0 ⟶ L ⟶ Ω^{2p}(M) ⊕ P ⟶ M ⟶ 0 ,

for some projective module P. It remains to prove the following. Lemma 9.6. In the preceding exact sequence, the projective module P is zero.

Proof. Because the module L is free as a kZ-module the sequence is split as a sequence of kZ-modules. So multiplication by z − 1 is an exact functor on this sequence. Hence we have a sequence

0 ⟶ (z − 1)^{p−1}L ⟶ (z − 1)^{p−1}Ω^{2p}(M) ⊕ (z − 1)^{p−1}P ⟶ (z − 1)^{p−1}M ⟶ 0 ,

that is

0 ⟶ \overline{L} ⟶ \overline{Ω^{2p}(M)} ⊕ \overline{P} ⟶ \overline{M} ⟶ 0 ,

which is a sequence of k\overline{G}-modules. Because V\overline{G}(\overline{L}) ∩ V\overline{G}(\overline{M}) = \{0\} by the previous lemma, we must have that the sequence splits. So

\overline{L} ⊕ \overline{M} ≅ \overline{Ω^{2p}(M)} ⊕ \overline{P} .

But \overline{L} ⊕ \overline{M} has no projective k\overline{G}-submodules by Lemma 9.4. Hence \overline{P} = \{0\} and therefore also P = {0}. □

Proof of Proposition 9.1. By Lemma 9.6, Dim Ω^{2p}(M) = Dim L + Dim M. By Lemma 4.4, Dim Ω^{2p}(k) = p^{3}(p + 1) + 1, and so Dim L = p^{3}(p + 1) by definition of L. Now by Lemma 9.2, Dim M ≥ p^{4}(p−1)+1 and Dim Ω^{−1}(M) ≥ p^{4}(p−1)−1. Hence we have that

Dim Ω^{2p}(M)+Dim Ω^{−1}(M) ≥ p^{3}(p+1)+p^{4}(p−1)+1+p^{4}(p−1)−1 = p^{3}(2p^{2}−p+1) .

This inequality, however, is a contradiction to Lemma 9.3 since we are assuming that p ≥ 3.

10. The general case in characteristic 2

We are now prepared to prove the general case by induction and complete the proof of the detection Theorem 1.4 when p = 2. Throughout, k has characteristic 2. Let G be an extraspecial or almost extraspecial group of order 2^{m+1}. The theorem that we are trying to prove is the following. It is equivalent to Theorem 1.4. Theorem 10.1. If G is an extraspecial or almost extraspecial 2-group and if G is not isomorphic to Q_{8}, then there are no non-trivial critical kG-modules.

Three cases have to be treated seperately, namely the groups of order at most 16 as well as D_{8} ∗ D_{8}. But these cases have been dealt with in Section 8. Therefore we can now assume that m ≥ 4 and that m > 4 for the groups of type 1. This allows us to use Corollary 3.6.

The strategy of the proof is expressed in the following.

Proposition 10.2. Let G be an extraspecial or almost extraspecial group of order 2^{m+1}, with m = 2n. Assume that m ≥ 4 and m > 4 if G is of type 1. Let t_{G} be the number of cohomology classes whose product vanishes, as described in Theorem 3.4, and let

σ_{G} = \binom{t_{G} + m − 4}{m − 2}|G| + 2  and  τ_{G} = \frac{|G|}{2} · \frac{|O_{G}|}{3^{n}} .

If τ_{G} > σ_{G} then there exists no non-trivial critical kG-module.

Proof. Let t = t_{G}. In view of Theorem 3.4, there exists non-zero elements η_{1}, …, η_{t} ∈ H^{1}(G, 𝔽_{2}) such that η_{1} … η_{t} = 0 and each η_{i} corresponds to a maximal subgroup H_{i}. Moreover each subgroup H_{i} is the centralizer of a non-central involution in G and by Theorem 3.5, H_{i} ≅ C_{2} × U where U has the same type as G. So H_{i} ≅ H_{1} for each i.

Suppose that M is a critical kG-module. Then by Theorem 7.2 with t = t_{G} and s = t − 1, we have

Dim M ≤ Dim Ω^{t−1}(M) + Dim M ≤ ∑_{i=1}^{t} Dim (Ω^{t−i}(k)↑_{H_{i}}^{G}) .

Since all the subgroups H_{i} are isomorphic to H_{1}, we obtain

Dim M ≤ ∑_{j=0}^{t−1} Dim (Ω^{j}(k)↑_{H_{1}}^{G}) .

Now by Corollary 3.6, which applies in view of our assumption on m (with m, in the Corollary, replaced by m − 2 and r = t_{G} − 1), we obtain

∑_{j=0}^{t−1} Dim (Ω^{j}(k)↑_{H_{1}}^{G}) ≤ |G| · \binom{m − 2 + t_{G} − 1 − 1}{m − 2} + 2 = σ_{G} .

It follows that Dim M ≤ σ_{G}.

If we assume now that there exists a non-trivial critical kG-module, then by Theorem 6.4, there exists a non-trivial critical kG-module M of dimension

Dim M > \frac{|G|}{2} · \frac{|O_{G}|}{|C|} ≥ \frac{|G|}{2} · \frac{|O_{G}|}{3^{n}} = τ_{G} > σ_{G} .

This contradicts the upper-bound obtained above. □

We have now reduced the problem to the proof that τ_{G} > σ_{G} for all the groups G as above. This is a purely numerical problem which only requires estimating the numbers τ_{G} and σ_{G}. We start with a lemma which will be useful for estimating σ_{G}. Lemma 10.3. Let t and m be integers with t ≥ 4 and m ≥ 6. Then

\begin{matrix}( & {2t + m − 2 \atop m} & ) \\ ( & {t + m − 4 \atop m − 2} & ) \end{matrix} < 2^{m−3} t^{2} .

Proof. Expanding the left hand side and eliminating the common factor (m−2)!, we get the following expression. Notice that we can bound each of the first m−5 fractions by 2 (using m ≥ 6), the next three by 3 (using t ≥ 4), and bound 1/m(m − 1) by 1/30. Thus we get the following.

\frac{2t + m − 2}{t + m − 4} · \frac{2t + m − 3}{t + m − 5} · . . . · \frac{2t + 4}{t + 2} · \frac{2t + 3}{t + 1} · \frac{2t + 2}{t} · \frac{2t + 1}{t − 1} · \frac{2t}{m} · \frac{2t − 1}{m − 1}

< 2^{m−5} 3^{3} \frac{4t^{2}}{30} < 2^{m−3}t^{2} .

□

For the proof that τ_{G} > σ_{G}, we proceed with cases.

10.1. Groups of type 1. Let G_{n} = D_{8} ∗ ⋯ ∗ D_{8} be the central product of n copies of D_{8}, with n ≥ 3. Remember that the cases n = 1 and n = 2 were treated in Propositions 8.1 and 8.2. For convenience, we write G = G_{n} and let σ_{n} = σ_{G_{n}} and τ_{n} = τ_{G_{n}}. We prove that σ_{n} < τ_{n} by induction, starting with two cases.

If n = 3, then t = t_{G} = 5 by Theorem 3.4 and we have that

τ_{3} = 2^{6} · \frac{2^{7} · 7 · 3 · 15}{3^{3}} > 2^{13} · 11 ,

σ_{3} = \binom{5 + 6 − 4}{6 − 2} · 2^{7} + 2 = 35 · 2^{7} + 2 < 2^{13} · 11 < τ_{3} .

If n = 4, then t = t_{G} = 9 by Theorem 3.4 and

τ_{4} = 2^{8} · \frac{2^{13} · 15 · 3 · 15 · 63}{3^{4}} = 2^{21} · 525 ,

σ_{4} = \binom{9 + 8 − 4}{8 − 2} · 2^{9} + 2 = 1716 · 2^{9} + 2 < τ_{4} .

