THE CLASSIFICATION OF ENDO-TRIVIAL MODULES

JON F. CARLSON^{∗} AND JACQUES THÉVENAZ

1. Introduction

This paper provides the final step for the classification of all endo-trivial kP - modules, where P is a finite p-group and k is a field of characteristic p. The classification was obtained in [CaTh2] in the case that every maximal elementary abelian subgroup of P has rank at least 3. The result had been known for a long time if P has a maximal elementary abelian subgroup of rank 1, i.e. if P cyclic or generalized quaternion. The purpose of this paper is to handle the only remaining case that P has at least one maximal elementary abelian subgroup of rank 2. This case is rather subtle and requires special treatment.

Let T (P) be the group of equivalence classes of endo-trivial kP -modules. The classification of endo-trivial kP -modules is equivalent to the complete description of the structure of T (P). It was proved in [CaTh2] that T (P) is free abelian if P is not cyclic, quaternion or semi-dihedral, and that T (P) ≅ ℤ in most cases, namely when every maximal elementary abelian subgroup of P has rank at least 3. Assuming now that P has at least one maximal elementary abelian subgroup of rank 2 (but P not semi-dihedral), we prove that T (P) is free abelian on r explicit generators, which are classes of suitable relative syzygies of the trivial module. The torsion-free rank of T (P), which is r, was described in group-theoretic terms by Alperin [Al]. The chief contribution of this paper is to show that the relative syzygies found by Alperin generate all of T (P) and not just a subgroup of finite index. This is the final step in the classification of the elements of T (P).

We should point out that there is another set of generators for T (P) found by the first author using a different approach [Ca]. These generators are constructed by direct decompositions of the ordinary syzygies of the trivial module using information from group cohomology. The generators must coincide with the relative syzygies used in this paper. However, we have no direct proof of the coincidence. Instead, the verification of the isomorphisms of the modules relies on the classification, and in particular, on the fact that T (P) is torsion free [CaTh2]. A third collection of elements constructed by Bouc and the second author using tensor induction [BoTh], generates a subgroup of T (P) of the correct rank, but which (in general) may not be all of T (P).

Our strategy is to solve first the problem for several small p-groups, namely all extraspecial and almost extraspecial p-groups which satisfy the condition of having a maximal elementary abelian subgroup of rank 2 (they have order at

Date: December 5, 2003.

* Partly supported by a grant from NSF.

2 JON CARLSON AND JACQUES THÉVENAZ

most p^{5}). Then a group-theoretic argument allows us to deduce the result in general. In the (almost) extraspecial case, we use a method which appears in a different form in [CaTh2] and which turns out to be applicable again with some modification. This consists of decomposing and tensoring endo-trivial modules according to the structure of the variety of a certain quotient of the module. The method was used in [CaTh2] in a proof by contradiction. Here we adapt the method and state positive results which are then used to handle our problem.

As a by-product of the main result, we obtain that T (P) is generated by relative syzygies, except for the case in which P is a generalized quaternion group and an exceptional module is known to exist. This confirms a recent conjecture and is the starting point needed by S. Bouc and N Mazza in their work [BM] on the Dade group of extraspecial p-groups. Another consequence of the property that T (P) is generated by relative syzygies is the fact that the classification is independent of the base field k (except for the quaternion group of order 8, see [CaTh1]). Thus, throughout this paper, k denotes an arbitrary field of characteristic p.

The paper is organized roughly as follows. The second and third sections develop some group theoretic information on p-groups with maximal elementary abelian subgroups of rank 2, and describe the restrictions to elementary abelian subgroups of the relative syzygies defined by Alperin. Sections four and five generalize results from [CaTh2] on decompositions and tensor products of endotrivial modules. In Section 6, we prove a classification of the endo-trivial modules for the extraspecial groups of rank 2. The final step in the classification comes in Section 7, using restriction to extraspecial subgroups and some group theoretic results.

The first author wishes to thank the Université de Lausanne and the École Polytechnique Fédérale de Lausanne for their kind support and hospitality during several visits while this and other papers in the series were being written.

2. Groups with maximal elementary subgroups of rank 2

We first recall some known results concerning p-groups P having maximal elementary abelian subgroups of rank 2. Let A be the poset of all elementary abelian subgroups of P of rank at least 2. Clearly the isolated vertices of A are precisely the maximal elementary abelian subgroups of rank 2. Let B be the subposet of A consisting of all elementary abelian subgroups of P of rank at least 3 as well as their subgroups of rank 2. The following group-theoretic result is well-known and not very hard to prove (see Lemma 10.21 in [GLS]).

Lemma 2.1. Let m(P) be the maximal rank of an elementary abelian subgroup of P.

(a) If m(P) = 2, then A consists of isolated vertices.

(b) If m(P) ≥ 3, then B is a connected component of A and all other connected components (if any) are isolated vertices.

We shall call B the big connected component of A.

By a theorem of Alperin [Al], the torsion-free rank of T (P) is equal to the number of conjugacy classes of connected components of A. We shall have to revisit Alperin’s theorem in the next section and, to this end, we need the following more precise lemma, which appears in slightly different forms in [Al] and [GLS]. For notation, let C_{m} denote a cyclic group of order m, D_{2m} a dihedral group of order 2m, and Q_{2^{k}} a generalized quaternion group of order 2^{k}.

Lemma 2.2. Let P be a p-group with m(P) ≥ 2. Let C_{1}, …, C_{r} be representatives of the conjugacy classes of connected components of A and assume that r ≥ 2. For a suitable choice of numbering, these representatives can be made to satisfy the following conditions:

(a) C_{1} contains a normal elementary abelian subgoup E of rank 2, unless p = 2 and P is dihedral of order ≥ 16. Moreover, if m(P) ≥ 3, then C_{1} is equal to the big component B.

(b) If 2 ≤ i ≤ r, then C_{i} = {E_{i}} where E_{i} is maximal elementary abelian of rank 2. Moreover, E_{i} contains a non-central subgroup S_{i} of order p, and we have that C_{P} (S_{i}) = S_{i} × L_{i} with L_{i} either cyclic or generalized quaternion.

(c) If 2 ≤ i ≤ r, then C_{P} (S_{i}) = C_{P} (E_{i}) is a subgroup of index p in N_{P} (E_{i}) and N_{P} (E_{i}) = (E·E_{1}) ∗ L_{i}, where ∗ denotes the central product. Moreover E·E_{1} is extraspecial of order p^{3} and exponent p if p is odd, and isomorphic to D_{8} if p = 2.

Proof. (a) If P = D_{8} or if P is not dihedral, then there exists a normal elementary abelian subgoup E of rank 2, by a well-known result of Hall, using also the fact that m(P) ≥ 2 and that P cannot be semi-dihedral because r ≥ 2 by assumption (see Theorem 5.4.10 in [Go]). We number the representatives C_{i} in such a way that E ∈ C_{1}.

Assume that m(P) ≥ 3 and let F be any elementary abelian subgoup of rank ≥ 3. Then C_{F} (E) has index 1 or p in F, because F/C_{F} (E) embeds in Aut(E), hence C_{F} (E) ∈ A. In view of the sequence of inclusions

E ≤ E · C_{F} (E) ≥ C_{F} (E) ≤ F ,

we conclude that E ∈ B. Therefore C_{1} = B, proving (a).

(b) If P is a dihedral 2-group, then there are 2 conjugacy classes of maximal elementary abelian subgroups of rank 2 (none of them is normal if |P | ≥ 16). For any such subgroup F, we have F = Z × C where Z = Z(P) and |C| = 2, and moreover C_{P} (C) = F and N_{P} (F) = D_{8}. So (b) and (c) hold in this case.

If P is not dihedral, let E be the normal elementary abelian subgoup of rank 2 considered in part (a). This subgroup E has the following properties. First E is non-central, otherwise E would be contained in every maximal elementary abelian subgoup, and the poset A would be connected, contrary to the assumption that r ≥ 2. Next Z = E ∩ Z(P) ≠ {1} because E is normal and so |Z| = p. Note that Z is contained in every maximal elementary abelian subgoup. Finally the subgroup M = C_{P} (E) is a proper subgroup of P, otherwise E would be central. Since N_{P} (E) = P and P/M embeds in Aut(E), M has index p in P.

