The classification of torsion endo-trivial modules
aa r X i v : . [ m a t h . G R ] J un Annals of Mathematics , (2005), 823–883 The classification of torsionendo-trivial modules By Jon F. Carlson ∗ and Jacques Th´evenaz
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 classificationof torsion endo-trivial modules for a finite p -group over a field of characteris-tic p . Our results also imply, at least when p is odd, the complete classificationof torsion endo-permutation modules.We refer to [CaTh] and [BoTh] for an overview of the problem and itsimportance in the representation theory of finite groups. Let us only mentionthat the classification of endo-trivial modules is the crucial step for under-standing the more general class of endo-permutation modules, and that endo-permutation modules play an important role in module theory, in particularas source modules, in block theory where they appear in the description ofsource algebras, and in both derived equivalences and stable equivalence ofblock 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 thata (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-trivialmodules are the Heller translates Ω n ( k ) of the trivial module. Any endo-trivial kG -module M is a direct sum M = M ⊕ L , where M is an indecomposableendo-trivial kG -module and L is free. Conversely, by adding a free moduleto an endo-trivial module, we always obtain an endo-trivial module. This de-fines an equivalence relation among endo-trivial modules and each equivalenceclass 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 mul-tiplication induced by tensor product, called simply the group of endo-trivial kG -modules. Since scalar extension of the coefficient field induces an injectivemap between the groups of endo-trivial modules, we can replace k by its alge-braic closure. So we assume that k is algebraically closed. We refer to [CaTh]for more details about T ( G ). ∗ The first author was partly supported by a grant from NSF. JON CARLSON AND JACQUES TH´EVENAZ
Dade [Da] proved that if A is a noncyclic abelian p -group then T ( A ) ∼ = Z ,generated by the class of Ω ( k ). For any p -group G , Puig [Pu] proved that theabelian group T ( G ) is finitely generated (but we do not use this here since itis 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 liesin the structure of the torsion subgroup T t ( G ).Let us first recall some important known cases (see [CaTh]). If G = 1or G = C , then T ( G ) = 0. If G = C p n is cyclic of order p n , with n ≥ p is odd and n ≥ p = 2, then T ( C p n ) ∼ = Z / Z (generated by theclass of Ω ( k )). If G = Q n is a quaternion group of order 2 n ≥
8, then T ( Q n ) = T t ( Q n ) ∼ = Z / Z ⊕ Z / Z . If G = SD n is a semi-dihedral groupof order 2 n ≥
16, then T (SD n ) ∼ = Z ⊕ Z / Z and so T t (SD n ) ∼ = Z / Z . Ourfirst main result asserts that these are the only cases where nontrivial torsionoccurs. Theorem 1.1.
Suppose that G is a finite p - group which is not cyclic , quaternion , or semi - dihedral. Then T t ( G ) = { } . As explained in [CaTh], the computation of the torsion subgroup T t ( G )is tightly connected to the problem of detecting nonzero elements of T ( G ) onrestriction to a suitable class of subgroups. A detection theorem was provedin [CaTh] and it was conjectured that the detecting family should actually onlyconsist of elementary abelian subgroups of rank at most 2 and, in addition when p = 2, cyclic groups of order 4 and quaternion subgroups Q of order 8. Thisconjecture is correct and the largest part of the present paper is concernedwith the proof of this conjecture.It is in fact only for the cases of cyclic, quaternion, and semi-dihedralgroups that one needs to include cyclic groups C p or C and quaternion sub-groups Q in the detecting family. For all the other cases, we are going toprove 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 Y E Res GE : T ( G ) −→ Y E T ( E ) ∼ = Y E Z is injective , where E runs through the set of all elementary abelian subgroupsof rank . 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 fromTheorem 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 HE CLASSIFICATION OF TORSION ENDO-TRIVIAL MODULES T ( G ) is a full lattice in a free abelian group A by showing that some explicitsubgroup S ( G ) of the same rank satisfies S ( G ) ⊆ T ( G ) ⊆ A . But there is stillthe problem of describing explicitly the finite group T ( G ) /S ( G ) ⊆ A/S ( G ).However, this additional problem only occurs if G contains maximal elementarysubgroups of rank 2 (see [Al2] or [BoTh] for details). In all other cases therank 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 maxi-mal elementary abelian subgroup has rank at least . Then T ( G ) ∼ = Z , generatedby the class of the module Ω ( k ) . For the proof of Theorem 1.2, we first use the results of [CaTh] which pro-vide 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 beeliminated from the detecting family. When p is odd, this was already donein [CaTh] for extraspecial p -groups of exponent p and almost extraspecial p -groups. So we are left with the remaining cases and we have to prove thefollowing theorem, which is in fact the main result we prove in the presentpaper. Theorem 1.4.
Suppose the following :(a) If p = 2, G is an extraspecial or almost extraspecial -group and G is notisomorphic to Q . (b) If p is odd , G is an extraspecial p -group of exponent p .Then the restriction homomorphism Y H Res GH : T ( G ) −→ Y 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 imme-diate consequences for the more general class of endo-permutation modules.The second goal of the present paper is to describe the consequences of themain results for the classification of torsion endo-permutation modules. Weprove a detection theorem for the Dade group of all endo-permutation mod-ules 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, by theresults of [BoTh].26 JON CARLSON AND JACQUES TH´EVENAZ
Theorem 1.5. If p is odd and G is a finite p -group , the torsion sub-group of the Dade group of all endo-permutation kG -modules is isomorphicto ( Z / Z ) s , where s is the number of conjugacy classes of nontrivial cyclicsubgroups of G . One set of s generators is described in [BoTh]. Since an element of or-der 2 corresponds to a self-dual module, we obtain in particular the followingcorollary. Corollary 1.6. If p is odd and G is a finite p -group , then an indecom-posable endo-permutation kG -module M with vertex G is self-dual if and onlyif 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 lyingin the Dade group (e.g. sources of simple modules) are either known or expectedto lie in the torsion subgroup, while it is not at all clear why the modules shouldbe self-dual.When p = 2, the situation is more complicated but we obtain that anytorsion element of the Dade group has order 2 or 4. Moreover, the detectionresult is efficient in some cases, but examples also show that it is not alwayssufficient to determine completely this torsion subgroup.Theorem 1.4 is the result whose proof requires most of the work. Theresult has to be treated separately when p = 2 or when p is odd. However, thestrategy is similar and many of the same methods are of use for the proof inboth cases. After a preliminary Section 2 and two sections about the cohomol-ogy 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 of support varieties of modules. The crucial role of Serre’stheorem on products of Bocksteins appears once again and we actually need abound for the number of terms in this product that was recently obtained byYal¸cin [Ya] for (almost) extraspecial groups. Also, the module-theoretic coun-terpart of Serre’s theorem described in [Ca2] plays a crucial role. All theseresults allow us to find an upper bound for the dimension of an indecompos-able 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 nontrivial critical modulesfor extraspecial and almost extraspecial 2-groups, except Q , and also none forextraspecial p -groups of exponent p (with p odd).The existence of a bound for the dimension of a critical module had beenknown for more than 20 years and was used by Puig [Pu] in his proof of thefinite generation of T ( G ). The new aspect is that we are now able to controlthis bound for (almost) extraspecial groups. One of the differences betweenthe case where p = 2 and the case where p is odd lies in the fact that the HE CLASSIFICATION OF TORSION ENDO-TRIVIAL MODULES p some more estimates are necessary.Another difference is due to the fact that we have three families of groups toconsider when p = 2, but only one when p is odd, because the other two werealready dealt with in [CaTh].The other main idea in the proof of Theorem 1.4 is the following. Un-der the assumption that there exists a nontrivial critical module M , we canconstruct many others using the action of Out( G ) (which is an orthogonal orsymplectic group since G is (almost) extraspecial), and then construct a verylarge critical module by taking tensor products. The dimension of this largemodule exceeds the upper bound mentioned above and we have a contradic-tion. It is this part in which the theory of varieties associated to modulesplays an essential role. We use it to analyze a suitable quotient module M which turns out to be periodic as a module over the elementary abelian group G = G/ Φ( G ).Once Theorem 1.4 is proved, the proof of Theorem 1.2 requires muchless 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 opinionsin the course of the writing of this paper. Special thanks are due to C´edricBonnaf´e, Roger Carter, Ian Leary, Gunter Malle, and Jan Saxl. The firstauthor also wishes to thank the Humboldt Foundation for supporting his stayin Germany while this paper was being written.