For n ≥ 4, we have t_{G_{n+1}} = 2t_{G_{n}} by Theorem 3.4, and this allows for an inductive argument. We assume that σ_{n} < τ_{n} and we prove that σ_{n+1} < τ_{n+1}. The course of our proof is to show that

\frac{σ_{n+1} − 2}{σ_{n} − 2} < 2^{4n} < \frac{τ_{n+1}}{τ_{n}}

from which we get σ_{n+1} − 2 < 2^{4n}σ_{n} − 2^{4n+1} < 2^{4n}τ_{n} − 2 < τ_{n+1} − 2 and we are done. So we are left with the proof of the two inequalities above.

¿From the value of τ_{n} given by Proposition 6.5, we obtain

\frac{τ_{n+1}}{τ_{n}} = \frac{|G_{n+1}|}{|G_{n}|} · \frac{2^{(n+1)n+1}}{2^{n(n−1)+1}} · \frac{2^{n+1} − 1}{2^{n} − 1} · \frac{2^{2n} − 1}{3}

> 2^{2} · 2^{2n} · 2 · \frac{2^{2n} − 1}{4} = 2^{4n+1} − 2^{2n+1} > 2^{4n} .

On the other hand, setting m = 2n and t_{G_{n}} = t_{n} = 2^{n−1} + 2^{n−4} (Theorem 3.4), we obtain by Lemma 10.3

\frac{σ_{n+1} − 2}{σ_{n} − 2} = \frac{|G_{n+1}|}{|G_{n}|} · \begin{matrix}( & {2t_{n} + m − 2 \atop m} & ) \\ ( & {t_{n} + m − 4 \atop m − 2} & ) \end{matrix}

< 4 · 2^{m−3} · t_{n}^{2} = 2^{2n−1}(2^{n−1} + 2^{n−4})^{2} < 2^{2n−1} 2^{2n} < 2^{4n} .

10.2. Groups of type 2. Let G_{n} = D_{8} ∗ ⋯ ∗ D_{8} ∗ Q_{8} be the central product of n − 1 copies of D_{8} and one of Q_{8}, with n ≥ 2. Let G = G_{n}, σ_{n} = σ_{G_{n}}, τ_{n} = τ_{G_{n}}, and t_{n} = t_{G_{n}} = 2^{n} + 2^{n−2} (see Theorem 3.4).

We start with the case n = 2, for which we need to replace τ_{2} by the slightly larger value

τ′_{2} = \frac{|G_{2}|}{2} · \frac{|O_{G}|}{|C|}

where C is a cyclic subgroup of O_{G} of maximal odd order. By Corollary 12.43 of Taylor’s book [Ta], O_{G} has a simple subgroup of index 2 isomorphic to P SL(2, 𝔽_{4}) (that is, A_{5} and in fact O_{G} is isomorphic to the symmetric group S_{5}). Therefore |C| = 5 and we get τ′_{2} = 384. On the other hand t_{2} = 5 and we have

σ_{2} = \binom{5 + 4 − 4}{4 − 2} · 2^{5} + 2 = 322 < τ′_{2} .

The argument of Proposition 10.2 goes through with τ′_{2} instead of τ_{2}. Now we prove that σ_{n} < τ_{n} by induction, starting with n = 3:

τ_{3} = 2^{6} · \frac{2^{7} · 9 · 3 · 15}{3^{3}} = 2^{13} · 15 ,

σ_{3} = \binom{10 + 6 − 4}{6 − 2} · 2^{7} + 2 = 495 · 2^{7} + 2 < τ_{3} .

If now n ≥ 3 the course of our proof is to show that

\frac{σ_{n+1} − 2}{σ_{n} − 2} < 2^{4n} − 2^{2n} < \frac{τ_{n+1}}{τ_{n}}

from which we conclude the proof as in the previous case. Here is the computation:

\frac{τ_{n+1}}{τ_{n}} = \frac{|G_{n+1}|}{|G_{n}|} · \frac{2^{(n+1)n+1}}{2^{n(n−1)+1}} · \frac{2^{n+1} + 1}{2^{n} + 1} · \frac{2^{2n} − 1}{3}

> 2^{2} · 2^{2n} · 1 · \frac{2^{2n} − 1}{4} = 2^{4n} − 2^{2n} .

On the other hand, we obtain by Lemma 10.3

\frac{σ_{n+1} − 2}{σ_{n} − 2} = \frac{|G_{n+1}|}{|G_{n}|} · \begin{matrix}( & {2t_{n} + m − 2 \atop m} & ) \\ ( & {t_{n} + m − 4 \atop m − 2} & ) \end{matrix}

< 4 · 2^{m−3} · t_{n}^{2} = 2^{2n−1}(2^{n} + 2^{n−2})^{2}
= 2^{4n−1} + 2^{4n−2} + 2^{4n−5} + 2^{2n} − 2^{2n} < 2^{4n} − 2^{2n} .

10.3. Groups of type 3. Let G_{n} = D_{8} ∗ ⋯ ∗ D_{8} ∗ C_{4} be the central product of n copies of D_{8} and one of C_{4}. Let G = G_{n}, σ_{n} = σ_{G_{n}}, τ_{n} = τ_{G_{n}}, and t_{n} = t_{G_{n}} = 2^{n} + 2^{n−2} (see Theorem 3.4). Note that m = 2n + 1 for type 3.

We prove that σ_{n} < τ_{n} by induction, starting with n = 2. Remember that the case in which n = 1 was treated in Proposition 8.4. First we have that

τ_{2} = 2^{5} · \frac{2^{4} · 3 · 15}{3^{2}} = 2560 ,

σ_{2} = \binom{5 + 5 − 4}{5 − 2} · 2^{6} + 2 = 1282 < τ_{2} .

If now n ≥ 2 we show that

\frac{σ_{n+1} − 2}{σ_{n} − 2} < 2^{4n+2} < \frac{τ_{n+1}}{τ_{n}}

from which we make our conclusions as in the previous cases. Here is the computation. First note that

\frac{τ_{n+1}}{τ_{n}} = \frac{|G_{n+1}|}{|G_{n}|} · \frac{2^{(n+1)^{2}}}{2^{n^{2}}} · \frac{2^{2(n+1)} − 1}{3}

> 2^{2} · 2^{2n+1} · \frac{2^{2n+1}}{4} = 2^{4n+2} .

On the other hand, we obtain by Lemma 10.3

\frac{σ_{n+1} − 2}{σ_{n} − 2} = \frac{|G_{n+1}|}{|G_{n}|} · \begin{matrix}( & {2t_{n} + m − 2 \atop m} & ) \\ ( & {t_{n} + m − 4 \atop m − 2} & ) \end{matrix}

< 4 · 2^{m−3} · t_{n}^{2} = 2^{2n}(2^{n} + 2^{n−2})^{2} < 2^{2n} 2^{2(n+1)} = 2^{4n+2} .

This completes the proof of Theorem 10.1 and hence also the proof of Theorem 1.4 when p = 2.

11. The general case in odd characteristic

In this section we complete the proof of Theorem 1.4 for odd p. We assume throughout that the field k has characteristic p and that G = G_{n} is an extraspecial group of order p^{2n+1} and exponent p. Our aim is to prove the following. Theorem 11.1. If G = G_{n}, then there are no non-trivial critical kG-modules.

If n = 1, the result follows from Section 9. Thus we can assume n ≥ 2. The proof follows the same basic pattern as in the last section. We define σ_{n} and τ_{n} such that σ_{n} is an upper bound for the dimension of any critical module and τ_{n} is a lower bound for the dimension of some non-trivial critical module if non-trivial critical modules exist. Then we prove that σ_{n} < τ_{n}. First we give the definitions.

For n ≥ 2 let

σ_{n} = 2|G_{n}|\binom{t_{n} + 2n − 3}{2n − 1}

where t_{n} = 2(p^{2} + p − 1)p^{n−2}. Let τ_{n} be given by the rule

τ_{n} = \frac{|G_{n}| |Sp(2n, 𝔽_{p})|}{c_{n}}

where c_{n} = (p + 1)^{n} except in the case in which p = 3 and n = 2. In that case let c_{n} = p^{2} + 1 = 10. Then we have the following. Proposition 11.2. If n ≥ 2 and τ_{n} > σ_{n}, then there exists no non-trivial critical kG_{n}-module.