4 JON CARLSON AND JACQUES THÉVENAZ

Since C_{1} = B whenever m(P) ≥ 3, it follows that in all cases, for 2 ≤ i ≤ r, we have C_{i} = {E_{i}} where E_{i} is maximal elementary abelian of rank 2. Moreover, as before, E_{i} cannot be central, otherwise A would be connected, and therefore E_{i} contains a non-central subgroup S_{i} of order p. Also Z ≤ E_{i} and so E_{i} = Z × S_{i}. Now S_{i} does not centralize E, otherwise E_{i} < E × S_{i}, contrary to the maximality of E_{i}. Therefore S_{i} ≰ M and since M has index p we have that P = S_{i}·M. Hence C_{P} (S_{i}) = S_{i}·C_{M}(S_{i}), that is, C_{P} (S_{i}) = S_{i} × L_{i} where L_{i} = C_{M}(S_{i}). If A is a maximal elementary abelian subgroup of L_{i}, then Z ≤ A and

E_{i} = Z × S_{i} ≤ A × S_{i} ,

hence Z = A by maximality of E_{i}. This shows that m(L_{i}) = 1 and therefore L_{i} is cyclic or generalized quaternion (Theorem 5.4.10 in [Go]). This completes the proof of (b).

(c) Clearly C_{P} (E_{i}) = C_{P} (S_{i}) because E_{i} = Z × S_{i}. Since E is normal and E ∩ E_{i} = Z, E·E_{i} is a group of order p^{3} and E_{i} has index p in E·E_{i}. Therefore E normalizes E_{i}. But we have seen that E does not centralize S_{i} and so |N_{P} (E_{i}) : C_{P} (E_{i})| ≠ 1. Since N_{P} (E_{i})/C_{P} (E_{i}) embeds in Aut(E_{i}), we have |N_{P} (E_{i}) : C_{P} (E_{i})| = p. Moreover, since E·E_{i} is non-abelian of order p^{3} and has two elementary abelian subgroups of rank 2, it has to be extraspecial of exponent p if p is odd and isomorphic to D_{8} if p = 2. Finally, we have

N_{P} (E_{i}) = E·C_{P} (E_{i}) = E·E_{i}·L_{i}

and we obtain a central product (E·E_{i}) ∗ L_{i} because (E·E_{i}) ∩ L_{i} = Z.

Remark 2.3. Note that Lemma 2.1 follows from the argument of the proof of (a) and that this argument also shows that if m(P) ≥ 3 then every normal elementary abelian subgoup of rank 2 belongs to B.

3. Alperin’s theorem revisited

We begin by recalling basic definitions and establishing some notation that is of use throughout the paper. The group T (P) has elements consisting of equivalence classes [M] of endo-trivial modules M. The equivalence relation is that [M] = [N] if M ⊕ Q_{1} ≅ N ⊕ Q_{2} for some projective modules Q_{1} and Q_{2}. We use an additive notation for the group law in T (P). That is the operation is given by

[M] + [M′] = [M ⊗ M′] .

For a kP -module M, let Γ(M) be the sum of all the indecomposable components of M having vertex P. Thus, M ≅ Γ(M) ⊕ U, where U is a direct sum of submodules with vertex properly contained in P. Both Γ(M) and U are well defined up to isomorphism. Let M^{⊗^{n}} denote the tensor product of n copies of M.

An endo-permutation module is a finitely generated module M whose endomorphism ring Hom_{k}(M, M) is a permutation module. In the Dade group D(P), the elements are classes [M] of endo-permutation modules M having at least one indecomposable summand with vertex P (unique up to isomorphism). Hence for M such a module [M] = [Γ(M)]. The operation is given by

[M] + [N] = [M ⊗ N] = [Γ(M ⊗ N)].

The group of endo-trivial modules T (G) injects into D(P) and can be considered as a subgroup of the Dade group in a natural way. In particular, we use the symbol [M] to denote the class of M in either D(P) or T (P), as appropriate. The reader should be warned that the classes do not mean the same thing in the two different groups. However, a class [M] in D(P) is in the image of a class in T (P) if and only if the module Γ(M) is an endo-trivial module. We emphasize also that Γ(M) is well defined up to isomorphism by its class [M] in D(P).

Alperin’s theorem asserts that the torsion free rank of T (P) is equal to the number r of conjugacy classes of connected components of the poset A of the previous section. The proof uses relative syzygies (see also [BoTh] for another approach using tensor induction). For later use, we need to revisit Alperin’s proof in more detail, taking also into account that T (P) has no torsion, by one of the main results of [CaTh2].

For any p-group P and for any finite P -set X, we let Ω_{X}^{1}(k) denote the first relative syzygy of the trivial kP -module, that is, the kernel of the map kX → k mapping every element of X to 1, where kX denotes the permutation module with basis X. We shall use the relative syzygies Ω_{P/Q}^{1}(k) for suitable subgroups Q. When Q = 1, we obtain the ordinary syzygy Ω_{P}^{1} (k) of the trivial kP -module, that is, the augmentation ideal of the group algebra kP. Also, Ω_{P}^{−1}(k) denotes the dual of Ω_{P}^{1} (k), whose class in T (P) is the inverse of the class of Ω_{P}^{1} (k). For any kP -module M, let Ω_{X}^{1}(M) denote the kernel of a relative kX-projective cover U ⟶ M. Then inductively, we define Ω_{X}^{n}(M) = Ω_{X}^{1}(Ω_{X}^{n−1}(M)). The usual calculus for these modules is that

Ω_{X}^{m}(M) ⊗ Ω_{X}^{n}(N) ≅ Ω_{X}^{m+n}(M ⊗ N) ⊕ W

where W is some relatively X-projective module. Note that if X = P, W is a projective kP -module.

We select maximal elementary abelian subgroups E_{2}, …, E_{r} of rank 2 as in Lemma 2.2 and we let S_{i} denote a non-central subgroup of E_{i}, for every i = 2, …, r. Moreover, we choose arbitrarily a subgroup E_{1} in the remaining connected component C_{1}. Recall that the group T (E_{i}) of endo-trivial modules for E_{i} is isomorphic to ℤ, generated by the class of Ω_{E_{i}}^{1}(k) (Dade [Da]). We define

α_{i} : T (P) ⟶^{Res_{E_{i}}^{P}} T (E_{i}) ⟶^{∼} ℤ.

That is, α_{i}([M]) = n if M↓_{E_{i}}^{P} ≅ Ω_{E_{i}}^{n}(k) ⊕ U for some projective kE_{i}-module U. Let

α = ∏_{i=1}^{r} α_{i} : T (P) ⟶ ∏_{i=1}^{r} ℤ = ℤ^{r} .

6 JON CARLSON AND JACQUES THÉVENAZ

Theorem 3.1. Let P be a finite p-group having a maximal elementary abelian subgroup of rank 2, and suppose that P is not semi-dihedral. Let α be defined as above.

(a) The map α is injective.

(b) (Alperin [Al]) The map α has a finite cokernel. In other words it is a rational isomorphism. In particular T (P) is free abelian of rank r.

(c) The classes of the modules Ω_{P}^{1} (k), N_{2}, …, N_{r} defined below form a rational basis of ℚ ⊗ T (P). If 2 ≤ i ≤ r, define the kP -module M_{i} = Ω_{P}^{−1}(k) ⊗ Ω_{P/S_{i}}^{1}(k). If C_{P} (S_{i})/S_{i} is cyclic of order ≥ 3, then N_{i} ≅ Γ(M_{i}^{⊗^{2}}) is endo-trivial and we have

α_{j}([N_{i}]) = \begin{cases}0 & if j ≠ i , \\ −2p & if j = i . \end{cases}

If p = 2 and C_{P} (S_{i})/S_{i} is cyclic of order 2, then N_{i} ≅ M_{i} is endo-trivial and we have

α_{j}([N_{i}]) = \begin{cases}0 & if j ≠ i , \\ −2 & if j = i . \end{cases}

If p = 2 and C_{P} (S_{i})/S_{i} is generalized quaternion, then N_{i} ≅ Γ(M_{i}^{⊗^{4}}) is endotrivial and we have

α_{j}([N_{i}]) = \begin{cases}0 & if j ≠ i , \\ −8 & if j = i . \end{cases}

Finally, α_{j}([Ω_{P}^{1} (k)]) = 1 for every j.

Proof. The group P cannot be cyclic nor quaternion and is not semi-dihedral by assumption. Therefore, by Theorem 1.2 of [CaTh2], the restriction map

Res : T (P) ⟶ ∏_{E} T (E) ≅ ∏_{E} ℤ

is an injection, where E runs through the set of all elementary abelian subgroups of P of rank 2. Clearly, it suffices to choose representatives of conjugacy classes of such subgroups. This proves (a) when m(P) = 2. Now if m(P) ≥ 3, then there is a big component B and the restriction map T (E) ⟶ T (F) is an isomorphism whenever E, F ∈ B with F ≤ E, because both groups are isomorphic to ℤ, generated by the class of Ω^{1}(k). Therefore it suffices to choose the single subgroup E_{1} ∈ B and (a) also follows in this case.