2. Preliminaries
Recall that G denotes a finite p -group, and k an algebraically closed fieldof characteristic p . In this section we write down some of the facts aboutmodules 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, andthat injective and projective modules coincide. Moreover, an indecomposable kG -module M is free if and only if t G · M = 0, where t G = P g ∈ G g (a generatorof the socle of kG ). More generally, if M is a kG -module and if m , . . . , m r ∈ M are such that t G m , . . . , t G m r are linearly independent, then m , . . . , m r generate a free submodule F of M of rank r . Moreover F is a direct summandof 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 and28 JON CARLSON AND JACQUES TH´EVENAZ define inductively Ω n ( M ) = Ω(Ω n − ( M )), for n >
1. Suppose that M µ −→ Q is an injective hull of M . Recall that Q is a projective as well as injectivemodule. Then we let Ω − ( M ) be the cokernel of µ . Again we have inductivelythat Ω − n ( M ) = Ω − (Ω − n +1 ( M )) for n >
1. The modules Ω n ( M ) are welldefined up to isomorphism and they have no nonzero projective submodules.In general we write M = Ω ( M ) ⊕ P where P is projective and Ω ( M ) has noprojective summands.The basic calculus of the syzygy modules Ω n ( M ) is expressed in the fol-lowing. 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 theaction of the group G defined diagonally, g ( m ⊗ n ) = gm ⊗ gn . The proof ofthe lemma is a consequence of the facts that M ⊗ k − and − ⊗ k N preserveexact sequences and that M ⊗ N is projective whenever either M or N is aprojective module.The cohomology ring H * ( G, k ) is a finitely generated k -algebra and forany kG -modules M and N , Ext ∗ kG ( M, N ) is a finitely generated module overH * ( G, k ) ∼ = Ext ∗ kG ( k, k ). We let V G ( k ) denote the maximal ideal spectrum ofH * ( G, k ). For any kG -module M , let J ( M ) be the annihilator in H * ( G, k ) ofthe cohomology ring Ext ∗ kG ( M, M ). Let V G ( M ) = V G ( J ( M )) be the closedsubset 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 supportvarieties in essential ways in the course of our proofs. See the general references[Be], [Ev] for more explanations and details. Theorem 2.2.
Let
L, M and N be kG -modules. (1) V G ( M ) = { } if and only if M is projective. (2) If → L → M → N → 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 ) = { } , 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 -dualof M . (5) If V G ( M ) = V ∪ V where V and V are nonzero closed subsets of V G ( k ) and V ∩ V = { } , then M ∼ = M ⊕ M where V G ( M ) = V and V G ( M )= V . HE CLASSIFICATION OF TORSION ENDO-TRIVIAL MODULES
A nonprojective module M is periodic ( i.e. for some n >
0, Ω n ( M ) ∼ =Ω ( M )) if and only if its variety V G ( M ) is a union of lines through theorigin in V G ( k ) . (7) Let ζ ∈ Ext nkG ( 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 −→ L −→ Ω n ( k ) ζ −→ k −→ . Then V G ( L ) = V G ( ζ ), the variety of the ideal generated by ζ , consistingof all maximal ideals containing ζ . We are particularly interested in the case in which the group G is anelementary abelian group. First assume that p = 2 and G = h x , . . . , x n i ∼ =( C ) n . Then H * ( G, k ) ∼ = k [ ζ , . . . , ζ n ] is a polynomial ring in n variables. Herethe elements ζ , . . . , ζ n are in degree 1 and by proper choice of generators wecan assume that res G, h x i i ( ζ j ) = δ ij · γ i where γ i ∈ H ( h x i i , k ) is a generator forthe cohomology ring of h x i i . Indeed if we assume that the generators are chosencorrectly, then for any α = ( α , . . . , α n ) ∈ k n , u α = 1 + P ni =1 α i ( x i − ∈ kG , U = h u α i , we have thatres G,U ( f ( ζ , . . . , ζ n )) = f ( α , . . . , α n ) γ tα where f is a homogeneous polynomial of degree t and γ α ∈ H ( U, k ) is agenerator of the cohomology ring of U .Now suppose that p is an odd prime and let G = h x , . . . , x n i ∼ = ( C p ) n .Then H * ( G, k ) ∼ = k [ ζ , . . . , ζ n ] ⊗ Λ( η , . . . , η n ) , where Λ is an exterior algebra generated by the elements η , . . . , η n in degree1 and the polynomial generators ζ , . . . , ζ n are in degree 2. We can assumethat each ζ j is the Bockstein of the element η j and that the elements can bechosen so that res G, h x i i ( ζ j ) = δ ij · γ i where γ i ∈ H ( h x i i , k ) is a generator for thecohomology ring of h x i i . Similarly, assuming that the generators are chosencorrectly, for any α = ( α , . . . , α n ) ∈ k n , u α = 1 + P ni =1 α i ( x i − ∈ kG , U = h u α i , we have thatres G,U ( f ( ζ , . . . , ζ n )) = f ( α p , . . . , α pn ) γ tα where f is a homogeneous polynomial of degree t and γ α ∈ H ( U, k ) is agenerator of the cohomology ring of U .Associated to a kG -module M we can define a rank variety V rG ( M ) = n α ∈ k n | M ↓ h u α i is not a free h u α i -module o ∪ { } where u α is given as above and where M ↓ h u α i denotes the restriction of M tothe subalgebra k h u α i of kG . Then we have the following result for any p .30 JON CARLSON AND JACQUES TH´EVENAZ
Theorem 2.3.