Proof. Let t = t_{n}/2 = (p^{2} + p − 1)p^{n−2}. By Theorem 4.1, we know that there exists non-zero elements η_{1}, …, η_{t} ∈ H^{1}(G, 𝔽_{2}) such that β(η_{1}) ⋯ β(η_{t}) = 0 and each η_{i} corresponds to a maximal subgroup H_{i}. Moreover each subgroup H_{i} is the centralizer of a non-central element of order p in G and by Theorem 4.7, H_{i} ≅ C_{p} × G_{n−1}. So H_{i} ≅ H_{1} for each i.

If M is a critical kG_{n}-module, then by Theorem 7.2 with r = 2t = t_{n} and s = t_{n} − 1, we have that

Dim M ≤ Dim Ω^{t_{n}−1}(M) + Dim M ≤ ∑_{i=1}^{t_{n}} Dim (Ω^{t_{n}−i}(k)↑_{H_{i}}^{G}) .

Since all the subgroups H_{i} are isomorphic to H_{1}, we obtain

Dim M ≤ ∑_{j=0}^{t_{n}−1} Dim (Ω^{j}(k)↑_{H_{1}}^{G}) .

By Corollary 4.8, we get that

∑_{j=0}^{t_{n}−1} Dim (Ω^{j}(k)↑_{H_{1}}^{G}) ≤ 2|G| · \binom{t_{n} − 1 + 2n − 2}{2n − 1} = σ_{n} .

It follows that Dim M ≤ σ_{n}.

On the other hand, if we assume that there exists a non-trivial critical kG_{n}- module, then by Theorem 6.4, there exists a non-trivial critical kG-module M of dimension

Dim M > |G| · \frac{|O_{G}|}{|C|}

where C is a cyclic p′-subgroup in Sp(2n, 𝔽_{p}) of maximal order. In the case that p = 3 and n = 2, we know from character tables or from direct analysis on Sp(4, 𝔽_{3}) that C has order at most 10. In all other cases we know by Proposition 6.5 that the order of C is at most (p + 1)^{n}. So in either case, Dim M > τ_{n}. Hence if σ_{n} < τ_{n} then we have a contradiction. □

So it remains to prove that τ_{n} > σ_{n}. We will proceed by induction beginning with the following. Lemma 11.3. τ_{2} > σ_{2}.

Proof. If p = 3, then σ_{2} = 860, 706 while τ_{2} = 1, 259, 712, so the lemma holds in that case (note that it is here that we need the special choice made for τ_{2}). So suppose that p ≥ 5. Then

\frac{σ_{2}}{p^{11}} = \frac{1}{p^{11}} 2p^{5}\binom{2(p^{2} + p − 1) + 1}{3}

= \frac{2}{3!}(2 + \frac{2}{p} − \frac{1}{p^{2}} )(2 + \frac{2}{p} − \frac{2}{p^{2}} )(2 + \frac{2}{p} − \frac{3}{p^{2}} ) < \frac{1}{3}3 · 3 · 3 = 9 .

On the other hand,

\frac{τ_{2}}{p^{11}} = \frac{p^{5}}{p^{11}} \frac{p^{4}(p^{2} − 1)(p^{4} − 1)}{(p + 1)^{2}} = p^{2} \frac{(1 − \frac{1}{p^{2}} )(1 − \frac{1}{p^{4}} )}{(1 + \frac{1}{p})^{2}} > p^{2} \frac{(\frac{4}{5})^{2}}{(\frac{4}{3})^{2}} = 9\frac{p^{2}}{25} > 9

using the fact that p ≥ 5 and hence 1 + 1/p < 4/3 and 1 − 1/p ≥ 4/5. So again τ_{2} > σ_{2}. □

Lemma 11.4. For n ≥ 2 we have that \frac{σ_{n+1}}{σ_{n}} < \frac{τ_{n+1}}{τ_{n}}.

Proof. Notice first that a special computation is needed if p = 3 and n = 2. In that case, by direct calculation, we have that σ_{3} = 49, 157, 255, 862 while τ_{3} = 313, 380, 128, 880. It is then easy to check the lemma in this particular case.

More generally, we calculate that

\frac{τ_{n+1}}{τ_{n}} = \frac{|G_{n+1}|}{|G_{n}|} \frac{p^{(n+1)^{2}}}{p^{n^{2}}} \frac{(p^{2} − 1) ⋯ (p^{2n+2} − 1)/(p + 1)^{n+1}}{(p^{2} − 1) ⋯ (p^{2n} − 1)/(p + 1)^{n}}

= p^{2} · p^{2n+1} · (p^{2n+2} − 1)/(p + 1) > \frac{1}{2}p^{4n+4}.

The above estimate is that, since p ≥ 3, we have 1/(p + 1) > 1/(\sqrt{2}p) and p^{2n+2} − 1 > p^{2n+2}/\sqrt{2}.

At the same time, setting t = t_{n} and noting that t_{n+1} = pt_{n}, we have

\frac{σ_{n+1}}{σ_{n}} = \frac{2p^{2n+3}}{2p^{2n+1}} \begin{matrix}( & {tp+2n−1 \atop 2n+1} & ) \\ ( & {t+2n−3 \atop 2n−1} & ) \end{matrix}

= \frac{p^{2}}{(2n + 1)(2n)}(tp + 2n − 1)(tp + 2n − 2)\frac{(tp + 2n − 3)}{(t + 2n − 3)} ⋯ \frac{tp}{t} \frac{(tp − 1)}{(t − 1)} .

Now we note that (tp+b)/(t+b) ≤ tp/t = p for all b ≥ 0. Also (tp−1)/(t−1) < \frac{3}{2}p because t ≥ 3. Moreover,

tp + 2n − 1 = 2(p^{2} + p − 1)p^{n−2}p + 2n − 1

= 2p^{n+1} (1 + \frac{1}{p} − \frac{1}{p^{2}} + \frac{2n − 1}{2p^{n+1}})

< 2p^{n+1}(2) = 4p^{n+1}

So we have that

\frac{σ_{n+1}}{σ_{n}} < \frac{p^{2}}{(2n + 1)(2n)}(4p^{n+1})(4p^{n+1})(p^{2n−2})(\frac{3}{2}p)

= \frac{16 · 3/2}{(2n + 1)(2n)}p^{4n+3} ≤ \frac{24}{20}p^{4n+3} < \frac{1}{2}p^{4n+4} .

Finally \frac{σ_{n+1}}{σ_{n}} < \frac{1}{2}p^{4n+4} < \frac{τ_{n+1}}{τ_{n}}, as required. □

Proof of Theorem 11.1. Remember that the case in which n = 1 was treated in Proposition 9.1. We have shown that τ_{2} > σ_{2} and that τ_{n+1}/τ_{n} > σ_{n+1}/σ_{n} for all n ≥ 2. So, by induction, assume that τ_{n} > σ_{n} We get that τ_{n+1} = (τ_{n+1}/τ_{n})τ_{n} > (σ_{n+1}/σ_{n})σ_{n} = σ_{n+1}. Therefore, τ_{n} > σ_{n} for all n. The Theorem follows from Proposition 11.2.

The proof of Theorem 1.4 is now complete in all cases.

12. The detection theorem and the vanishing theorem

Having now settled Theorem 1.4, we can move to the main detection theorem (Theorem 1.2) and the vanishing theorem (Theorem 1.1). Recall that they assert that if G is not cyclic, quaternion or semi-dihedral, then T (G) is detected on restriction to all elementary abelian subgroups E of rank 2, and that the torsion subgroup of T (G) is trivial.

Let us first prove a general version of the detection theorem. Theorem 12.1. For any p-group G, the restriction homomorphism

∏_{H} Res_{H}^{G} : T (G) ⟶ ∏_{H} T (H)

is injective, where H runs through the set of all subgroups of G which are elementary abelian of rank 2, cyclic of order p with p odd, cyclic of order 4, and quaternion of order 8.