It is clear that (c) implies (b). Actually (c) is implicit in Alperin’s paper [Al] and all the ideas come from that paper. We now indicate how to prove (c) and refer to Bouc’s paper [Bo] for a detailed exposition of the machinery concerning relative syzygies. We remind the reader of the notational convention regarding the classes [M] in D(P) and T (P), that was stated at the beginning of the section.

By Theorem 1 of [Al] or Lemma 2.3.3 of [Bo], the module Ω_{P/S_{i}}^{1}(k) is an endopermutation module. Hence, so is M_{i}. In order to prove that a suitable multiple in the Dade group of [Ω_{P/S_{i}}^{1}(k)] is endo-trivial, it suffices to prove that its class in D(P) lies in the kernel of the deflation–restriction map

Def_{N_{P} (Q)/Q}^{N_{P} (Q)} Res_{N_{P} (Q)}^{P} : D(P) ⟶ D(N_{P} (Q)/Q)

for every non-trivial subgroup Q of P (see 2.1.2 in [Pu]). Now by Lemma 4.2.1 in [Bo],

Def_{N_{P} (Q)/Q}^{N_{P} (Q)} Res_{N_{P} (Q)}^{P}([Ω_{P/S_{i}}^{1}(k)]) = [Ω_{(P/S_{i})^{Q}}^{1}(k)]

where (P/S_{i})^{Q} is the set of Q-fixed points in P/S_{i}, viewed as an N_{P} (Q)/Q-set. If Q is not conjugate to S_{i}, then (P/S_{i})^{Q} is empty and therefore

Def_{N_{P} (Q)/Q}^{N_{P} (Q)} Res_{N_{P} (Q)}^{P}([Ω_{P/S_{i}}^{1}(k)]) = 0 .

If Q is conjugate to S_{i} then we can assume that Q = S_{i} and we have that

Def_{N_{P} (S_{i})/S_{i}}^{N_{P} (S_{i})} Res_{N_{P} (S_{i})}^{P}([Ω_{P/S_{i}}^{1}(k)]) = [Ω_{N_{P} (S_{i})/S_{i}}^{1}(k)] .

But we have N_{P} (S_{i})/S_{i} = C_{P} (S_{i})/S_{i} ≅ L_{i} by Lemma 2.2 and [Ω_{L_{i}}^{1}(k)] has order 2 if L_{i} is cyclic of order ≥ 3, order 1 if L_{i} is cyclic of order 2, and order 4 if L_{i} is generalized quaternion. This shows that the corresponding multiple of [Ω_{P/S_{i}}^{1}(k)] lies in the kernel of the deflation–restriction map. Consequently, the class of the corresponding multiple of M_{i} in the Dade group is also the class of an endo-trivial module.

In order to compute the image of N_{i} under the map α_{j}, we need to compute the restriction M_{i}↓_{E_{j}}^{P}, and also Ω_{P/S_{i}}^{1}(k)↓_{E_{j}}^{P}. By Corollary 4.1.2 in [Bo], we have that

[Ω_{P/S_{i}}^{1}(k)↓_{E_{j}}^{P}] = [Ω_{Res_{E_{j}}^{P} (P/S_{i})}^{1}(k)] = \begin{cases}[Ω_{E_{j}}^{1}(k)] & if j ≠ i , \\ [Ω_{Res_{E_{i}}^{P}(P/S_{i})}^{1}(k)] & if j = i . \end{cases}

Now the Mackey decomposition shows that Res_{E_{i}}^{P}(P/S_{i}) is the disjoint union of a free E_{i}-set and of |C_{P} (E_{i}) : E_{i}| copies of

⨆_{g∈[N_{P} (E_{i})/C_{P} (E_{i})]}E_{i}/S_{i}^{g} = ⨆_{s=1}^{p} E_{i}/S_{i}^{g_{s}} ,

where g_{1}, …, g_{p} are representatives of N_{P} (E_{i})/C_{P} (E_{i}) (see Lemma 2.2). By Lemma 3.2.7 in [Bo], we deduce that

[Ω_{P/S_{i}}^{1}(k)↓_{E_{i}}^{P}] = [Ω_{⨆_{s=1}^{p} E_{i}/S_{i}^{gs}}^{1}(k)] .

Now by repeated applications of Lemma 5.2.1 of [Bo] (which is also the method used on page 9–10 of [Al]), we obtain

[Ω_{⨆_{s=1}^{p} E_{i}/S_{i}^{gs}}^{1}(k)] = −(p − 1)[Ω_{E_{i}}] + ∑_{s=1}^{p} [Ω_{E_{i}/S_{i}^{gs}}^{1}(k)] .

But E_{i}/S_{i}^{g_{s}} is cyclic of order p, so [Ω_{E_{i}/S_{i}^{gs}}^{1}(k)] = 0 if p = 2 and 2[Ω_{E_{i}/S_{i}^{gs}}^{1}(k)] = 0 if p is odd. This shows that

α_{j}([Γ(Ω_{P/S_{i}}^{1}(k)^{⊗^{m}})]) = \begin{cases}m & if j ≠ i , \\ −m(p − 1) & if j = i , \end{cases}

8 JON CARLSON AND JACQUES THÉVENAZ

where we take m = 1 (resp. 2, resp. 4) if C_{P} (S_{i})/S_{i} ≅ L_{i} is cyclic of order 2 (resp. cyclic of order ≥ 3, resp. generalized quaternion). The class [N_{i}] is now obtained by adding [Ω_{P}^{−1}(k)^{⊗^{m}}] = m[Ω_{P}^{−1}(k)]), whose image under α is (−m, −m, …, −m). The result of the computation for α_{j}([N_{i}]) follows.

Theorem 3.1 raises the question of describing explicitly the image of α and this is the main purpose of the present paper. We prove that the basis appearing in Theorem 3.1 is actually an integral basis of T (P).

Note that we can assume that the integer r is at least 2, for if r = 1 then Res_{E_{1}}^{P} : T (P) ⟶ T (E_{1}) ≅ ℤ is an isomorphism and T (P) is generated by the class of the ordinary syzygy Ω_{P}^{1} (k). One of the effects of the assumption that r ≥ 2 is that all E_{i}’s intersect in the subgroup Z of order p considered in Section 2, namely the unique central subgroup of order p. This motivates the situation considered in the next two sections, where Z will denote a central subgroup of order p.

Remark 3.2. In the first case of (c), the module N_{i} has the form of a translate of a relative syzygy, N_{i} ≅ Ω^{−2}(Ω_{P/S_{i}}^{2}(k)). Likewise, in the third case, that p = 2 and C_{P} (S_{i})/S_{i} is generalized quaternion, we have that N_{i} ≅ Ω^{−4}(Ω_{P/S_{i}}^{4}(k)). That is, in all cases, the module N_{i} is a translate of a relative syzygy of the trivial module.

4. Decomposing endo-trivial modules

The purpose of this section is to show that Theorem 5.6 of [CaTh2] is still valid under different assumptions. The hypotheses in [CaTh2] is about a critical endotrivial module, but it was shown later in the paper that no such module exists. So the result needs to be revisited in order to be used in a different context.

Throughout this section, we let P be a p-group, Z be a central subgroup of order p generated by z, and we write \overline{P} = P/Z. Let M be a kP -module such that M↓_{Z}^{P} ≅ k ⊕ (free). Recall that projective modules are free because P is a p-group. We define M′ = {m ∈ M | (z − 1)^{p−1}m = 0} and we set

\overline{M} = M/M′ .

Since (z − 1)\overline{M} = 0, the module \overline{M} can be viewed as a k\overline{P} -module. Moreover, by Lemma 5.2 in [CaTh2], we have Dim(M) = p Dim(\overline{M}) + 1 and

\overline{M} ≅ (z − 1)^{i}M/(z − 1)^{i+1}M ≅ (z − 1)^{p−1}M , (i = 1, …, p − 1) .

In fact, (z − 1)M is free as a module over kZ/(z − 1)^{p−1}.

Lemma 4.1. Let M be a kP -module such that M↓_{Z}^{P} ≅ k ⊕ (free). Then M ≅ k ⊕ (free) as a kP -module if and only if \overline{M} is a free k\overline{P} -module. More precisely, M has a free summand with r generators as a kP -module if and only if \overline{M} has a free summand with r generators as a k\overline{P} -module. In particular, if M is indecomposable, then \overline{M} has no projective summands.

Proof. This is Lemma 5.3 in [CaTh2]. The assumption in [CaTh2] is slightly different but the proof can be applied without change.

We refer to [CaTh2] for the definition and properties of the variety V\overline{P} (\overline{L}) attached to any k\overline{P} -module \overline{L}.