Let M be any kG -module. If p = 2 then , V rG ( M ) = V G ( M ) as subsets of k n . If p > then the map V G ( M ) −→ V rG ( M ) given by α α p = ( α p , . . . , α pn ) is an inseparable isogeny ( both injective and surjective ). Inparticular , for α = 0, α p ∈ V G ( M ) ( α ∈ V G ( M ) if p = 2) if and only if M ↓ h u α i is not a free k h u α i -module. We should emphasize that if v is a unit in kG such that v ≡ u α mod(Rad( kG ) )then M ↓ h v i is a free k h v i -module if and only if α p V G ( M ) ( α V G ( M ) if p = 2). So for example the element x x x fails to act freely on M if and onlyif (1 , , , , . . . , ∈ V G ( M ).
3. Extraspecial groups in characteristic 2
In this section and the next, we are interested in the structure and coho-mology of extraspecial and almost extraspecial p -groups. These are preciselythe p -groups G with the property that G has a unique normal subgroup Z oforder p such that G/Z is elementary abelian. Note that the dihedral group D of order 8 and, more generally, the Sylow p -subgroup of GL(3 , p ) are extraspe-cial p -groups. The quaternion group Q of order 8 and the cyclic group C p of order p also have the required property. Indeed, for p = 2 any extraspecialor almost extraspecial group is constructed from copies of D , Q and C bytaking central products. In this section we concentrate on the case p = 2 andlook more deeply into the structure of the extraspecial and almost extraspecialgroup and their cohomology.Suppose that G and G are 2-groups with the property that each has aunique normal subgroup of order 2. Let h z i i ∈ G i be the subgroups. Then thecentral product G ∗ G is defined by G ∗ G = ( G × G ) / h ( z , z ) i . It is not difficult to check that D ∗ D ∼ = Q ∗ Q and that D ∗ C ∼ = Q ∗ C .Moreover, C ∗ C has a central elementary abelian subgroup of order 4 andhence is not of interest to us (it is neither extraspecial nor almost extraspecial).We are left with three types. They are:Type 1. G = D ∗ D ∗ · · · ∗ D of order 2 n +1 where n is the number offactors in the central product.Type 2. G = D ∗ · · · ∗ D ∗ Q of order 2 n +1 where n is the number offactors in the central product.Type 3. G = D ∗ · · · ∗ D ∗ C of order 2 n +2 where n is the number offactors isomorphic to D .The groups of type 1 and 2 are the extraspecial groups (see [Go1]) whilethe groups of type 3 are what we call the almost extraspecial groups. HE CLASSIFICATION OF TORSION ENDO-TRIVIAL MODULES −→ Z −→ G µ −→ E −→ Z = h z i is the unique central normal subgroup of order 2 and E ∼ = F m is elementary abelian. Recall that a quadratic form on E (as a vector spaceover F ) is a map q : E −→ F with the property that q ( x + y ) = q ( x ) + q ( y ) + b ( x, y )where b : E × E −→ F 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˜ x , ˜ y are elements of G and if µ (˜ x ) = x and µ (˜ y ) = y , then˜ x = z q ( x ) and [˜ x, ˜ y ] = z b ( x,y ) . Notice here that we are writing the operation in G as multiplication. Given thestructure of the groups, it is not difficult to write down the associated quadraticforms. With respect to a choice of basis, E can be identified with F m and inthe sequel we make this identification. Thus we write x = ( x , . . . , x m ) for theelements of E . Lemma 3.1.
Let G be an extraspecial or almost extraspecial group oforder m +1 . Then the quadratic form q associated to G is given on x =( x , . . . , x m ) ∈ F m = E as follows.For type q ( x ) = x x + · · · + x n − x n ( m = 2 n ) .For type q ( x ) = x x + · · · + x n − x n − + x n − + x n − x n + x n ( m = 2 n ) .For type q ( x ) = x x + · · · + x n − x n + x n +1 ( m = 2 n + 1) . Now on the k -vector space V = k m of dimension m , let q, b denote the sameforms but with the field of coefficients expanded from F to k . Let F : k → k be the Frobenius homomorphism, F ( a ) = a . If ν = ( x , . . . , x m ) ∈ V , let F act on ν by F ( ν ) = ( x , x , . . . , x m ). Recall that a subspace W ⊆ V is isotropicif 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 sub-space of V . The values of h for the quadratic forms associated to the abovegroups are : h = n for G of type m = 2 n ), h = n + 1 for G of type m = 2 n ) or type m = 2 n + 1). Moreover h is the index in G of a maximal elementary abelian subgroup. JON CARLSON AND JACQUES TH´EVENAZ
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 extraspecialgroup of order m +1 . If ν = ( x , . . . , x m ), then H * ( G, k ) = k [ x , . . . , x m ] / ( q ( ν ) , b ( ν, F ( ν )) , . . . , b ( ν, F h − ( ν ))) ⊗ k [ δ ] where δ is an element of degree h that restricts to a nonzero element of Z .Moreover the elements q ( ν ) , b ( ν, F ( ν )) , . . . , b ( ν, F h − ( ν )) form a regular se-quence in k [ x , . . . , 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 -group.Define t = t G to be the natural number given as follows. If G is of type oforder n +1 , let t G = ( n − + 1 for n ≤ , n − + 2 n − for n ≥ . If G is of type of order n +1 or of type of order n +2 , then let t G = ( for n = 1 , n + 2 n − for n ≥ . Then there exist nonzero elements ζ , . . . , ζ t ∈ H ( G, F ) such that ζ . . . ζ t = 0 . Moreover , in the isomorphism H ( G, F ) ∼ = Hom( G, F ), each ζ i corre-sponds to a homomorphism whose kernel is a maximal subgroup of G and isthe 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 ofnonzero elements in H ( G, F ) such that the product of those elements is zero(see [Ya, Th. 1.3]).Now, suppose that G has type 2 or 3. Then t G in our theorem is equal tothe cardinality s ( G ) of a representing set in G (see [Ya, Props. 6.2 and 6.3]).A representing set for G is a collection of elements of G that contains at leastone noncentral element from each elementary abelian subgroup of G . But nowProposition 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 maximalelementary abelian subgroup of G is contained in the centralizers of some ele-ments in a representing set. Because the cohomology ring H ∗ ( G, F ) is Cohen-Macaulay (see Theorem 3.3), any element whose restriction to the centralizer ofevery 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 HE CLASSIFICATION OF TORSION ENDO-TRIVIAL MODULES
Theorem 3.5.