Proof. First note that that there is nothing to prove if G is cyclic of order 1 or 2, because T (G) = {0}. There is also nothing to prove if G is in the detecting family of the statement. So we can assume that G is not elementary abelian of rank 2, C_{p}, C_{4}, nor Q_{8}. By an obvious induction argument, it suffices to prove that

∏_{H} Res_{H}^{G} : T (G) ⟶ ∏_{H} T (H)

is injective, where H runs through the set of all maximal subgroups of G.

If G = C_{p^{n}} is cyclic (with n ≥ 2 for p odd and n ≥ 3 if p = 2), then Res_{C_{p^{n−1}}}^{C_{p^{n}}} : T (C_{p^{n}}) ⟶ T (C_{p^{n−1}}) is an isomorphism (both groups are isomorphic to ℤ/2ℤ generated by the class of Ω^{1}(k)). If G is extraspecial or almost extraspecial, the result follows from the main theorem of this paper (Theorem 1.4) if either p = 2 or p is odd and G has exponent p. If p is odd and G has exponent p^{2} (extraspecial or almost extraspecial), then the result was proved in Section 4 of [CaTh].

So we can assume that G is neither cyclic, nor elementary abelian of rank 2, nor extraspecial, nor almost extraspecial. In that case, the result was proved as Theorem 3.2 of [CaTh]. □

This theorem provides a direct proof of the following result, which was first proved by Puig [Pu] using an argument of commutative algebra. Corollary 12.2. The abelian group T (G) is finitely generated.

As observed by Puig, this easily implies the finite generation of the Dade group of all endo-permutation modules (see Corollary 2.4 in Puig [Pu]).

Proof. T (H) is finitely generated whenever H is in the detecting family. Now a subgroup of a finitely generated group is finitely generated. □

Theorem 12.1 is the intermediate statement which we need for our inductive proof of Theorem 1.2. We first need to prove the result in two special cases. Proposition 12.3. Suppose that G ≅ Q_{8} × C_{2} or G ≅ D_{8} ∗ C_{4}. Then T (G) is detected on restriction to all elementary abelian subgroups E of rank 2.

Proof. Suppose that M is a non-trivial endo-trivial module such that M↓_{E}^{G} ≅ k ⊕ (free) for every elementary abelian subgroup E of rank 2. Assume that M has minimal dimension among such modules. On restriction to a maximal subgroup of the form C_{4}×C_{2}, we must have that M↓_{C_{4}×C_{2}}^{G} ≅ k ⊕ (free), because T (C_{4}×C_{2}) ⟶ T (E) is an isomorphism for E = C_{2}×C_{2} ⊂ C_{4}×C_{2}. It follows that Dim(M) ≡ 1 (mod 8). It also follows that M↓_{C_{4}}^{G} ≅ k ⊕ (free) for any cyclic subgroup C_{4}, because C_{4} is contained in a maximal subgroup of the form C_{4}×C_{2}.

Since M is non-trivial, it must be detected on some restriction (Theorem 12.1). So there exists a quaternion subgroup H ≅ Q_{8} in G such that M↓_{H}^{G} is nontrivial. Then M↓_{H}^{G} ≅ Ω^{2}(k_{H})⊕(free), because Ω^{2}(k_{H}) is the only indecomposable endo-trivial kH-module other than k_{H} itself whose dimension is congruent to 1 modulo 8 (see [CaTh], Section 6).

Let z be the generator of the center of H (which is also central in G). We consider the variety V\overline{G}(\overline{M}) ⊆ V\overline{G}(k) ≅ k^{3} where, as in Section 5, \overline{G} = G/⟨z⟩ and \overline{M} ≅ (z − 1)M. On restriction to H, we have V\overline{H}(\overline{M}) = V\overline{H}( \overline{Ω^{2}(k_{H})} ) and \overline{Ω^{2}(k_{H})} is a periodic k\overline{H}-module by Lemma 5.3. Since Ω^{2}(k_{H}) is invariant under Galois automorphisms, so is \overline{Ω^{2}(k_{H})}, and therefore V\overline{H}(\overline{M}) is a union of lines permuted by Galois automorphisms. But these lines are not 𝔽_{2}-rational (by Lemma 6.1 applied to the kH-module Ω^{2}(k_{H}), which is critical), hence not fixed by Galois automorphisms. It follows that there are at least two lines in V\overline{H}(\overline{M}) (and in fact exactly two, which are 𝔽_{4}-rational, because this is the only possibility for the 4-dimensional module \overline{Ω^{2}(k_{H})} ). Now V\overline{G}(\overline{M}) also contains at least two lines since it contains res_{\overline{G},\overline{H}}^{∗}(V\overline{H}(\overline{M})). So \overline{M} ≅ \overline{M_{1}} ⊕ \overline{M_{2}} where V\overline{G}(\overline{M_{1}}) is one of the two lines. Now following the procedure as in Theorem 5.6 we can construct a non-trivial endo-trivial kG-module N_{1} such that \overline{N_{1}} = \overline{M_{1}}. Moreover N_{1} is trivial on restriction to every elementary abelian subgroup. But Dim(N_{1}) < Dim(M), contrary to the choice of M. □

We also need a group-theoretical lemma. Lemma 12.4. Let G be a semi-direct product G = Q_{2^{n}} ⋊ C_{2} for some n ≥ 3 and some action of C_{2} on Q_{2^{n}}. Then one of the following properties holds:

(a) G contains a semi-dihedral subgroup S such that S ⊇ Q_{8} ⊆ Q_{2^{n}}.

(b) G contains a subgroup Q_{8} ∗ C_{4} with Q_{8} ⊆ Q_{2^{n}}.

(c) G contains a subgroup Q_{8} × C_{2} with Q_{8} ⊆ Q_{2^{n}}.

Proof. Let u be a generator of C_{2}. We use induction on n and first consider the case n = 3. If the action of u on Q_{8}/Z(Q_{8}) is non-trivial, then we can choose two generators x and y of Q_{8} such that uxu^{−1} = y. In that case G is semi-dihedral and we are in case (a). If now u acts trivially on Q_{8}/Z(Q_{8}), then u fixes each of the three cyclic subgroups of order 4 of Q_{8}. If u acts trivially on Q_{8}, then G = Q_{8} × C_{2} and we are in case (c). Otherwise it easy to see that u must invert two of the cyclic subgroups of order 4 and fix pointwise the third one, say ⟨x⟩. But then the actions of u and x coincide, so ux^{−1} acts trivially and G = Q_{8} ∗ C_{4}, which is case (b).

Assume now that n ≥ 4. Let x and y be generators of Q_{2^{n}} with x^{2^{n−1}} = 1, y^{2} = x^{2^{n−2}} and yxy^{−1} = x^{−1}. All elements of the form x^{b}y have order 4 (where 0 ≤ b ≤ 2^{n−1}). Conjugation by u must satisfy uxu^{−1} = x^{a} for some odd integer a and uyu^{−1} = x^{b}y for some b. Since u^{2} = 1, we must have the following congruences modulo 2^{n−1} :

a ≡ ±1 , 2^{n−2} ± 1 and (a + 1)b ≡ 0 .

If a ≡ −1 and b is odd, we can replace x by x^{b} and we get a standard presentation of the semi-dihedral group SD_{2^{n+1}}, so we are in case (a). Otherwise b must be even, because this is forced by the condition (a + 1)b ≡ 0 if a ≢ −1. Therefore conjugation by u stabilizes the subgroup Q_{2^{n−1}} generated by x^{2} and y. The result now follows by induction applied to the group Q_{2^{n−1}} ⋊ ⟨u⟩. □

Now we come to the detection theorem (Theorem 1.2 of the introduction). Theorem 12.5. Suppose that G is a p-group which is not cyclic, quaternion, or semi-dihedral. Then T (G) is detected on restriction to all elementary abelian subgroups E of rank 2.

Proof. We use induction on the order of G. First recall that the result is known if G is abelian or dihedral (see [CaTh]), so we assume that G is neither abelian nor dihedral.