Theorem 4.2. Let M be a kP -module such that M↓_{Z}^{P} ≅ k ⊕ (free). Suppose that \overline{M} = \overline{M}_{1} ⊕ \overline{M}_{2} where \overline{M}_{1} and \overline{M}_{2} are k\overline{P} -submodules such that the varieties satisfy

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

(a) There exist kP -modules N_{1} and N_{2} such that N_{i}↓_{Z}^{P} ≅ k ⊕ (free) and \overline{N}_{i} ≅ \overline{M}_{i} for 1 ≤ i ≤ 2.

(b) If moreover \overline{M}_{2} is a free k\overline{P} -module, then there exists a submodule N_{1} of M satisfying the conditions of (a) and such that M = N_{1} ⊕ (free).

Proof. (a) The result is similar to Theorem 5.6 in [CaTh2] and the proof is almost identical. Therefore we only indicate where the proof needs to be adapted. The construction of N_{1} and N_{2} is the same and we use the same notation. We have (z − 1)M = W_{1} ⊕ W_{2} with W_{i}/(z − 1)W_{i} ≅ \overline{M}_{i} for i = 1, 2 (and it is here that the condition on the varieties is used). We let M_{i} ⊆ M be the inverse image of \overline{M}_{i} under the quotient map M ⟶ M/M′ = \overline{M} and we define N_{1} = M_{1}/W_{2} and N_{2} = M_{2}/W_{1}. Then we have Dim(N_{i}) = p Dim(\overline{M}_{i}) + 1.

Next we prove that N_{1}↓_{Z}^{P} ≅ k ⊕ (free) (by an argument similar to, but much easier than the one used in [CaTh2]). We have (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. Now (z − 1)^{p−1}M_{1} ≅ \overline{M}_{1} because multiplication by (z −1)^{p−1} induces an isomorphism \overline{M} ≅ (z −1)^{p−1}M. Therefore

( ∑_{i=0}^{p−1} z^{i} )N_{1} = (z − 1)^{p−1}N_{1} ≅ \overline{M}_{1}

and it follows that N_{1}↓_{Z}^{P} has a free submodule of dimension p Dim(\overline{M}_{1}), that is, Dim(N_{1}) − 1. The only way this can happen is if N_{1}↓_{Z}^{P} ≅ k ⊕ (free).

Finally we have to show that \overline{N}_{1} ≅ \overline{M}_{1}, and similarly for N_{2}. The proof is exactly the same as in [CaTh2]: \overline{N}_{1} appears naturally as a quotient of \overline{M}_{1} and they have the same dimension.

(b) This additional part, which was not proved in [CaTh2], is based on the construction of N_{1}. Let \overline{x}_{1}, …, \overline{x}_{m} be a basis of \overline{M}_{2} as a k\overline{P} -module, let x_{i} ∈ M_{2} be a lift of \overline{x}_{i} and let F be the kP -submodule of M_{2} generated by x_{1}, …, x_{m}. By Lemma 4.1, F is free of dimension m|P | = pm|\overline{P} | = p Dim(\overline{M}_{2}). Now

(z − 1)^{p−1}F ⊆ (z − 1)^{p−1}M_{2} = (z − 1)^{p−2}W_{2} ≅ \overline{M}_{2} ≅ F/(z − 1)F ,

and therefore (z − 1)^{p−1}F = (z − 1)^{p−2}W_{2}. It follows that W_{1} ∩ F = {0} because if we choose w ∈ W_{1} ∩ F, then we can assume that (z − 1)w = 0 (replacing w by (z − 1)^{i}w for some i), so that w ∈ (z − 1)^{p−1}F which intersects W_{1} trivially. Hence W_{1} ∩ (z − 1)F = \{0\}. But Dim(F) = p Dim(\overline{M}_{2}) = p Dim((z − 1)^{p−2}W_{2}), hence

Dim((z − 1)F) = (p − 1) Dim((z − 1)^{p−2}W_{2}) = Dim(W_{2}) .

Since (z − 1)M = W_{1} ⊕ W_{2}, we must also have (z − 1)M = W_{1} ⊕ (z − 1)F and so we can choose W_{2} = (z − 1)F for the construction of N_{1} = M_{1}/W_{2}.

10 JON CARLSON AND JACQUES THÉVENAZ

Since F ⊆ M_{2}, we have M_{1} ∩ F = M_{1} ∩ M_{2} ∩ F = M′ ∩ F. But the elements of F lying in the kernel of multiplication by (z − 1)^{p−1} are precisely the multiples of (z − 1) and therefore M′ ∩ F = (z − 1)F = W_{2}. Hence M_{1} ∩ F = W_{2}. On the other hand, since M′ ⊆ M_{1}, we have

M_{1} + F = M_{1} + M′ + F = M_{1} + M_{2} = M .

It follows that

M/F = (M_{1} + F)/F ≅ M_{1}/(M_{1} ∩ F) = M_{1}/W_{2} = N_{1} .

Since F is free, hence injective (because kP is self-injective), F is a direct summand of M and we obtain M = F ⊕N′_{1}, with N′_{1} isomorphic to N_{1}, as required.

Now we apply this theorem to the case of an endo-trivial module over a pgroup P having a maximal elementary abelian subgroup of rank 2. We wish to prove that the modules N_{1} and N_{2} are again endo-trivial in this case and that we have precise information about their restrictions to elementary abelian subgroups.

Let E_{1}, …, E_{r} be elementary abelian subgroups chosen as in Theorem 3.1. If M is any endo-trivial kP -module, then Res_{E_{1}}^{P}[M] = [Ω_{E_{1}}^{s}(k)] for some s, where [M] denotes the class of M in the group T (P) of endo-trivial kP -modules. Therefore

Res_{E_{1}}^{P}[Ω_{P}^{−s}(M)] = Res_{E_{1}}^{P}[Ω_{P}^{−s}(k) ⊗ M] = [Ω_{E_{1}}^{−s}(k) ⊗ Ω_{E_{1}}^{s}(k)] = [k] .

It follows that, up to a shift, we can always choose M such that Res_{E_{1}}^{P}[M] is trivial, that is, M↓_{E_{1}}^{P} ≅ k ⊕ (free).

As noticed at the end of Section 2, we can assume that r ≥ 2 and therefore all E_{i}’s intersect in a central subgroup Z of order p. In particular M↓_{Z}^{P} ≅ k ⊕ (free) and we can use Theorem 4.2.

Theorem 4.3. Let P be a p-group having a maximal elementary abelian subgroup of rank 2 and let E_{1}, …, E_{r} be elementary abelian subgroups chosen as in Theorem 3.1. We assume that r ≥ 2 and we let Z be the unique central subgroup of order p (contained in every E_{j}). Let M be an endo-trivial kP -module such that M↓_{E_{1}}^{P} ≅ k ⊕ (free). Suppose that \overline{M} = \overline{M}_{1} ⊕ \overline{M}_{2} where \overline{M}_{1} and \overline{M}_{2} are k\overline{P} -submodules such that their varieties V_{1} = V\overline{P} (\overline{M}_{1}) and V_{2} = V\overline{P} (\overline{M}_{2}) satisfy

V_{1} ∩ V_{2} = {0} .

(a) There exist endo-trivial kP -modules N_{1} and N_{2} such that N_{i}↓_{Z}^{P} ≅ k ⊕(free) and \overline{N}_{i} ≅ \overline{M}_{i} for 1 ≤ i ≤ 2.

(b) For E = E_{1} and for any elementary abelian subgroup E lying in the big component, we have N_{i}↓_{E}^{P} ≅ k ⊕ (free) for 1 ≤ i ≤ 2.

(c) For E = E_{j} where j = 2, …, r, one of the two modules N_{1}, N_{2} is trivial on restriction to E. More precisely, setting \overline{E} = E/Z, we have the following cases:

1. V\overline{E}(\overline{M}) = \{0\} in which case N_{i}↓_{E}^{P} ≅ k ⊕ (free) for 1 ≤ i ≤ 2;

2. V\overline{E}(\overline{M}) = V\overline{E}(k) and res_{\overline{P} ,\overline{E}}^{∗}(V\overline{E}(k)) ⊆ V_{1} in which case V\overline{E}(\overline{N}_{1}) = V\overline{E}(k), M↓_{E}^{P} ≅ N_{1}↓_{E}^{P} ⊕ (free), and N_{2}↓_{E}^{P} ≅ k ⊕ (free); 3. V\overline{E}(\overline{M}) = V\overline{E}(k) and res_{\overline{P} ,\overline{E}}^{∗}(V\overline{E}(k)) ⊆ V_{2} in which case V\overline{E}(\overline{N}_{2}) = V\overline{E}(k),

M↓_{E}^{P} ≅ N_{2}↓_{E}^{P} ⊕ (free), and N_{1}↓_{E}^{P} ≅ k ⊕ (free).