Let G be an extraspecial or almost extraspecial group oforder m +3 and let H be the centralizer of a noncentral involution in G . Then H ∼ = C × U where U is an extraspecial or almost extraspecial group of or-der m +1 of the same type as G . Assume that m ≥ and , if m = 2, that U = D . Then for ≤ r ≤ t G ,Dim H r ( H, k ) ≤ (cid:18) m + rr (cid:19) − (cid:18) m + r − r − (cid:19) . Proof . The structure of the centralizer H can be verified directly fromwhat we know of G . For one thing it can be checked that all noncentralinvolutions 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 coho-mology of the group U , so that H ∗ ( U, k ) is generated by x , . . . , x m and δ , withdeg( 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 , k )and moreover we know that H ∗ ( C , k ) ∼ = k [ y ] is a polynomial ring in onevariable y in degree 1. We want to focus on the polynomial ring S generatedby x , . . . , x m , y . We have a homomorphism from S to H ∗ ( H, k ) whose kernelcontains the elements q ( ν ) and β ( ν, F ( ν )) where ν = ( x , . . . , x m ). Let Q denote the image of S in H ∗ ( H, k ). For this argument, let S = S/ ( q ( ν )) andlet 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 = 0if r < S r = (cid:0) m + rm (cid:1) = (cid:0) m + rr (cid:1) . Because q ( ν ) and β ( ν, F ( ν ))are two terms of a regular sequence of elements in S we must have thatDim S r = Dim S r − Dim S r − and Dim S r = Dim S r − Dim S r − 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 ≤ t U (with equality in most cases) and moreover,by Lemma 3.2, we see that t U < h in all cases. The choice that r ≤ t G nowmeans that r ≤ t G ≤ t U < · h = 2 · deg( δ )34 JON CARLSON AND JACQUES TH´EVENAZ 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 < δ ).Notice also that deg( δ ) = 2 h ≥ m ≥ U = D (if U ∼ = D , then h = 1 and deg( δ ) = 2). Hence we have thatDim H r ( H, k ) ≤ Dim Q r + Dim Q r − deg( ζ ) ≤ Dim S r − Dim S r − + Dim S r − deg( ζ ) − Dim S r − deg( ζ ) − ≤ Dim S r − Dim S r − + Dim S r − deg( ζ ) ≤ Dim S r = (cid:18) m + rr (cid:19) − (cid:18) m + r − r − (cid:19) . The last inequality follows from the facts that r − deg( δ ) ≤ r − S s is an increasing function of s . Corollary 3.6.
Suppose that G and H are as in the theorem. If ≤ r ≤ t G , then r X i =0 Dim Ω i ( k H ) ↑ GH ≤ (cid:18) m + r − m (cid:19) | G | + 2 . Proof . For any i we have an exact sequence0 −→ Ω i +1 ( k H ) −→ P i −→ Ω i ( k H ) −→ P i is the degree i term in a minimal kH -projective resolution of thetrivial kH -module k H . Recall that Dim P i = Dim H i ( H, k ) · | H | . Then by thetheorem, for r = 2 s + 1, r X i =0 Dim Ω i ( k H ) = s X j =0 (cid:0) Dim Ω j +1 ( k H ) + Dim Ω j ( k H ) (cid:1) = s X j =0 Dim P j ≤ Dim P + s X j =1 h(cid:18) m + 2 j j (cid:19) − (cid:18) m + 2 j − j − (cid:19)i | H | = | H | + h(cid:18) m + 2 s s (cid:19) − (cid:18) m (cid:19)i | H | = (cid:18) m + r − r − (cid:19) | H | = (cid:18) m + r − m (cid:19) | H | . On the other hand if r = 2 s is even, then we use the fact that Dim P = (cid:0) m +11 (cid:1) | G | and we obtain similarly r X i =0 Dim Ω i ( k H ) = Dim k + Dim P + s X j =2 Dim P j − ≤ (cid:18) m + 2 s − s − (cid:19) | H | = 1 + (cid:18) m + r − m (cid:19) | H | . HE CLASSIFICATION OF TORSION ENDO-TRIVIAL MODULES H to G , the dimension of Ω i ( k H ) ↑ GH is doubledand the result follows.
4. Extraspecial groups in odd characteristic
Our aim in this section is to get results similar to those of the last sectionfor extraspecial p -groups in the case that the prime p is not 2. As in thecharacteristic 2 case, for any positive integer n there are two isomorphismtypes of extraspecial groups of order p n +1 and one isomorphism type of almostextraspecial group of order p n +2 . For each n , one of the two nonisomorphicgroups of order p n +1 has exponent p and the other one has exponent p . Inthe earlier paper [CaTh] we showed that Theorem 1.4 holds for extraspecialgroups of exponent p and almost extraspecial groups (i.e. for these groupsthere are no nontrivial critical modules). As a consequence, the only groupsof interest to us are the extraspecial groups of order p n +1 and exponent p .Up to isomorphism, there is exactly one extraspecial group G of order p and exponent p . It is generated by elements x , y and z , which satisfy therelations that z is in the center of G , z p = x p = y p = 1 and [ x, y ] = z . It isisomorphic to the Sylow p -subgroup of the general linear group GL(3 , p ). For n >
1, the extraspecial group of order p n +1 is a central product G n = G ∗ G ∗ . . . ∗ G of n copies of G as in the last section. That is, G n is the quotient groupobtained by taking the direct product of n copies of G and then identifyingthe centers (see [Go1]). The center of G n is a cyclic subgroup Z = h z i of order p and G n /Z is an elementary abelian p -group of order p n .We need an analogue to Theorem 3.4 for our case. Theorem 4.1.
For G = G , let t G = 2( p + 1), while for G = G n , n > let t G = ( p + p − p n − . Then there exist nonzero elements η , . . . , η t ∈ H ( G, F p ) such that β ( η ) . . . β ( η t ) = 0 where t = t G . Moreover , in the iso-morphism H ( G, F p ) ∼ = Hom( G, F p ), each η i corresponds to a homomorphismwhose kernel is a maximal subgroup of G and is the centralizer of a noncentralelement of order p in G .Proof . The proof of the theorem is contained in the paper by Yal¸cin asTheorem 1.2 of [Ya]. In this case the dimension of H ( G, F p , ) is the same asthat of Hom( G, F p ) which is 2 n .As in the last section we are going to need estimates on the dimensionsof the cohomology groups H r ( G n , k ) where k is a field of characteristic p . Webegin with the case of the extraspecial group G = G of order p . Ian Leary[Le1] has given a complete description of the cohomology ring H * ( G, k ) except36
JON CARLSON AND JACQUES TH´EVENAZ that he did not fully compute the Poincar´e series, which is something that weneed. The calculation is, of course, implicit in his work, and he did calculateit in the special case that p = 3 [Le2]. Note that our results agree with his inthat situation. Theorem 4.2.