Let M be an endo-trivial module such that Res_{E}^{G}[M] = 0 for every elementary abelian subgroup E of rank 2, where [M] denote the class of M in T (G). It suffices to prove that Res_{H}^{G}[M] = 0 for every maximal subgroup H of G, because then [M] = 0 by Theorem 12.1. For every maximal subgroup H which is not cyclic, quaternion, or semi-dihedral, M↓_{H}^{G} satisfies the same assumption as M, so Res_{H}^{G}[M] = 0 by induction. So we are left with the cases where the maximal subgroup H is cyclic, quaternion, or semi-dihedral.

Assume first that H ≅ C_{p^{n}} is cyclic. By a well-known result of group theory (see Theorem 4.4 in Chapter 5 of [Go1]), G is either abelian, or isomorphic to a group P to be described below, or in addition when p = 2, isomorphic to D_{2^{n+1}}, Q_{2^{n+1}}, or SD_{2^{n+1}}. The cases of the cyclic group C_{p^{n+1}}, the quaternion group Q_{2^{n+1}}, or the semi-dihedral group SD_{2^{n+1}}, are excluded by our hypothesis. The cases of an abelian group or a dihedral group D_{2^{n+1}} have been already dealt with. So we are left with the case G = P = H ⋊ C_{p}, with respect to the action uxu^{−1} = x^{1+p^{n−1}}, where x is a generator of H and u is a generator of C_{p}. This case occurs if n ≥ 2 when p is odd and n ≥ 3 when p = 2. Now G also contains a maximal subgroup K = ⟨x^{p}⟩ × ⟨u⟩ ≅ C_{p^{n−1}} × C_{p} and we already know that Res_{K}^{G}[M] = 0. Therefore

Res_{C_{p^{n−1}}}^{G}[M] = Res_{C_{p^{n−1}}}^{K} Res_{K}^{G}[M] = 0 .

But we also have Res_{C_{p^{n−1}}}^{G} = Res_{C_{p^{n−1}}}^{H} Res_{H}^{G} and

Res_{C_{p^{n−1}}}^{H} : T (H) ⟶ T (C_{p^{n−1}})

is an isomorphism since both T (H) and T (C_{p^{n−1}}) are cyclic of order 2 generated by the class of Ω^{1}(k) (because n ≥ 2 and n ≥ 3 if p = 2). It follows that Res_{H}^{G}[M] = 0.

Assume now that H ≅ SD_{2^{n}} is semi-dihedral. We know that the torsion subgroup T_{t}(H) is cyclic of order 2 generated by the class of an endo-trivial module whose dimension is congruent to 1 modulo 2^{n−1} (see [CaTh], Section 7). This class cannot be in the image of Res_{H}^{G}, because all endo-trivial modules for G have dimension congruent to ±1 modulo 2^{n}, by Lemma 2.10 in [CaTh]. It follows that the image of Res_{H}^{G} is contained in ⟨ [Ω_{H}^{1}(k)] ⟩ ≅ ℤ, because T (H) = T_{t}(H) ⊕ ⟨ [Ω_{H}^{1}(k)] ⟩. But now the restriction map

Res_{E}^{H} : ⟨ [Ω_{H}^{1}(k)] ⟩ ⟶ T (E) = ⟨ [Ω_{E}^{1}(k)] ⟩

is an isomorphism where E is an elementary abelian subgroup of rank 2. Since Res_{E}^{H} Res_{H}^{G}[M] = 0, we must have Res_{H}^{G}[M] = 0 as required. Note that the same argument shows that Res_{S}^{G}[M] = 0 for any semi-dihedral subgroup S of G.

Assume finally that H ≅ Q_{2^{n}} is quaternion. Since G is neither cyclic nor quaternion, its 2-rank cannot be 1 (see Chapter 5 of [Go1]) and so there exists an element of order 2 outside H. Therefore G ≅ Q_{2^{n}} ⋊ C_{2} for some action of C_{2} on Q_{2^{n}}. By Lemma 12.4, G contains a subgroup R which is isomorphic to Q_{8}∗C_{4}, Q_{8} × C_{2}, or semi-dihedral, and such that R ⊇ Q_{8} ⊆ H. In the first two cases we have Res_{R}^{G}[M] = 0 by Proposition 12.3 and in the third we have Res_{R}^{G}[M] = 0 by the argument above. It follows that

Res_{Q_{8}}^{H} Res_{H}^{G}[M] = Res_{Q_{8}}^{G}[M] = Res_{Q_{8}}^{R} Res_{R}^{G}[M] = 0 .

We know that T (H) ≅ ℤ/4ℤ ⊕ ℤ/2ℤ, where ℤ/4ℤ is generated by the class of Ω_{H}^{1}(k) and ℤ/2ℤ is generated by the class of an endo-trivial module of dimension 2^{n−1} + 1 (see [CaTh], Section 6). Again this class cannot be in the image of Res_{H}^{G}, because all endo-trivial modules for G have dimension congruent to ±1 modulo 2^{n}. Thus the image of Res_{H}^{G} is contained in ℤ/4ℤ = ⟨ [Ω_{H}^{1}(k)] ⟩. But now the restriction map

Res_{Q_{8}}^{H} : ⟨ [Ω_{H}^{1}(k)] ⟩ ⟶ ⟨ [Ω_{Q_{8}}^{1}(k)] ⟩

is an isomorphism. Since Res_{Q_{8}}^{H} Res_{H}^{G}[M] = 0, we must have Res_{H}^{G}[M] = 0 as required. □

We immediately deduce the vanishing theorem (Theorem 1.1 of the introduction). Corollary 12.6. If G is not cyclic, quaternion or semi-dihedral, then the torsion subgroup of T (G) is trivial.

Proof. By the theorem, we know that T (G) is embedded in a product of copies of T (E) ≅ ℤ, where E is elementary abelian of rank 2. □

We can now prove Corollary 1.3 of the introduction. Corollary 12.7. Suppose that G is a finite p-group for which every maximal elementary subgroup has rank at least 3. Then T (G) ≅ ℤ, generated by the class of the module Ω^{1}(k).

Proof. The assumption implies that G cannot be cyclic, quaternion or semidihedral. Therefore, by the theorem, T (G) is detected on restriction to elementary abelian subgroups of rank 2. The rest of the proof follows Alperin [Al2] and we recall the argument (also used in [BoTh]). The partially ordered set of all elementary abelian subgroups of rank at least 2 is connected, in view of the assumption and using a well-known result of the theory of p-groups. For any such subgroup H, the restriction map T (H) ⟶ T (E) ≅ ℤ to an elementary abelian subgroup of rank 2 is an isomorphism. It follows that all restrictions to such rank 2 subgroups E are equal. □

13. The Dade group

In this section, we prove detection theorems for the Dade group D(G) of all endo-permutation modules and we determine its torsion subgroup when p is odd. We refer to [BoTh] for details about D(G). Let us only mention that the torsionfree rank of D(G) has been determined in [BoTh] so that we are particularly interested in the torsion subgroup D_{t}(G). We first state an easy consequence of Theorem 12.1. Theorem 13.1. Let G be a finite p-group.

(a) The product of all restriction-deflation maps

∏_{K/H} Def_{K/H}^{K} Res_{K}^{G} : D(G) ⟶ ∏_{K/H} D(K/H)

is injective, where K/H runs through the set of all sections of G which are elementary abelian of rank 2, cyclic of order p with p odd, cyclic of order 4, or quaternion of order 8.

(b) For the torsion subgroup, the product of all restriction-deflation maps

∏_{K/H} Def_{K/H}^{K} Res_{K}^{G} : D_{t}(G) ⟶ ∏_{K/H} D_{t}(K/H)

is injective, where K/H runs through the set of all sections of G which are cyclic of order p if p is odd, quaternion of order 8 or cyclic of order 4 if p = 2.

Proof. The argument is exactly the same as the one given in Theorem 1.6 of [BoTh] or in Theorem 10.1 of [CaTh]. □

We now deduce Corollary 1.6 of the introduction. Corollary 13.2. Let G be a finite p-group.