Proof. (a) We use Theorem 4.2 for the existence of N_{1} and N_{2} and moreover we have now to prove that N_{1} and N_{2} are endo-trivial. This will follow if we show that they are endo-trivial on restriction to every elementary abelian subgroup of P, by Lemma 2.9 of [CaTh1]. Thus it suffices to prove (b) and (c), because (b) and (c) show that N_{i}↓_{E}^{P} is isomorphic, up to addition or deletion of free summands, either to k or to M↓_{E}^{P}, for every maximal elementary subgroup E of P.

(b) Since M↓_{E_{1}}^{P} ≅ k ⊕ (free) by assumption, we also have M↓_{E}^{P} ≅ k ⊕ (free) for any elementary abelian subgroup E lying in the big component (see the beginning of the proof of Theorem 3.1). Thus \overline{M}↓\overline{P} ; \overline{E} is free by Lemma 4.1. Therefore \overline{M}_{1}↓\overline{P} ; \overline{E} and \overline{M}_{2}↓\overline{P} ; \overline{E} are free and so N_{1}↓_{E}^{P} ≅ k⊕(free) and N_{2}↓_{E}^{P} ≅ k⊕(free), by Lemma 4.1 again.

(c) If V\overline{E}(\overline{M}) = \{0\}, then \overline{M}↓\overline{P} ; \overline{E} is free and we conclude as in case (b). Since E has rank 2, \overline{E} is cyclic of order p and the only other possibility is V\overline{E}(\overline{M}) = V\overline{E}(k), which is a line. Then res_{\overline{P} ,\overline{E}}^{∗}(V\overline{E}(k)) must be contained either in V_{1} or in V_{2}. We can assume that res_{\overline{P} ,\overline{E}}^{∗}(V\overline{E}(k)) ⊆ V_{1} and we must show that case (2) occurs.

Since res_{\overline{P} ,\overline{E}}^{∗}(V\overline{E}(k)) ⊆ V_{1}, we have res_{\overline{P} ,\overline{E}}^{∗}(V\overline{E}(k)) ∩ V_{2} = \{0\} and since V_{2} =

V\overline{P} (\overline{N}_{2}), we must have V\overline{E}(\overline{N}_{2}↓\overline{P} ; \overline{E}) = \{0\}. Therefore \overline{N}_{2}↓\overline{P} ; \overline{E} is free and N_{2}↓_{E}^{P} ≅ k ⊕ (free) by Lemma 4.1. On the other hand, part (b) of Theorem 4.2 applied to M↓_{E}^{P} shows that M↓_{E}^{P} ≅ N_{1}↓_{E}^{P} ⊕ (free).

5. Tensoring endo-trivial modules

In this section we show that Theorem 5.7 of [CaTh2] is still valid under different assumptions. As in the previous section, the result needs to be revisited in order to be used in a different context. However, there is only a cosmetic change to be made. We assume again that P is a p-group with a central subgroup Z of order p.

Theorem 5.1. Let M_{1} and M_{2} be indecomposable endo-trivial kP -modules such that (M_{j})↓_{Z}^{P} ≅ k ⊕ (free) for j = 1, 2. Suppose that the varieties of \overline{M}_{1} and \overline{M}_{2} satisfy

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

Then M_{1} ⊗ M_{2} ≅ M ⊕ (free) where M is an indecomposable endo-trivial kP - module such that M↓_{Z}^{P} ≅ k ⊕ (free) and \overline{M} ≅ \overline{M}_{1} ⊕ \overline{M}_{2}.

Proof. The proof is exactly the same as in Theorem 5.7 of [CaTh2], except for the following minor changes. In [CaTh2], the modules M_{1} and M_{2} were assumed to be critical and then M was shown to be critical. Here we only assume that M_{j} is indecomposable endo-trivial such that (M_{j})↓_{Z}^{P} ≅ k⊕(free) for j = 1, 2 and we only have to show that M is indecomposable endo-trivial such that M↓_{Z}^{P} ≅ k ⊕ (free). The new assumption has no effect on the proof and therefore only the new aspect of the conclusion needs to be proved.

12 JON CARLSON AND JACQUES THÉVENAZ

Explicitly, the proof that M_{1} ⊗ M_{2} ≅ M ⊕ (free) applies without change and so does the proof that \overline{M} ≅ \overline{M}_{1} ⊕ \overline{M}_{2}. In order to prove the new conclusion, it suffices to notice that if M_{1} and M_{2} are endo-trivial then so are M_{1} ⊗ M_{2} and M, and moreover if (M_{j})↓_{Z}^{P} ≅ k ⊕ (free) for j = 1, 2, then (M_{1} ⊗ M_{2})↓_{Z}^{P} ≅ k ⊕ (free), and hence M↓_{Z}^{P} ≅ k ⊕ (free). Since M_{j} is indecomposable, the k\overline{P} -module \overline{M}_{j} does not have free summands by Lemma 4.1. Thus \overline{M} has no free summand and it follows that M has no free summand by Lemma 4.1 again. This shows that M is indecomposable.

6. Endo-trivial modules for small extraspecial groups

In this section we describe explicitly all endo-trivial modules for a p-group P which is either extraspecial or almost extraspecial. We refer to [Go] for extraspecial groups and [CaTh2] for almost extraspecial groups. If every maximal elementary abelian subgroup has rank ≥ 3, then T (P) ≅ ℤ by [CaTh2] and the only indecomposable endo-trivial modules are the syzygies Ω_{P}^{s} (k) with s ∈ ℤ. So we can assume that there is at least one maximal elementary abelian subgroup of rank 2 and this forces P to be one of the following groups:

1. P is extraspecial of order p^{3} and exponent p with p odd. Here P has p + 1 maximal subgroups, which are elementary abelian of rank 2.

2. P is extraspecial of order p^{3} and exponent p^{2} with p odd. Here P has a unique maximal elementary abelian subgroup of rank 2. Therefore this case is of no interest for us since we know that T (P) ≅ ℤ, generated by Ω_{P}^{1} (k). Actually, P is metacyclic and so this case is also covered by the work of Mazza [Ma].

3. P is almost extraspecial of order p^{4} with p odd, that is, P = Q ∗ C_{p^{2}}, where Q is extraspecial of order p^{3} and exponent p and C_{p^{2}} is cyclic of order p^{2}. Here P has p + 1 maximal elementary abelian subgroups of rank 2.

4. p = 2 and P = D_{8} is the dihedral group of order 8. This case has been described in detail in [CaTh1]. There are two non-central subgroups S_{1} and S_{2} of order 2, up to conjugacy, and T (P) is free abelian of rank 2, generated by the classes of Ω_{P}^{1} (k) and Ω_{P/S_{i}}^{1}(k) where i is either 1 or 2.

5. p = 2 and P = D_{8} ∗ Q_{8}, the central product of the dihedral group of order 8 and the quaternion group of order 8 (so P is extraspecial of order 32). Here P has 5 maximal elementary abelian subgroups of rank 2.

6. p = 2 and P is almost extraspecial of order 16, that is, P = D_{8} ∗ C_{4}, the central product of the dihedral group of order 8 and the cyclic group of order 4. Here P has 3 maximal elementary abelian subgroups of rank 2.

Note that the list is complete because the next cases are the two extraspecial groups of order p^{5} with p odd and the group D_{8} ∗ D_{8} of order 32, whose maximal elementary abelian subgroups have rank at least 3. All other extraspecial and almost extraspecial groups contain one of these groups of order p^{5} and therefore do not have maximal elementary abelian subgroups of rank 2.

We are going to treat case (1) in detail and then case (3) will follow easily. Cases (5) and (6) will be handled next.

Theorem 6.1. Let p be an odd prime and let P be the extraspecial group of order p^{3} and exponent p. Let E_{1}, …, E_{p+1} be the maximal subgroups of P (which are elementary abelian of rank 2) and for each i let S_{i} be a non-central subgroup of order p in E_{i}. Then the group T (P) is free abelian generated by the classes of the modules Ω_{P}^{1} (k), N_{2}, …, N_{p+1}, where N_{i} = Γ((Ω_{P}^{−1}(k) ⊗ Ω_{P/S_{i}}^{1}(k))^{⊗^{2}}) for every i = 2, …, p + 1.

Proof. Consider the injective map α : T (P) ⟶ ℤ^{p+1} of Theorem 3.1. We know that α([Ω_{P}^{1} (k)]) = (1, 1, …, 1) and α([N_{i}]) = −2p·e_{i}, where e_{i} denotes the i-th element of the canonical basis of ℤ^{p+1} (for every i = 2, …, p + 1). We have to prove that any class of endo-trivial modules is a linear combination of these elements.