The Poincar ´ e series for the cohomology ring of the group G = G is given by the rational function ∞ X n =0 Dim H n ( G, k ) t n = 1 + t + 2 t + 2 t + t + t + · · · + t p − (1 − t )(1 − t p ) . Proof . We will not repeat the long list of relations given by Leary (The-orem 6 of [Le1]). However we will use exactly the notation of that paper andthe interested reader can follow the computation. The strategy is first to ig-nore the contribution of the regular element z in degree 2 p . This element is anondivisor of zero as it restricts nontrivially to the center of G . We also knowthat it is regular from the given relation and from the fact that it is representedon the E of the spectral sequence, by an element in E , p which survives tothe E ∞ page of the spectral sequence. Consequently, the Poincar´e series f ( t )of H * ( G, k ) is obtained by multiplying 1 / (1 − t p ) times the Poincar´e series ofthe 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 gen-erated by x and x ′ . Note that x and x ′ are in degree 2 and satisfy therelation x p x ′ − xx ′ p = 0 in degree 2 p + 2. So the Poincar´e series for R is f = (1 − t p +2 ) / (1 − t ) . This is also the series for the R -submodule M generated by the element 1 in degree 0. The first thing that needs to be es-tablished from the relations is that the R -generators are the elements of thesequence S = [1 , y, y ′ , Y, Y ′ , X, X ′ , yY ′ , XY ′ , XX ′ , d , c , d , . . . , c p − , d p ]of length 2 p + 3. Let M i be the R -submodule generated by the first i elementsof the sequence, and let f i be the Poincar´e series for M i /M i − . Then thedesired Poincar´e series for A is f + f + · · · + f p +3 . Note that f has beencalculated. • For f , we note that xy ′ = x ′ y and x p y ′ = x ′ p y . So x ′ ( x p − − x ′ p − ) y = 0.Therefore f = t (1 − t p ) / (1 − t ) . • Since xy ′ = x ′ y ∈ M , we have that f = t/ (1 − t ). • Similarly to the calculation for f , we have that f = t (1 − t p ) / (1 − t ) and f = t (1 − t p ) / (1 − t ) . • For f , note that x Y ′ = xx ′ Y ∈ M and xx ′ Y ′ ∈ M . Therefore f = t + t / (1 − t ). HE CLASSIFICATION OF TORSION ENDO-TRIVIAL MODULES • The calculation for f is similar to that for f and we get that f = t / (1 − t ). • For i := 8 , . . . , p + 3, it should be checked that xS i , x ′ S i ∈ M i − 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 = 3 while j i = i − i ≥ f + f + · · · + f p +3 = (1 + t + 2 t + 2 t + t + · · · + t p − ) / (1 − t )by routine but tedious calculation.We need to derive two facts from the above theorem. The first is an upperbound which is not optimal but will be sufficient for our purposes. Corollary 4.3.
For G = G ,Dim H r ( G, k ) ≤ r + 1) = 2 (cid:18) r + 11 (cid:19) . Moreover , Dim H r ( G, k ) = 2 r if ≤ r ≤ and Dim H r ( G, k ) = r + 3 if ≤ r ≤ p − .Proof . Consider the series expansion g ( t ) = 1 + t + 2 t + 2 t + t + · · · + t p − − t = ∞ X r =0 a r t r . A routine computation yields the value of the coefficients a = 1, a r = 2 r if1 ≤ r ≤ a r = r + 3 if 4 ≤ r ≤ p −
1, and a r = 2 p + 2 if r ≥ p −
1. ThePoincar´e series for the cohomology ring of G is obtained by multiplying g ( t )with − t p = P ∞ i =0 t ip . Therefore Dim H r ( G, k ) = a r for r ≤ p − r , writing r = j + q (2 p ) with 0 ≤ j < p , we have thatDim H r ( G, k ) = a j + qa p ≤ ( j + 3) + q (2 p + 2) ≤ r + 1) . Corollary 4.4.
For G = G , Dim Ω p ( k ) = p ( p + 1) + 1 .Proof . If P j is the j -th term of a minimal projective resolution of k , wehave 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 corol-lary, we obtain Dim Ω ( k ) = p + 1 and then by induction Dim Ω j − ( k ) =( j + 1) p − j ( k ) = ( j + 1) p + 1 for 2 ≤ j ≤ p .In the rest of the section, we require the following well known combinato-rial identity.38 JON CARLSON AND JACQUES TH´EVENAZ
Lemma 4.5.
For all integers c, i, j ≥ X a + b = c (cid:18) a + ii (cid:19)(cid:18) b + jj (cid:19) = (cid:18) c + i + j + 1 i + j + 1 (cid:19) . Proof . Recall that if P is a polynomial ring in n variables, then thenumber of monomials of degree r is (cid:0) r + n − n − (cid:1) . Now the tensor product of apolynomial ring in i + 1 variables with a polynomial ring in j + 1 variablesyields a polynomial ring in i + j + 2 variables. The identity follows by countingthe number of monomials of degree c .We also need to know the dimension of the cohomology groups of elemen-tary 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 ) = (cid:0) r + m − m − (cid:1) .Proof . Recall that H ∗ ( E, k ) ∼ = k [ ζ , . . . , ζ m ] ⊗ Λ( η , . . . , η m ) where ζ , . . . , ζ m are in degree 2 and η , . . . , η m are in degree 1. A basis of H r ( E, k ) consistsof the elements ζ a . . . , ζ a m m η e , . . . , η e m m where 0 ≤ a i ≤ r/
2, 0 ≤ e i ≤ P mi =1 (2 a i + e i ) = r . This basis is in bijection with the set of mono-mials of degree r in k [ x , . . . , x m ] by mapping the above basis element to x a + e . . . x a m + e m m . Now the number of monomials of degree r is (cid:0) r + m − m − (cid:1) .Our main result in this section gives estimates for the dimensions of thecohomology of the centralizers of p -elements. Theorem 4.7.
Let G = G n be an extraspecial group of order p n +1 andexponent p . Let H be the centralizer of a noncentral element of order p in G .Then H ∼ = C p × G n − . Moreover ,Dim H m ( H, k ) ≤ (cid:18) m + 2 n − n − (cid:19) . 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 elementsof 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 groupsof the group G n − for n ≥
1. The estimate in Corollary 4.3 will serve in thecase that n = 2. Let N be a normal subgroup of G n − such that N ∼ = G . Wecan take N to be the first factor in the central product that expresses G n − .Then G n − /N ∼ = C n − p , an elementary abelian group of order p n − . TheLyndon-Hochschild-Serre spectral sequence of the extension of G n − /N by N HE CLASSIFICATION OF TORSION ENDO-TRIVIAL MODULES E term E r,s = H r ( G n − /N, H s ( N, k )) ⇒ H r + s ( G n − , k ) . As k -vector spaces, it is true that E r,s ∼ = H r ( G n − /N, k ) ⊗ H s ( N, k ) because N commutes with the other factors of the central product. So we have thatDim H m ( G n − , k ) ≤ X r + s = m Dim( E r,s )= X r + s = m Dim H r ( G n − /N, k ) Dim H s ( N, k ) ≤ X r + s = m (cid:18) r + 2( n − − n − − (cid:19) (cid:18) s + 11 (cid:19) = 2 (cid:18) m + 2 n − n − (cid:19) , using Lemma 4.6, Corollary 4.3 and the combinatorial identity of Lemma 4.5.Now H m ( H, k ) ∼ = L r + s = m H r ( G n − , k ) ⊗ H s ( C p , k ). Therefore,Dim H m ( H, k ) = X r + s = m Dim H r ( G n − , k ) · Dim H s ( C p , k ) ≤ X r + s = m (cid:18) r + 2 n − n − (cid:19)(cid:18) s (cid:19) = 2 (cid:18) m + 2 n − n − (cid:19) , again by Corollary 4.3 and Lemma 4.5. Corollary 4.8.