(a) If p is odd, any non-trivial torsion element in D(G) has order 2. In other words, for any indecomposable endo-permutation kG-module M with vertex G, the class of M is a torsion element if and only if M is self-dual.

(b) If p = 2, any non-trivial torsion element in D(G) has order 2 or 4.

Proof. The non-trivial elements of D(C_{p}) have order 2, while those of D(Q_{8}) and D(C_{4}) have order 2 or 4. Moreover, an element of order 2 corresponds to a self-dual module by definition of the group law. □

If p is odd, the detection theorem above allows for a complete description of the torsion subgroup of D(G) (Theorem 1.5 of the introduction), using the partial results already obtained in [BoTh]. Theorem 13.3. If p is odd and G is a finite p-group, the torsion subgroup of D(G) is isomorphic to (ℤ/2ℤ)^{s}, where s is the number of conjugacy classes of non-trivial cyclic subgroups of G.

Note that explicit generators are described in [BoTh].

Proof. Theorem 6.2 in [BoTh] asserts that a certain quotient \overline{D_{t}(G)} of the torsion subgroup D_{t}(G) is isomorphic to (ℤ/2ℤ)^{s}, where s is as above. So we only have to prove that \overline{D_{t}(G)} = D_{t}(G). But by definition, \overline{D_{t}(G)} = D_{t}(G)/ Ker(ψ), where ψ is the product of all restriction-deflation maps

ψ = ∏_{K/H} Def_{K/H}^{K} Res_{K}^{G} : D_{t}(G) ⟶ ∏_{K/H} D_{t}(K/H)

where K/H runs through the set of all sections of G which are cyclic of order p. Now ψ is injective by Theorem 13.1 and the result follows. □

Our purpose now is to improve Theorem 13.1 by restricting the kind of section needed on the right hand side. However, we will also change the target by including all groups having torsion endo-trivial modules, namely cyclic, quaternion, and semi-dihedral groups.

If S = ⟨x⟩ is cyclic of order p^{n}, then

D(S) = D_{t}(S) ≅ ∏_{i=1}^{n} T_{t}(S/⟨x^{p^{i}}⟩) ,

and we let π_{S} : D_{t}(S) → T_{t}(S) denote the projection onto the factor indexed by i = n. The situation is easier if S is a quaternion or semi-dihedral group, since D_{t}(S) = T_{t}(S) by [CaTh], Section 10. In this case, we write π_{S} : D_{t}(S) → T_{t}(S) for the identity map. Theorem 13.4. Let G be a finite p-group. If p is odd, let X be the class of all subgroups H of G such that N_{G}(H)/H is cyclic. If p = 2, let X be the class of all subgroups H of G such that N_{G}(H)/H is cyclic of order ≥ 4, quaternion of order ≥ 8, or semi-dihedral of order ≥ 16. Let [X /G] be a system of representatives of conjugacy classes of subgroups in X. Then the map

∏_{H∈[X /G]} π_{N_{G}(H)/H} Def_{N_{G}(H)/H}^{N_{G}(H)} Res_{N_{G}(H)}^{G} : D_{t}(G) ⟶ ∏_{H∈[X /G]} T_{t}(N_{G}(H)/H)

is injective.

Proof. Let φ denote the map in the statement and let a ∈ Ker(φ), so that π_{N_{G}(H)/H} Defres_{N_{G}(H)/H}^{G}(a) = 0 for every H ∈ X, where we write for simplicity Defres_{K/H}^{G} = Def_{K/H}^{K} Res_{K}^{G} for every section K/H. By Theorem 13.1 above, it suffices to prove that Defres_{K/H}^{G}(a) = 0 for every section K/H isomorphic to C_{p}, C_{4} or Q_{8}. We are going to show that Defres_{N_{G}(H)/H}^{G}(a) = 0 and the result will follow from this since Defres_{K/H}^{G} = Res_{K/H}^{N_{G}(H)/H} Defres_{N_{G}(H)/H}^{G}. For simplicity of notation, we write now L = N_{G}(H).

We use induction on the index |G : H|. If H has index p, there is nothing to prove because L = G, π_{G/H} = id, and

Defres_{G/H}^{G}(a) = π_{G/H} Defres_{G/H}^{G}(a) = 0 ,

by assumption if p is odd and by the fact that D(G/H) = {0} if p = 2. Let F be a subgroup such that H < F ≤ L. By induction, Defres_{N_{G}(F)/F}^{G} (a) = 0 and consequently Defres_{N_{L}(F)/F}^{G} (a) = 0. This holds for every such F and therefore

Defres_{L/H}^{G}(a) ∈ ⋂_{H<F ≤L} Ker(Defres_{N_{L}(F)/F}^{L/H}) = T (L/H) .

The last equality is a well-known characterization of T (L/H) as a subgroup of D(L/H) (see Lemma 2.1 in [CaTh] and note that this characterization is also at the heart of the proof of Theorem 13.1). Since a was chosen to be a torsion element in D(G), we have proved that Defres_{L/H}^{G}(a) ∈ T_{t}(L/H).

If L/H is not cyclic, quaternion, or semi-dihedral, then T_{t}(L/H) = {0} by Theorem 1.1 and so Defres_{L/H}^{G}(a) = 0. The same holds if L/H is cyclic of order 2. If L/H is quaternion or semi-dihedral, then π_{L/H} is the identity map and π_{L/H} Defres_{L/H}^{G}(a) = 0 by assumption, so Defres_{L/H}^{G}(a) = 0. If L/H is cyclic of order ≥ 3, then π_{L/H} : D_{t}(L/H) → T_{t}(L/H) restricts to the identity on T_{t}(L/H).

Since π_{L/H} Defres_{L/H}^{G}(a) = 0 by assumption, we obtain again Defres_{L/H}^{G}(a) = 0. □

In order to illustrate the efficiency of Theorem 13.4 compared to Theorem 13.1, suppose that G is abelian. Then there are numerous sections of G isomorphic to C_{p} or C_{4} and the map in Theorem 13.1 is an injection in a much larger group, whereas the map in Theorem 13.4 hits exactly every cyclic quotient of G and is an isomorphism (Dade’s theorem).

If G is a dihedral 2-group, there are many sections of G isomorphic to C_{4}, but N_{G}(H)/H is never cyclic of order ≥ 4, quaternion, or semi-dihedral, so that X is empty and D_{t}(G) = {0}, a result also obtained in [CaTh], Section 10.

Theorem 13.4 allows us to handle also a case where the structure of D_{t}(G) was not previously known. Proposition 13.5. Suppose that G is an extraspecial 2-group of type 1, that is, a central product of copies of D_{8}. Then D_{t}(G) = {0}.

Proof. We claim that X is empty and so D_{t}(G) = {0}. If H is a subgroup of G containing Z(G), then H is a normal subgroup, G/H is elementary abelian, and H ∉ X. If H does not contain Z(G), then for any g ∈ N_{G}(H), we have that [g, h] ∈ H ∩ Z(G) = {1}. Thus N_{G}(H) = C_{G}(H) and in particular H is abelian, actually elementary abelian since the square of every element of H belongs to H ∩ Z(G) = {1}. Using the quadratic form on G/Z(G), it is not hard to prove that if n is the number of copies of D_{8} in the central product and if H = (C_{2})^{k}, then C_{G}(H) = H × L where L is a central product of n−k copies of D_{8} (possibly n−k = 0 and L = Z(G)). Therefore N_{G}(H)/H is extraspecial and H ∉ X. This proves that X is empty. □

14. Two examples

Theorem 13.4 is not sufficient to determine D_{t}(G) in all cases when p = 2. This seems to be in contrast with the case of an odd prime, for which the solution of the detection conjecture for T (G) allows for a complete description of D_{t}(G) (Theorem 1.5).

Our purpose is to illustrate the situation with the extraspecial groups of type 2 and the almost extraspecial groups (type 3). For simplicity, we shall only deal with the smallest of the groups, namely D_{8} ∗ Q_{8} and D_{8} ∗ C_{4}, but our results can easily be generalized to the other groups of type 2 and 3.