Let M be any endo-trivial kP -module. By subtracting a multiple of [Ω_{P}^{1} (k)], we can assume that α_{1}([M]) = 0 and we then have to prove that [M] is a linear combination of [N_{2}], …, [N_{p+1}]. In other words

α([M]) = (0, s_{2}, s_{3}, …, s_{p+1})

and we have to prove that s_{i} is a multiple of 2p for every i = 2, …, p + 1. We proceed by induction on the number of non-zero components in α([M]). We can assume that M is indecomposable by deleting all free summands of M.

Suppose first that α([M]) has a unique non-zero component s_{q} where q is some fixed integer with 2 ≤ q ≤ p + 1, and write s = s_{q} for simplicity. By subtracting a multiple of [N_{q}], we can assume that 0 < s ≤ 2p and we now have to prove that s = 2p. Since s_{1} = α_{1}([M]) = 0, [M↓_{E_{1}}^{P}] = 0, hence M↓_{E_{1}}^{P} ≅ k ⊕ (free) and Dim(M) ≡ 1 (mod p^{2}). Now α_{q}([M]) = s, that is,

M↓_{E_{q}}^{P} ≅ Ω_{E_{q}}^{1}(k)^{⊗^{s}} ⊕ (free) .

Because Ω_{E_{q}}^{1}(k) has dimension p^{2} − 1, we have that Dim(M) ≡ (−1)^{s} (mod p^{2}) and it follows that s is even.

Now we claim that the automorphism group of P acts transitively on the set of maximal subgroups of P. By Lemma 6.3 in [CaTh2], the subgroup of Out(P) fixing Z(P) pointwise is isomorphic to Sp_{2}(𝔽_{p}) = SL_{2}(𝔽_{p}). This acts transitively on the lines of (𝔽_{p})^{2} ≅ \overline{P}, which correspond to the maximal subgroups of P. If now β_{i} is an automorphism of P mapping E_{q} to E_{i}, then the conjugate module L_{i} = M^{β_{i}} satisfies

α_{j}([L_{i}]) = \begin{cases}s & if j = i , \\ 0 & otherwise . \end{cases}

Since M is indecomposable, so is L_{i} for each i. By repeated applications of Theorem 5.1, we get that

L_{1} ⊗ L_{2} ⊗ … ⊗ L_{p+1} = U ⊕ (free)

where U is an indecomposable endo-trivial module such that, with the notation of Section 4,

\overline{U} = \overline{L}_{1} ⊕ \overline{L}_{2} ⊕ … ⊕ \overline{L}_{p+1} .

14 JON CARLSON AND JACQUES THÉVENAZ

Now for all j we have that

α_{j}([U]) = α_{j}([L_{1} ⊗ L_{2} ⊗ … ⊗ L_{p+1}]) = α_{j}([L_{1}] + [L_{2}] + … + [L_{p+1}]) = s ,

and

α_{j}([Ω_{P}^{s} (k)] = α_{j}([Ω_{P}^{1} (k)^{⊗^{s}}] = α_{j}(s[Ω_{P}^{1} (k)]) = s .

Therefore [U] = [Ω_{P}^{s} (k)] by injectivity of α and so U ≅ Ω_{P}^{s} (k) because both modules are indecomposable. Since \overline{M}↓\overline{P} ; \overline{E_{1}} is free, Dim(\overline{M}) = mp for some m (hence Dim(M) = mp^{2} + 1). It follows that Dim(\overline{U}) = (p + 1)mp, hence

Dim(Ω_{P}^{s} (k)) = Dim(U) = (p + 1)mp^{2} + 1 .

But the mod-p cohomology of P is known (see Section 4 in [CaTh2]) and hence also the dimensions of the syzygies of k. Explicitly, by Corollary 4.4 in [CaTh2] and its proof, we have

Dim(Ω_{P}^{2j}(k)) = (j + 1)p^{3} + 1 if 2 ≤ j ≤ p .

Remembering that s is even, we write s = 2j and we obtain

(p + 1)mp^{2} + 1 = (j + 1)p^{3} + 1 ,

hence (p + 1)m = (j + 1)p. Therefore p divides m and the only way this can happen is if j = p (and m = p). This proves that s = 2p, as required.

Suppose now that the number of non-zero components in α([M]) is at least 2. Because α_{1}([M]) = 0, we have that M↓_{E_{1}}^{P} ≅ k ⊕ (free), and we are in a situation where we can apply Theorem 4.3. Since M is indecomposable, the k\overline{P} -module \overline{M} has no free summands (Lemma 4.1). However, \overline{M} is free on restriction to \overline{E_{1}}, because M↓_{E_{1}}^{P} ≅ k ⊕ (free), and since E_{1} is a maximal subgroup of P we deduce that \overline{M} is a periodic k\overline{P} -module (see Lemma 5.3 in [CaTh2] for details).

We use standard properties of varieties, as listed for instance in Theorem 2.2 of [CaTh2]. Since \overline{M} is periodic, the variety of \overline{M} is a union of lines. Let ℓ be one of these lines and let W be the union of all the others, so that

V\overline{P} (\overline{M}) = ℓ ∪ W , ℓ ∩ W = \{0\} .

Then \overline{M} = \overline{M}_{1} ⊕\overline{M}_{2} with V\overline{P} (\overline{M}_{1}) = ℓ and V\overline{P} (\overline{M}_{2}) = W. By Theorem 4.3, there exist endo-trivial modules N_{1} and N_{2} satisfying the properties of that theorem. Let us write L = N_{1} for simplicity. In particular we have V\overline{P} (\overline{L}) = ℓ and

L↓_{E_{1}}^{P} ≅ k ⊕ (free) , V\overline{E_{1}}(\overline{L}) = \{0\} .

So for j = 2, …, p + 1, we have either

L↓_{E_{j}}^{P} ≅ k ⊕ (free) , V\overline{E_{j}}(\overline{L}) = \{0\} , res_{\overline{P} ,\overline{E_{j}}}^{∗}(V\overline{E_{j}}(k)) ⊆ W ,

or

L↓_{E_{j}}^{P} ≅ k̸ ⊕ (free) , V\overline{E_{j}}(\overline{L}) = V\overline{E_{j}}(k) , res_{\overline{P} ,\overline{E_{j}}}^{∗}(V\overline{E_{j}}(k)) ⊆ ℓ .

We cannot have that L↓_{E_{j}}^{P} ≅ k ⊕ (free) for all j, because otherwise α([L]) = 0 and L ≅ k ⊕ (free) by injectivity of α. However, this would imply that \overline{L} is free (Lemma 4.1) and V\overline{P} (\overline{L}) = \{0\}, a contradiction. Without loss of generality, we can assume that for the index j = 2 we have

L↓_{E_{2}}^{P} ≅ k̸ ⊕ (free) , V\overline{E_{2}}(\overline{L}) = V\overline{E_{2}}(k) , res_{\overline{P} ,\overline{E_{2}}}^{∗}(V\overline{E_{2}}(k)) ⊆ ℓ .

Actually the latter inclusion is an equality because \overline{P} is elementary abelian and res_{\overline{P} ,\overline{E_{2}}}^{∗}(V\overline{E_{2}}(k)) is a line in the plane V\overline{P} (k). Moreover, Theorem 4.3 also asserts that M↓_{E_{2}}^{P} ≅ L↓_{E_{2}}^{P} ⊕ (free) and therefore α_{2}([L]) = α_{2}([M]) = s_{2}. Now if j ≥ 3, res_{\overline{P} ,\overline{E_{j}}}^{∗}(V\overline{E_{j}}(k)) is a line different from ℓ and therefore the other case occurs,

namely L↓_{E_{j}}^{P} ≅ k ⊕ (free). Thus we have proved that

α_{j}([L]) = \begin{cases}s_{2} & if j = 2 , \\ 0 & otherwise . \end{cases}

By the first step of induction, we know that s_{2} is a multiple of 2p.

Now let M′ be an indecomposable endo-trivial module in the class of [M]−[L]. Then α([M′]) has fewer non-zero components than α([M]) and so they are all multiples of 2p by induction. This shows that s_{j} is a multiple of 2p for j ≥ 3 and completes the proof.

The following corollary is not actually needed for the proof of the main theorem. However, we include a proof for the sake of completeness.

Corollary 6.2. Let P = Q∗C_{p^{2}} be an almost extraspecial of order p^{4} with p odd, where Q is extraspecial of order p^{3} and exponent p and C_{p^{2}} is cyclic of order p^{2}.

(a) Res_{Q}^{P} : T (P) ⟶ T (Q) is an isomorphism.

(b) Let E_{1}, …, E_{p+1} be the maximal subgroups of Q (which are elementary abelian of rank 2) and for each i let S_{i} be a non-central subgroup of order p in E_{i}. Then the group T (P) is free abelian generated by the classes of the modules Ω_{P}^{1} (k), N_{2}, …, N_{p+1}, where N_{i} = Γ((Ω_{P}^{−1}(k)⊗Ω_{P/S_{i}}^{1}(k))^{⊗^{2}}) for i = 2, …, p + 1.