Suppose that G and H are as in the theorem. If r ≥ then r X i =0 Dim Ω i ( k H ) ↑ GH ≤ p n +1 (cid:18) r + 2 n − n − (cid:19) . Proof . Suppose that · · · → P → P → k → kH -projective resolution of the trivial module k . Then we know that Dim Ω ( k ) +Dim Ω ( k ) = Dim P . For j ≥
2, Ω j ( k H ) is a submodule of P j − . The dimen-sion of P j is precisely | H | Dim H j ( H, k ) and the dimension of Ω j ( k H ) ↑ GH is p times the dimension of Ω j ( k H ). So from the theorem we have that r X i =0 Dim Ω i ( k H ) ↑ GH ≤ p | H | r − X i =0 Dim H i ( H, k ) ≤ p n +1 r − X i =0 (cid:18) i + 2 n − n − (cid:19)(cid:18) r − − i (cid:19) = 2 p n +1 (cid:18) r + 2 n − n − (cid:19) , by the identity 4.5.40 JON CARLSON AND JACQUES TH´EVENAZ
5. New endo-trivial modules from old endo-trivial modules
Here we start the proof of Theorem 1.4. Suppose that G is an extraspecialor almost extraspecial p -group and that G = Q . Let Z = h z i be the Frattinisubgroup of G , of order p , with elementary abelian quotient G = G/Z ofrank m . Let x , . . . , x m ∈ G such that G = h x , . . . , x m i . Recall that Z is theunique 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 sectionhold more generally if G has a Frattini subgroup Z of order p , but we leavethis generalization to the reader.Let M be an endo-trivial kG -module whose class in T ( G ) lies in the kernelof the restriction to proper subgroups. This means that M ↓ GH ∼ = k ⊕ (free) forevery maximal subgroup H of G . For the purpose of the proof of Theorem 1.4(Sections 5–11), we make the following definition: Definition
We say that a kG -module M is critical if it is an inde-composable endo-trivial module such that M ↓ GH ∼ = k ⊕ (free) for every maximalsubgroup H of G .Actually, the last condition implies that the module M is endo-trivialbecause 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 torsionendo-trivial module by a theorem of Puig [Pu], but we do not need this fact inour 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 doso. We shall often omit to mention this indecomposability condition, to theeffect that we shall usually only prove that a module satisfies the condition onrestriction to maximal subgroups in order to deduce that it is critical. Sinceour aim is to prove that the kernel above is trivial, we have to show thatany critical kG -module M is isomorphic to k as a kG -module. We will oftenassume, by contradiction, the existence of a nontrivial critical kG -module.In this section, we prove several results concerning the structure of acritical module M and the construction of new modules with the same property.For some of the results, we only need to assume that M ↓ GH ∼ = k ⊕ (free) for asingle subgroup H of G .For any critical kG -module M , and more generally for any kG -module M such that M ↓ GZ ∼ = k ⊕ (free), we let M ′ = { m ∈ M | ( z − p − m = 0 } and weset M = M/M ′ . We let − : M −→ M be the quotient map. Since ( z − M = 0, the module M can be viewed as a kG -module. A large part of this paper is devoted to ananalysis of the properties of the module M . HE CLASSIFICATION OF TORSION ENDO-TRIVIAL MODULES
Lemma 5.2.
Let M be a kG -module. Suppose that M ↓ GZ ∼ = k ⊕ (free) . (a) The module M has two filtrations K ⊂ K ⊂ . . . ⊂ K p − ⊂ K p = M ∪ ∪ ∪{ } ⊂ I p − ⊂ I p − ⊂ . . . ⊂ I where K i = { m ∈ M | ( z − i m = 0 } is the kernel of multiplication by ( z − i ( in particular K p − = M ′ ) and I i = ( z − i M is the image ofmultiplication by ( z − i . (b) K i /I p − i ∼ = k for any i = 1 , . . . , p − . Moreover K p − /I p − ∼ = k ⊕ ( I /I p − ) . (c) The module I = ( z − M is free as a module over the ring kZ/ ( z − p − .Moreover , I i /I i +1 ∼ = M for any i = 1 , . . . , p − . (d) The module
M/K is isomorphic to I . In particular it is free as a moduleover the ring kZ/ ( z − p − and K i +1 /K i ∼ = M for any i = 1 , . . . , p − . (e) Dim( M ) = p Dim( 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 − p = 0. The filtrations are clear.(b) In order to prove (b), it suffices to restrict to the subgroup Z . But wehave M ↓ GZ = k ⊕ F for some free kZ -module F , and therefore K i = k ⊕ ( z − p − i F , I p − i = ( z − p − i F .
Moreover it is clear that K p − /I p − = K /I p − ⊕ ( I /I p − ) ∼ = k ⊕ ( I /I p − ).(c) Multiplication by ( z − i induces a map M −→ ( z − i M/ ( z − i +1 M = I i /I i +1 and we claim that its kernel is M ′ . Again, in order to prove this, it suffices torestrict to the subgroup Z and consider the decomposition M ↓ GZ = k ⊕ F asabove. Then the kernel is k ⊕ ( z − F = M ′ . It is also clear that( z − M = ( z − F ∼ = F/ ( z − p − F and this is free over the ring kZ/ ( z − p − .(d) Multiplication by ( z −
1) induces an isomorphism
M/K ∼ = I .(e) Since M ↓ GZ = k ⊕ F , we have that Dim( M ) = Dim( F/ ( z − F ) =Dim( F ) /p and Dim( M ) = p Dim( M ) + 1. Lemma 5.3.
Let M be a kG -module. Suppose that there is a maximalsubgroup H of G such that M ↓ GH ∼ = k ⊕ (free) . JON CARLSON AND JACQUES TH´EVENAZ (a) M ∼ = k ⊕ (free) as a kG -module if and only if M is a free kG -module.More precisely , M has a free summand with r generators as a kG -moduleif and only if M has a free summand with r generators as a kG -module.In particular , if M is indecomposable , then M has no projective sum-mands. (b) M = k ⊕ (free) as a kG -module if and only if M is a periodic kG -module.Proof . (a) It is easy to see that if M has a free summand L ∼ = ( kG ) r as a kG -module then M has a free summand L/ ( z − L ∼ = ( kG ) r as a kG -module.The converse is essentially contained in Lemma 3.3 of [CaTh] and we recallthe argument. Assume that M = N ⊕ L where L is free and N has no freesummands. Then t G · N = 0 where t G = X g ∈ G g = ( z − p − m Y i =1 ( x i − p − ,x i being a lift in G of the generator x i of G . Let X = m Y i =1 ( x i − p − . If N has a free submodule then X · N = 0, since X = t G . But if X · N = 0then, via the isomorphism N ∼ = ( z − p − N of Lemma 5.2, we would obtain( z − p − X · N = t G · N = 0, which is a contradiction.(b) The hypothesis on M ↓ GH implies that M is free on restriction to H/Z .But H = H/Z is a maximal subgroup of G = G/Z , so
G/H is a cyclic groupof order p . Tensoring with M the exact sequence0 −→ k −→ k [ G/H ] −→ k [ G/H ] −→ k −→ , we obtain an exact sequence with M at both ends and free kG -modules in themiddle, because k [ G/H ] ⊗ M ∼ = M ↓ GH ↑ GH . If now M = k ⊕ (free), then M is not zero and is not free as a kG -module, by part (a), so M is periodic. Ifconversely M is periodic, then M is not free and M = k ⊕ (free) by part (a). Lemma 5.4.