If follows from Theorem 13.4 that the product of all restriction-deflation maps

∏_{H∈[X /G]} Def_{N_{G}(H)/H}^{N_{G}(H)} Res_{N_{G}(H)}^{G} : D_{t}(G) ⟶ ∏_{H∈[X /G]} D_{t}(N_{G}(H)/H)

is injective. In the opposite direction, there is the sum of all maps obtained by composing inflation maps Inf_{N_{G}(H)/H}^{N_{G}(H)} and tensor induction Ten_{N_{G}(H)}^{G}, namely

∑_{H∈[X /G]} Ten_{N_{G}(H)}^{G} Inf_{N_{G}(H)/H}^{N_{G}(H)} : ⨁_{H∈[X /G]} D_{t}(N_{G}(H)/H) ⟶ D_{t}(G) .

We let D_{t}^{0}(G) by the image of this map. The question of the surjectivity of this map does not seem to be easy and this is why we have to introduce the subgroup D_{t}^{0}(G). In similar situations for odd primes, or for the Dade group tensored with ℚ, we can prove the surjectivity of the map (see Section 4 and 6 of [BoTh]), so it seems natural to conjecture that D_{t}^{0}(G) = D_{t}(G). In our two examples, we shall be able to compute D_{t}^{0}(G) but it is not easy to know if D_{t}(G) is larger or not.

In order to compute the image by restriction-deflation of elements of D_{t}^{0}(G), we need a technical formula which is derived from the results of [BoTh]. There is a general formula describing the restriction-deflation of an element of the form Ten_{K}^{G} Inf_{K/H}^{K}(x), but for simplicity we only consider two very special cases. The Frobenius map λ ↦ λ^{p^{n}} is an endomorphism of k and we let

γ_{p^{n}} : D(G) ⟶ D(G)

be the group homomorphism induced by the Frobenius map, as defined in Section 3 of [BoTh]. Lemma 14.1. Let G be a p-group and let K and H be subgroups of G such that H is a normal subgroup of K.

(a) Let P and R be subgroups of G such that R is a normal subgroup of P. Assume that K and P satisfy KP = G (a single double coset). Assume further that the inclusions P ∩ K → K and P ∩ K → P induce isomorphisms

(P ∩ K)/(R ∩ H) ⟶^{∼} K/H  and  (P ∩ K)/(R ∩ H) ⟶^{∼} P/R

respectively. Then the following maps from D(K/H) to D(P/R) are equal:

Def_{P/R}^{P} Res_{P}^{G} Ten_{K}^{G} Inf_{K/H}^{K} = γ_{|R:R∩H|} Iso_{(P ∩K)/(R∩H)}^{P/R} (Iso_{(P ∩K)/(R∩H)}^{K/H})^{−1} ,

where the two latter maps are induced by the isomorphisms (P ∩ K)/(R ∩ H) ⟶^{∼} P/R and (P ∩ K)/(R ∩ H) ⟶^{∼} K/H respectively.

(b) Let L be a normal subgroup of K. Then the following maps from D(K/H) to D(K/L) are equal:

Def_{K/L}^{K} Inf_{K/H}^{K} = Inf_{K/HL}^{K/L} Def_{K/HL}^{K/H} .

Proof. (a) Since there is a single double coset, the Mackey formula implies that

Def_{P/R}^{P} Res_{P}^{G} Ten_{K}^{G} Inf_{K/H}^{K} = Def_{P/R}^{P} Ten_{P ∩K}^{P} Res_{P ∩K}^{K} Inf_{K/H}^{K} .

Now Proposition 3.10 in [BoTh] asserts that

Def_{P/R}^{P} Ten_{Q}^{P} = γ_{|R:Q∩R|} Ten_{QR/R}^{P/R} Iso_{Q/Q∩R}^{QR/R} Def_{Q/Q∩R}^{Q} .

Applying this with Q = P ∩ K, we have that QR = P and Q ∩ R = R ∩ H, because of the assumed isomorphism (P ∩ K)/(R ∩ H) ⟶^{∼} P/R, and therefore

Def_{P/R}^{P} Ten_{P ∩K}^{P} = γ_{|R:R∩H|} Iso_{P ∩K/R∩H}^{P/R} Def_{P ∩K/R∩H}^{P ∩K} .

Composing on the right with Res_{P ∩K}^{K} Inf_{K/H}^{K}, it is easy to see that

Def_{P ∩K/R∩H}^{P ∩K} Res_{P ∩K}^{K} Inf_{K/H}^{K} = (Iso_{P ∩K/R∩H}^{K/H})^{−1} ,

using either the definitions of the maps or the methods of Corollary 3.9 in [BoTh]. It follows that

Def_{P/R}^{P} Ten_{P ∩K}^{P} Res_{P ∩K}^{K} Inf_{K/H}^{K} = γ_{|R:R∩H|} Iso_{P ∩K/R∩H}^{P/R} (Iso_{P ∩K/R∩H}^{K/H})^{−1} ,

and the result follows.

(b) This follows either from the definitions of the maps or from the methods of Corollary 3.9 in [BoTh]. □

Now we can start with our first example D_{8} ∗ C_{4}. Let S_{1}, S_{2}, S_{3} be representatives of the 3 conjugacy classes of non-central subgroups of order 2 (the 2 classes in D_{8} and the product of a generator of C_{4} with an element of order 4 in D_{8}).

Proposition 14.2. Let G = D_{8} ∗ C_{4} be the almost extraspecial group of order 16. Then D_{t}^{0}(G) is cyclic of order 2, generated by the class of the module Ten_{S_{1}×C_{4}}^{G} Inf_{S_{1}×C_{4}/S_{1}}^{S_{1}×C_{4}}(Ω_{S_{1}×C_{4}/S_{1}}^{1}(k)).

Proof. We have that N_{G}(S_{i}) = S_{i} × C_{4} and so N_{G}(S_{i})/S_{i} ≅ C_{4} and S_{i} is in the class X of Theorem 13.4. These are the only subgroups in X (because every other non-trivial subgroup H contains the Frattini subgroup and G/H is elementary abelian). Therefore Theorem 13.4 yields an injective map

∏_{i=1}^{3} Def_{S_{i}×C_{4}/S_{i}}^{S_{i}×C_{4}} Res_{S_{i}×C_{4}}^{G} : D_{t}(G) ⟶ ∏_{i=1}^{3} D_{t}(S_{i} × C_{4}/S_{i}) ≅ (ℤ/2ℤ)^{3} ,

each factor D_{t}(S_{i}×C_{4}/S_{i}) ≅ D_{t}(C_{4}) being cyclic of order 2 generated by the class of Ω_{S_{i}×C_{4}/S_{i}}^{1}(k). Now by definition D_{t}^{0}(G) is generated by the three elements

Ten_{S_{i}×C_{4}}^{G} Inf_{S_{i}×C_{4}/S_{i}}^{S_{i}×C_{4}}(Ω_{S_{i}×C_{4}/S_{i}}^{1}(k)) (1 ≤ i ≤ 3) .

We claim that they are all equal and have order 2. This will complete the proof of the proposition.

In order to prove the claim, we show that the image of any of these three elements by the injective map above is equal to the “diagonal element”

(Ω_{S_{1}×C_{4}/S_{1}}^{1}(k) , Ω_{S_{2}×C_{4}/S_{2}}^{1}(k) , Ω_{S_{3}×C_{4}/S_{3}}^{1}(k) ) .