Proof. Let R be the set of all elements of order p. We first claim that R is a subgroup and that R = Q. Let u, v ∈ R and z = v^{−1}u^{−1}vu. Since P is almost extraspecial, z is central and therefore (uv)^{p} = u^{p}v^{p}z^{\binom{p}{2}} = z^{\binom{p}{2}}. But z has order 1 or p and p divides \binom{p}{2} because p is odd. Therefore uv has order p and it follows that R is a subgroup. Now R is a proper subgroup containing the maximal subgroup Q, so R = Q.

Since every elementary abelian subgroup is contained in R, the subgroups E_{1}, …, E_{p+1} are the maximal elementary abelian subgroups of P. By Theorem 3.1, the map

Res : T (P) ⟶ ∏_{i=1}^{p+1} T (E_{i}) ≅ ℤ^{p+1}

is injective. Since this factors through T (Q), we deduce that Res_{Q}^{P} : T (P) ⟶ T (Q) is injective.

16 JON CARLSON AND JACQUES THÉVENAZ

Both (a) and (b) will follow if we prove that the classes [Ω_{P}^{1} (k)], [N_{2}], …, [N_{p+1}] are mapped under Res_{Q}^{P} onto the corresponding generators of T (Q), namely

[Ω_{Q}^{1}(k)], [N′_{2}], …, [N′_{p+1}], where N′_{i} = (Ω_{Q}^{−1}(k)⊗Ω_{Q/S_{i}}^{1}(k))^{⊗^{2}} for i = 2, …, p + 1. It is clear that Res_{Q}^{P}([Ω_{P}^{1} (k)]) = [Ω_{Q}^{1}(k)], so we only need to prove that

Res_{Q}^{P}([Ω_{P/S_{i}}^{1}(k)]) = [Ω_{Q/S_{i}}^{1}(k)]

(these are classes in the Dade groups D(P) and D(Q) respectively). By an easy argument (see also Corollary 4.1.2 in [Bo])

Res_{Q}^{P}([Ω_{P/S_{i}}^{1}(k)]) = [Ω_{Res_{Q}^{P}(P/S_{i})}^{1}(k)]

and Res_{Q}^{P}(P/S_{i}) is a disjoint union of p copies of Q/S_{i}, because we can choose central elements in C_{p^{2}} as representatives of P/Q. The multiplicity of the Qsets does not change the class of relative syzygies in the Dade group D(Q), by Lemma 3.2.7 in [Bo]. Thus we are left with a single orbit Q/S_{i} and it follows that Res_{Q}^{P}([Ω_{P/S_{i}}^{1}(k)]) = [Ω_{Q/S_{i}}^{1}(k)].

We now turn to the prime 2. The first extraspecial group to consider is the dihedral group D_{8} of order 8. But this was done explicitly in [CaTh1]. The group D_{8} has two elementary abelian subgroups E_{1}, E_{2} of rank 2 and the image of the injective map

α : T (D_{8}) ⟶ ℤ^{2}

is generated by α([Ω_{D_{8}}^{1}(k)]) = (1, 1) and α([Ω_{D_{8}}^{−1}(k) ⊗ Ω_{D_{8}/S}^{1}(k)]) = (0, −2), where S is non-central of order 2. Thus the rational basis in Theorem 3.1 is actually an integral basis of T (D_{8}).

We do not have to consider the quaternion group of order 8 since it has no elementary abelian subgroup of rank 2. Actually T (Q_{8}) has an exceptional behaviour described in [CaTh1]. The group D_{8}∗D_{8} has maximal elementary abelian subgroups of rank 3 and therefore T (D_{8} ∗ D_{8}) ≅ ℤ. Thus our next extraspecial group is D_{8} ∗ Q_{8}.

Theorem 6.3. Let p = 2 and let P = D_{8} ∗ Q_{8} be the central product of the dihedral group of order 8 and the quaternion group of order 8 (so P is extraspecial of order 32). Let E_{1}, …, E_{5} be the five maximal elementary abelian subgroups of rank 2, and for each i let S_{i} be a non-central subgroup of order 2 in E_{i}. Then the group T (P) is free abelian generated by the classes of the modules Ω_{P}^{1} (k), N_{2}, N_{3}, N_{4}, N_{5}, where N_{i} = Γ((Ω_{P}^{−1}(k) ⊗ Ω_{P/S_{i}}^{1}(k))^{⊗^{4}}) for 2 ≤ i ≤ 5.

Proof. The strategy is exactly the same as that of the proof of Theorem 6.1. Let u, v be two elements of order 2 generating D_{8} and let a, b be two generators of Q_{8}. Then, up to conjugation, u, v, uva, uvb, uvab are the five non-central elements of order 2 and they generate subgroups S_{1}, …, S_{5} respectively. Moreover, together with the central element a^{2} = b^{2}, every S_{i} generates one of the five maximal elementary abelian subgroups E_{i} of rank 2. We have C_{P} (S_{i}) ≅ S_{i} × Q_{8} so we need 4-th tensor powers in the description of the generators of ℚ⊗T (P) given by Theorem 3.1. More precisely, Theorem 3.1 asserts that the map α : T (P) ⟶ ℤ^{5} is injective and that α([Ω_{P}^{1} (k)]) = (1, 1, 1, 1, 1) and α([N_{i}]) = −8·e_{i}, where e_{i} denotes the i-th element of the canonical basis of ℤ^{5} (with 2 ≤ i ≤ 5). We have to prove that any class of endo-trivial modules is a linear combination of these.

We follow closely the proof of Theorem 6.1 and indicate the necessary changes. Let M be any endo-trivial kP -module. By subtracting a multiple of [Ω_{P}^{1} (k)], we can assume that

α([M]) = (0, s_{2}, s_{3}, s_{4}, s_{5})

and we have to prove that s_{i} is a multiple of 8 for every i = 2, …, 5. We can assume that M is indecomposable and we proceed by induction on the number of non-zero components in α([M]).

Suppose first that s_{q} is the unique non-zero component of α([M]), where q is some fixed integer with 2 ≤ q ≤ 5, and write s = s_{q} for simplicity. By subtracting a multiple of [N_{q}], we can assume that 0 < s ≤ 8, and we now have to prove that s = 8. We also know that s is even. The automorphism group of P acts transitively on the five maximal elementary abelian subgroups of P, so if β_{i} is an automorphism of P mapping E_{q} to E_{i}, then the conjugate module L_{i} = M^{β_{i}} satisfies

α_{j}([L_{i}]) = \begin{cases}s & if j = i , \\ 0 & otherwise . \end{cases}

By repeated applications of Theorem 5.1, we obtain

L_{1} ⊗ L_{2} ⊗ L_{3} ⊗ L_{4} ⊗ L_{5} = U ⊕ (free)

where U is an indecomposable endo-trivial module such that

\overline{U} = \overline{L}_{1} ⊕ \overline{L}_{2} ⊕ \overline{L}_{3} ⊕ \overline{L}_{4} ⊕ \overline{L}_{5} .

We obtain [U] = [Ω_{P}^{s} (k)] by injectivity of α, hence U ≅ Ω_{P}^{s} (k) because both modules are indecomposable. For some m, we have that Dim(\overline{M}) = 2m, hence Dim(\overline{U}) = 10m and Dim(Ω_{P}^{s} (k)) = Dim(U) = 20m + 1.

Calculations using the MAGMA computer algebra system [BoCa] yield

Dim(Ω_{P}^{2} (k)) = 3 · 32 + 1 ,
Dim(Ω_{P}^{4} (k)) = 9 · 32 + 1 ,
Dim(Ω_{P}^{6} (k)) = 14 · 32 + 1 ,
Dim(Ω_{P}^{8} (k)) = 15 · 32 + 1

(actually the mod-2 cohomology of P can be computed using Quillen’s theorem [BeCa] and the dimensions of the cohomology groups turn out to be 1, 4, 9, 15, 21, 26, 29, 30, …). We know that Dim(Ω_{P}^{s} (k)) = 20m + 1 but the only value of s for which 5 divides Dim(Ω_{P}^{s} (k)) − 1 is s = 8. This proves the first step of induction.

For the second part of the proof, we have to assume that the number of nonzero components in α([M]) is at least 2. An important difference in the argument concerns the periodicity of \overline{M}. We have S_{1} < E_{1} < S_{1} × Q_{8} and we know that [M↓_{E_{1}}^{P}] = 0. But

Res_{E_{1}}^{S_{1}×Q_{8}} : T (S_{1} × Q_{8}) ⟶ T (E_{1})

is an isomorphism because E_{1} is the only maximal elementary abelian subgroup of S_{1} × Q_{8} (Theorem 3.1). Therefore [M↓_{S_{1}×Q_{8}}^{P}] = 0, that is,

M↓_{S_{1}×Q_{8}}^{P} ≅ k ⊕ (free) .