Suppose that p = 2 and that M is a nontrivial critical kG -module. Then the number of generators of M is the same as the numberof generators of M and is equal to M ) / | G | . Moreover Dim(Ω( M )) =Dim(Ω − ( M )) = Dim( M ) − .Proof . Let H be a maximal subgroup of G . Since M ↓ GH ∼ = k ⊕ (free), weknow that M is free as a module over kH . Thus, the number of generators of M as a kH -module is Dim( M ) / | H | . Our first claim is that G acts trivially on M /
Rad( kH ) M . Thus, the number of generators of M as a kG -module is alsoDim( M /
Rad( kH ) M ) = Dim( M ) / | H | . In order to prove the claim, we note HE CLASSIFICATION OF TORSION ENDO-TRIVIAL MODULES
G/H acts on
M /
Rad( kH ) M . If there were a free summandgenerated by the class of an element m , then m would generate a free summandof M as a module over kG , contrary to part (c) of the previous lemma. Sincethe group G/H has order 2, the only possibility is that
G/H acts trivially on
M /
Rad( kH ) M .Now our second claim is that, given a set of generators of M , some liftsof those generators in M will generate M . If we asume this, it follows thatthe number of generators of M is Dim( M ) / | H | = 4 Dim( M ) / | G | . If r =4 Dim( M ) / | G | , then the projective cover of M is the free module ( kG ) r . UsingLemma 5.2 we obtainDim(Ω( M )) = Dim(( kG ) r ) − Dim( M )= 4 Dim( M ) − M ) − M ) − M ∗ also satisfies the assumptions ofthe lemma, we have thatDim(Ω − ( M )) = Dim(Ω − ( M ) ∗ )= Dim(Ω( M ∗ )) = Dim( M ∗ ) − M ) − L be the submod-ule of M generated by some lifts in M of the generators of M . Assume bycontradiction that L = M . Since M ↓ GH = k ⊕ F for some free kH -module F ,we have M ↓ GH = F/ ( z − F and so we can choose the lifts of the generatorsof M so that L ↓ GH = F . Now for any other maximal subgroup H ′ of G , wehave M ↓ GH ′ = k ⊕ F ′ for some free kH ′ -module F ′ . The subgroup H ∩ H ′ isnontrivial because it contains Z and there are two decompositions M ↓ GH ∩ H ′ = T ↓ HH ∩ H ′ ⊕ F ↓ HH ∩ 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 H ∩ H ′ · M in both decompositions, we see that T ′ ↓ H ′ H ∩ H ′ cannot be containedin F ↓ HH ∩ H ′ and therefore M ↓ GH ∩ H ′ = T ′ ↓ H ′ H ∩ H ′ ⊕ F ↓ HH ∩ H ′ (see Lemma 8.2 in [CaTh] for details). Since F is the restriction of a kG -submodule, this is a decomposition of M as a kH ′ -module, namely M ↓ GH ′ = T ′ ⊕ L ↓ GH ′ . By the Krull-Schmidt theorem, we deduce that L ↓ GH ′ is free. Since this holds forany maximal subgroup H ′ and since G is not elementary abelian, Chouinard’stheorem (see [Be] or [Ev]) implies that L is free as a kG -module and so M ∼ = k ⊕ L . But M is indecomposable and nontrivial by assumption. This contradic-tion completes the proof of the claim.44 JON CARLSON AND JACQUES TH´EVENAZ
For our next theorem, we first need a technical lemma.
Lemma 5.5.
Let W be a kG -module satisfying the following two condi-tions :(a) W/ ( z − W = U ⊕ U where U and U are kG -submodules such thatthe varieties satisfy V G ( U ) ∩ V G ( U ) = { } . (b) For some r ≤ p , there is ( z − r W = 0 and W is free as a module overthe ring kZ/ ( z − r .Then W = W ⊕ W where W and W are kG -submodules of W such that W i / ( z − W i ∼ = U i for i = 1 , .Proof . We use induction on r . There is nothing to prove if r = 1 so weassume r ≥
2. By induction, W/ ( z − r − W = V ⊕ V where V and V are kG -submodules of W/ ( z − r − W such that V i / ( z − V i ∼ = U i for i = 1 , W is free as a module over kZ/ ( z − r , multiplication by ( z − W/ ( z − r − W ∼ = ( z − W and we write L i for theimage of V i . So ( z − W = L ⊕ L .Let π : W → W/ ( z − W = U ⊕ U be the canonical surjection. Passingto the quotient by L , we obtain a short exact sequence0 −→ L −→ W/L e π −→ U ⊕ U −→ e π is induced by π . Let K = { x ∈ W/L | ( z − x = 0 } . We claim that e π ( K ) = U . Let x ∈ K and let w ∈ W be a lift of x . Then ( z − w ∈ L . Sincemultiplication by ( z −
1) induces an isomorphism W/ ( z − r − W ∼ = ( z − W ,the class of w in W/ ( z − r − W is in V . It follows that π ( w ) ∈ U , hence e π ( x ) ∈ U , proving the claim.Therefore we obtain a short exact sequence0 −→ ( z − r − L −→ K e π −→ U −→ L ∩ Ker( z −
1) = ( z − r − L . This is a sequence of kG -modulessince ( z − K = 0 by construction. Now multiplication by ( z − r − inducesan isomorphism W/ ( z − W ∼ = ( z − r − W mapping U onto ( z − r − L .By applying our assumption on the varieties of U and U we deduce that thesequence splits (see Theorem 2.2). Let σ be a section of e π : K → U and let W be the inverse image of σ ( U ) in W , so that W /L = σ ( U ). We haveobtained a short exact sequence0 −→ L −→ W π −→ U −→ . We can construct similarly a submodule W and a short exact sequence0 −→ L −→ W π −→ U −→ . HE CLASSIFICATION OF TORSION ENDO-TRIVIAL MODULES π ( W ∩ W ) = 0, so that W ∩ W ⊆ Ker( π ) = L ⊕ L . But since W i ∩ Ker( π ) = L i , we obtain W ∩ W = 0. For reasons of dimensions (or bya direct argument), the direct sum W ⊕ W must be the whole of W . Theorem 5.6.