This follows from a straightforward application of Lemma 14.1. If i ≠ j, we obtain

Def_{S_{j}×C_{4}/S_{j}}^{S_{j}×C_{4}} Res_{S_{j}×C_{4}}^{G} Ten_{S_{i}×C_{4}}^{G} Inf_{S_{i}×C_{4}/S_{i}}^{S_{i}×C_{4}}(Ω_{S_{i}×C_{4}/S_{i}}^{1}(k))
= γ_{|S_{j}:1|} Iso_{C_{4}/1}^{S_{j}×C_{4}/S_{j}} (Iso_{C_{4}/1}^{S_{i}×C_{4}/S_{i}})^{−1}(Ω_{S_{i}×C_{4}/S_{i}}^{1}(k))
= γ_{2}(Ω_{S_{j}×C_{4}/S_{j}}^{1}(k)) = Ω_{S_{j}×C_{4}/S_{j}}^{1}(k) ,

using the fact that Ω^{1}(k) is defined over the prime field 𝔽_{2} and hence is fixed by the Frobenius map γ_{2}. In the case i = j, we have for any k[S_{i} × C_{4}]-module M,

Res_{S_{i}×C_{4}}^{G} Ten_{S_{i}×C_{4}}^{G}(M) = M ⊗ ^{g}M ,

where g is a representative of the non-trivial class of G/S_{i} × C_{4} and ^{g}M denotes the conjugate module. Therefore, ignoring inflation for simplicity, we obtain

Def_{S_{i}×C_{4}/S_{i}}^{S_{i}×C_{4}} Res_{S_{i}×C_{4}}^{G} Ten_{S_{i}×C_{4}}^{G} Inf_{S_{i}×C_{4}/S_{i}}^{S_{i}×C_{4}}(Ω_{S_{i}×C_{4}/S_{i}}^{1}(k))
= Def_{S_{i}×C_{4}/S_{i}}^{S_{i}×C_{4}} (Ω_{S_{i}×C_{4}/S_{i}}^{1}(k) ⊗ ^{g}(Ω_{S_{i}×C_{4}/S_{i}}^{1}(k)))
= Def_{S_{i}×C_{4}/S_{i}}^{S_{i}×C_{4}} (Ω_{S_{i}×C_{4}/S_{i}}^{1}(k)) ⊗ Def_{S_{i}×C_{4}/S_{i}}^{S_{i}×C_{4}} (Ω_{S_{i}×C_{4}/gS_{i}g^{−1}}^{1}(k))
= Ω_{S_{i}×C_{4}/S_{i}}^{1}(k) ⊗ Def_{S_{i}×C_{4}/S_{i}}^{S_{i}×C_{4}}(Ω_{S_{i}×C_{4}/gS_{i}g^{−1}}^{1}(k)) .

But the second factor is trivial because, by part (b) of Lemma 14.1 with K = S_{i} × C_{4}, we have

Def_{K/S_{i}}^{K} Inf_{K/gS_{i}g^{−1}}^{K} = Inf_{K/S_{i}(gS_{i}g^{−1})}^{K/S_{i}} Def_{K/S_{i}(gS_{i}g^{−1})}^{K/gS_{i}g^{−1}}

and a deflation of the class of Ω^{1}(k) is trivial (see Lemma 1.3 of [BoTh]). □

In this example, we see that D_{t}(G) embeds in three copies of ℤ/2ℤ and that D_{t}^{0}(G) ≅ ℤ/2ℤ. So in order to prove the conjectural equality D_{t}^{0}(G) = D_{t}(G), we would have to improve Theorem 13.4 by showing the injectivity of the restrictiondeflation map to a single section S_{i} × C_{4}/S_{i}. In this specific example, we have been able to do this by a rather delicate argument which we do not give here.

The methods are similar with our second example D_{8} ∗ Q_{8}, but another complication occurs. Recall that D_{t}(Q_{8}) is generated by the class of Ω_{Q_{8}}^{1}(k), which has order 4, and the class of a certain 5-dimensional module M, which has order 2 (see [CaTh], Section 6). Moreover M is defined over the field 𝔽_{4} (so we assume here that k contains 𝔽_{4}) and M is not invariant under the Galois automorphism γ_{2}. Actually γ_{2}(M) ≅ Ω^{2}(M), another 5-dimensional module, and Ω^{2}(k), M, Ω^{2}(M) are the 3 elements of order 2 in D_{t}(Q_{8}) ≅ ℤ/4ℤ ⊕ ℤ/2ℤ.

Let S_{1}, …, S_{5} be representatives of the 5 conjugacy classes of non-central subgroups of order 2 (the 2 classes in D_{8} and the product of an element of order 4 in D_{8} with one of the three possible elements of order 4 in Q_{8}). Proposition 14.3. Let G = D_{8}∗Q_{8} be the extraspecial group of order 32 (type 2). Then

D_{t}^{0}(G) ≅ ℤ/4ℤ ⊕ ℤ/2ℤ

generated by the class of the module Ten_{N_{G}(S_{1})}^{G} Inf_{N_{G}(S_{1})/S_{1}}^{N_{G}(S_{1})}(Ω_{N_{G}(S_{1})/S_{1}}^{1}(k)) (or-

der 4) and by the class Ten_{N_{G}(S_{1})}^{G} Inf_{N_{G}(S_{1})/S_{1}}^{N_{G}(S_{1})}(M_{N_{G}(S_{1})/S_{1}}) (order 2), where we denote by M_{N_{G}(S_{1})/S_{1}} the module M viewed as a module for the group N_{G}(S_{1})/S_{1}, which is isomorphic to Q_{8}.

Proof. We have that N_{G}(S_{i}) = S_{i} × C (for some subgroup C isomorphic to Q_{8}) and so N_{G}(S_{i})/S_{i} ≅ Q_{8} and S_{i} is in the class X of Theorem 13.4. These are the only subgroups in X, because every other non-trivial subgroup H contains the Frattini subgroup and G/H is elementary abelian. Therefore, by Theorem 13.4, the map

∏_{i=1}^{5} Def_{N_{G}(S_{i})/S_{i}}^{N_{G}(S_{i})} Res_{N_{G}(S_{i})}^{G} : D_{t}(G) ⟶ ∏_{i=1}^{5} D_{t}(N_{G}(S_{i})/S_{i}) ≅ (D_{t}(Q_{8}))^{5}

is injective. Now by definition D_{t}^{0}(G) is generated by the classes of the modules

Ten_{N_{G}(S_{i})}^{G} Inf_{N_{G}(S_{i})/S_{i}}^{N_{G}(S_{i})}(X) (1 ≤ i ≤ 5) ,

where X is either Ω^{1}(k) or M (viewed in D_{t}(N_{G}(S_{i})/S_{i}) ).

If X = Ω^{1}(k), we always obtain the same element, independently of i, mapping to the diagonal element consisting of Ω^{1}(k) in each component under the injective map above. The proof of this follows exactly the same argument as the one used in the proof of Proposition 14.2, with the following minor modification. For every pair S_{i}, S_{j} with i ≠ j, the group generated by S_{i} and S_{j} is isomorphic to D_{8}. Its centralizer C is isomorphic to Q_{8}, and we have N_{G}(S_{i}) = S_{i} × C and N_{G}(S_{j}) = S_{j}×C. It follows that we can use Lemma 14.1 (with P/Q = N_{G}(S_{i})/S_{i}, K/H = N_{G}(S_{j})/S_{j}, P ∩ K = C). The rest of the argument is similar to what was used in Proposition 14.2.

If now X = M, we use again Lemma 14.1, but the computation changes because of the presence of the Galois automorphism γ_{2} which does not fix the class of M. Moreover, for each i, we need to fix a choice of isomorphism N_{G}(S_{i})/S_{i} ≅ Q_{8} in order to be able to make a consistent computation. We skip the details and only give the result. It turns out that, under the injective map above, the image of Ten_{N_{G}(S_{i})}^{G} Inf_{N_{G}(S_{i})/S_{i}}^{N_{G}(S_{i})}(M) is the 5-tuple (M, M, M, M, M), again independently of i. It follows that we obtain just one extra generator of D_{t}^{0}(G), of order 2. □

In this example, D_{t}(G) is sandwiched between D_{t}^{0}(G) ≅ ℤ/4ℤ ⊕ ℤ/2ℤ and (ℤ/4ℤ ⊕ ℤ/2ℤ)^{5}. The question of the equality D_{t}(G) = D_{t}^{0}(G) remains open.

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J. F. C.: Department of Mathematics, University of Georgia, Athens, Georgia 30602, USA

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

J. T.: Institut de Mathématiques, Université de Lausanne, CH-1015 Lausanne, Switzerland

  E-mail address: Jacques.Thevenaz@ima.unil.ch