By Lemma 4.1, it follows that \overline{M} is free on restriction to \overline{S_{1} × Q_{8}}. Since S_{1}×Q_{8} is a maximal subgroup of P we deduce that \overline{M} is a periodic k\overline{P} -module (Lemma 5.3 in [CaTh2]).

The rest of the proof is exactly the same as that of Theorem 6.1. We apply Theorem 4.3 to construct a module L such that α([L]) has a unique non-zero component and we apply induction to [L] and [M] − [L].

Finally we consider almost extraspecial groups for the prime 2. The only case to treat is the following group of order 16.

Theorem 6.4. Let p = 2 and let P = D_{8}∗C_{4} be the central product of the dihedral group of order 8 and the cyclic group of order 4 (so P is almost extraspecial of order 16). Let E_{1}, E_{2}, E_{3} be the three maximal elementary abelian subgroups of rank 2, and for each i let S_{i} be a non-central subgroup of order 2 in E_{i}. Then the group T (P) is free abelian generated by the classes of the modules Ω_{P}^{1} (k), N_{2}, N_{3}, where N_{i} = Γ((Ω_{P}^{−1}(k) ⊗ Ω_{P/S_{i}}^{1}(k))^{⊗^{2}}) for 2 ≤ i ≤ 3.

Proof. The strategy is exactly the same as that of the proof of Theorem 6.3 and we only indicate the changes. Let u, v be two elements of order 2 generating D_{8} and let a be a generator of C_{4}. Then, up to conjugation, u, v, uva are the three noncentral elements of order 2 and they generate subgroups S_{1}, S_{2}, S_{3} respectively. Moreover, together with the central element a^{2}, every S_{i} generates one of the three maximal elementary abelian subgroups E_{i} of rank 2. We have C_{P} (S_{i}) ≅ S_{i} × C_{4} so we need second tensor powers in the description of the generators of ℚ ⊗ T (P) given by Theorem 3.1.

More precisely, Theorem 3.1 asserts that the map α : T (P) ⟶ ℤ^{3} is injective and that α([Ω_{P}^{1} (k)]) = (1, 1, 1), α([N_{2}]) = (0, −4, 0), and α([N_{3}]) = (0, 0, −4). We have to prove that any class of endo-trivial modules is a linear combination of these elements.

In the first case where α([M]) has a unique non-zero component s, we can assume that s is even and s ≤ 4 and we have to prove that s = 4. We construct again an indecomposable endo-trivial module U such that

\overline{U} = \overline{L}_{1} ⊕ \overline{L}_{2} ⊕ \overline{L}_{3} ,

where L_{i} is a conjugate of M by an automorphism of P. We then obtain U ≅ Ω_{P}^{s} (k). We have Dim(\overline{M}) = 2m for some m, hence Dim(\overline{U}) = 6m and Dim(Ω_{P}^{s} (k)) = Dim(U) = 12m + 1. On the other hand, by direct computation (e.g. using the MAGMA computer algebra system) we obtain Dim(Ω_{P}^{2} (k)) = 33 and Dim(Ω_{P}^{4} (k)) = 49. Thus the only possibility is s = 4.

In the second part of the proof, we assume that the number of non-zero components in α([M]) is at least 2. We have again [M↓_{E_{1}}^{P}] = 0 and we need to prove that M is periodic. But we have S_{1} < E_{1} < S_{1} × C_{4} and

Res_{E_{1}}^{S_{1}×C_{4}} : T (S_{1} × C_{4}) ⟶ T (E_{1})

is an isomorphism because E_{1} is the only maximal elementary abelian subgroup of S_{1} × C_{4}. Therefore [M↓_{S_{1}×C_{4}}^{P}] = 0, that is,

M↓_{S_{1}×C_{4}}^{P} ≅ k ⊕ (free) .

By Lemma 4.1, it follows that \overline{M} is free on restriction to \overline{S_{1} × C_{4}}. Since S_{1} × C_{4} is a maximal subgroup of P we deduce that \overline{M} is a periodic k\overline{P} -module. The rest of the proof is exactly the same as that of Theorem 6.3.

7. The general case

The following is the main theorem of the paper. It, together with the results of [CaTh2], give a complete classification of endo-trivial modules for p-groups.

Theorem 7.1. Let P be a finite p-group having a maximal elementary abelian subgroup of rank 2 and suppose that P is not semi-dihedral. Fix the same notation as in Theorem 3.1. Then the group T (P) is free abelian generated by the classes of the modules Ω_{P}^{1} (k), N_{2}, …, N_{r}.

Proof. Consider the injective map α : T (P) ⟶ ℤ^{r} of Theorem 3.1. We know that α([Ω_{P}^{1} (k)]) = (1, 1, …, 1) and that

α([N_{i}]) = −n_{i}p·e_{i} .

Here, e_{i} = (0, …, 0, 1, 0, …, 0) the i-th element of the canonical basis of ℤ^{r}. We set n_{i} = 1 (resp. 2, resp. 4) if C_{P} (S_{i})/S_{i} ≅ L_{i} is cyclic of order 2 (resp. cyclic of order ≥ 3, resp. generalized quaternion). We have to prove that any class of endo-trivial modules is a linear combination of these elements.

Let M be any endo-trivial kP -module. By subtracting a multiple of [Ω_{P}^{1} (k)], we can assume that α_{1}([M]) = 0 and we then have to prove that [M] is a linear combination of [N_{2}], …, [N_{r}]. In other words

α([M]) = (0, s_{2}, s_{3}, …, s_{r})

and we have to prove that s_{i} is a multiple of n_{i}p for every i = 2, …, r. Without loss of generality, we can take i = 2.

By Lemma 2.2, we know that N_{P} (E_{2}) = R ∗ L, where R = E·E_{2} is extraspecial of order p^{3} and L = L_{2} is either cyclic or generalized quaternion. Recall that R is extraspecial of exponent p if p is odd and that R = D_{8} if p = 2. Clearly L contains a subgroup L′ defined as follows:

L′ = \begin{cases}C_{p} & if p is odd , \\ C_{2} & if p = 2 and L = C_{2} , \\ C_{4} & if p = 2 and L = C_{2^{m}} , m ≥ 2 , \\ Q_{8} & if p = 2 and L = Q_{2^{m}} , m ≥ 3 . \end{cases}

Then N_{P} (E_{2}) contains a subgroup Q = R ∗ L′ which is of the following type:

Q = \begin{cases}R & if p is odd , \\ D_{8} & if p = 2 and L = C_{2} , \\ D_{8} ∗ C_{4} & if p = 2 and L = C_{2^{m}} , m ≥ 2 , \\ D_{8} ∗ Q_{8} & if p = 2 and L = Q_{2^{m}} , m ≥ 3 . \end{cases}

The restriction map T (P) → T (E_{2}) factors through T (Q) and so does the map α. Thus we need only consider [M↓_{Q}^{P}] ∈ T (Q). The map of Theorem 3.1 for the group Q is an injection α_{Q} : T (Q) ⟶ ℤ^{r′} (for a certain r′) and the first two components of α_{Q}([M↓_{Q}^{P}]) are the same as those of α([M]), that is, 0 and s_{2}. We now conclude that s_{2} is a multiple of n_{2}p by applying the results of Section 6 for the group Q in each of the appropriate cases. More precisely, if p is odd, then Q = R is extraspecial of order p^{3} and exponent p, we have n_{2} = 2 and Theorem 6.1 applies. If p = 2, then we have to apply Theorem 6.4 if Q = D_{8} ∗ C_{4} (with n_{2} = 2), and apply Theorem 6.3 provided Q = D_{8} ∗ Q_{8} (with n_{2} = 4). We apply the remarks before Theorem 6.3 in the case that Q = D_{8} (with n_{2} = 1).

Corollary 7.2. Let P be a finite p-group. Then T (P) is generated by the classes of relative syzygies of the trivial module, except in the case that p = 2 and P is generalized quaternion.

Proof. If P is generalized quaternion, it is known that there are exceptional endotrivial modules (see [CaTh1]), so we exclude this case. The result is well-known (and easy) if P is cyclic and it can be found in Theorem 7.1 of [CaTh1] if P is semidihedral. If now P has a maximal elementary abelian subgroup of rank 2, then the result follows from Theorem 7.1 above. If all maximal elementary abelian subgroups have rank ≥ 3, then T (P) is generated by [Ω_{P}^{1} (k)] (Corollary 1.3 in [CaTh2]).

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

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

J. T.: Institut de géométrie, algèbre et topologie, EPFL, CH-1015 Lausanne, Switzerland

  E-mail address: Jacques.Thevenaz@epfl.ch