Let M be a critical kG -module and suppose that M = M ⊕ M where M and M are kG -submodules. Suppose that the varietiessatisfy V G ( M ) ∩ V G ( M ) = { } . Then there exist critical kG -modules N and N such that N i ∼ = M i for ≤ i ≤ .Proof . As before, let M ′ = { m ∈ M | ( z − p − m = 0 } . Let M ⊆ M be the inverse image of M under the quotient map M −→ M/M ′ = M . Let M be the inverse image of M . Then M ′ = M ∩ M and M /M ′ ∼ = M , M /M ′ = M .By Lemma 5.2, ( z − M is free over kZ/ ( z − p − and( z − M/ ( z − M ∼ = M/M ′ = M = M ⊕ M . Therefore Lemma 5.5 applies and we have ( z − M = W ⊕ W such that W i / ( z − W i ∼ = M i for i = 1 ,
2. Now define N = M /W and N = M /W .If r i = Dim( M i ), then Dim( M ) = r + r and by Lemma 5.2 we obtainDim( M ) = pr + pr + 1 and Dim(( z − M ) = ( p − r + ( p − r . Thereforewe have Dim( M ) = pr + ( p − r + 1 and Dim( M ) = ( p − r + pr + 1.Also Dim( W i ) = ( p − r i ; hence Dim( N i ) = pr i + 1 for i = 1 , N ↓ GH ∼ = k ⊕ (free) for every maximal subgroup H of G (andsimilarly for N ). Let H = h z, y , . . . , y m − i where y , . . . , y m − are generatorsof H = H/Z . The assumption on M ↓ GH implies that M is free as a kH -module.Therefore M and M must be free as kH -modules. Let Y = m − Y i =1 ( y i − p − so that Y = t H and Y ( z − p − = t H . Then we getDim( M ) = | H | · Dim( Y · M ) . Now ( z − p − N ∼ = ( z − p − M because N = M /W and ( z − p − W = 0.Therefore t H · N = Y ( z − p − N ∼ = Y ( z − p − M ∼ = Y · M = Y · M . It follows that | H | Dim( t H · N ) = p · | H | · Dim( Y · M ) = p · Dim( M ) = pr = Dim( N ) − . Therefore N ↓ GH has a free submodule of dimension Dim( N ) −
1. The onlyway this can happen is if N ↓ GH ∼ = k ⊕ (free).46 JON CARLSON AND JACQUES TH´EVENAZ
Now we prove that N ∼ = M (and similarly for N ). We have to computethe submodule N ′ = { x ∈ N | ( z − p − x = 0 } . But N = M /W andwe have W ⊆ M ′ ⊆ M and ( z − p − M ′ = 0. Therefore M ′ /W ⊆ N ′ and N = N /N ′ is a quotient of N / ( M ′ /W ) ∼ = M /M ′ = M . In order toprove that this is not a proper quotient, it suffices to prove that N and M have the same dimension. But by the previous part of the proof, we know that N ↓ GH ∼ = k ⊕ (free) for every maximal subgroup H . By Lemma 5.2 this impliesDim( N ) = Dim( N ) − p = r = Dim( M ) , as was to be shown.Finally we conclude that N is critical. Indeed, since M has no freesummand as a kG -module, N cannot have a free summand and therefore N has no free summand as a kG -module by Lemma 5.3. This implies that N iscritical since we know that N ↓ GH ∼ = k ⊕ (free) for every maximal subgroup H . Theorem 5.7.
Let M and M be critical kG -modules and suppose thatthe varieties satisfy V G ( M ) ∩ V G ( M ) = { } . Then M ⊗ M ∼ = M ⊕ (free) where M is a critical kG -module such that M ∼ = M ⊕ M .Proof . Let r j = Dim( M j ) for j = 1 ,
2. Thus Dim( M j ) = pr j + 1. Considerthe filtration of M as in Lemma 5.2 { } ⊂ ( z − p − M ⊂ K ⊂ · · · ⊂ K p − ⊂ K p = M , where K i = { m ∈ M | ( z − i m = 0 } . This induces a filtration on M ⊗ M { } ⊂ ( z − p − M ⊗ M ⊂ K ⊗ M ⊂ · · · ⊂ K p − ⊗ M ⊂ M ⊗ M , with all quotients but one isomorphic to M ⊗ M . We need to prove thefollowing. Lemma 5.8. M ⊗ M = F ⊕ L where L ∼ = M and F is a free kG -moduleof dimension pr r such that ( z − p − F = M ⊗ ( z − p − M .Proof . By hypothesis V G ( M ) ∩ V G ( M ) = { } and hence M ⊗ M isprojective as a kG -module. Choose elements m , . . . , m r ∈ M ⊗ M such that m , . . . , m r is a free kG -basis for M ⊗ M . Here m i = m i + ( M ⊗ M ′ ) denotesthe class of m i in M ⊗ M = ( M ⊗ M ) / ( M ⊗ M ′ ).As before, let X = m Y i =1 ( x i − p − so that X = t G and X ( z − p − = t G . Then Xm , . . . , Xm r are linearly independent in M ⊗ M . Since z acts HE CLASSIFICATION OF TORSION ENDO-TRIVIAL MODULES M , multiplication by ( z − p − induces an isomorphism M ⊗ M ∼ = M ⊗ ( z − p − M and it follows that X ( z − p − m , . . . , X ( z − p − m r are linearly independent in M ⊗ ( z − p − M . Therefore t G m , . . . , t G m r are linearly independent in M ⊗ M . So the kG -submodule F of M ⊗ M generated by m , . . . , m r is a free kG -module. Moreover we have ( z − p − F = M ⊗ ( z − p − M .Consider now the exact sequence of kG -modules0 −→ M ⊗ ( z − p − M −→ M ⊗ ( M ) Z −→ M ⊗ k −→ , where ( M ) Z = { x ∈ M | ( z − x = 0 } . Since the kernel is free over kG ,the sequence splits and we have M ⊗ ( M ) Z ∼ = ( z − p − F ⊕ L where L isa submodule isomorphic to M . If we had F ∩ L = 0, then we would haveSoc( F ) ∩ L = 0; hence ( z − p − F ∩ L = 0, a contradiction. Therefore F ∩ L = 0and M ⊗ M contains a submodule F ⊕ L .We now show that F ⊕ L = M ⊗ M by proving that both modules havethe same dimension. We have Dim( F ) = p · Dim( M ⊗ M ) = pr r andthereforeDim( F ⊕ L ) = pr r + r = r ( pr + 1) = Dim( M ) Dim( M ) , as was to be shown.Now, continuing with the proof of the theorem, we note that each quo-tient ( K i +1 ⊗ M ) / ( K i ⊗ M ) is isomorphic to M ⊗ M , hence contains afree submodule F i of dimension pr r by the lemma. Now remember thatprojective modules are also injective and, as a result, if a projective module isa direct summand of a section of a module V , then it is a direct summand of V .Thus we can lift the free module F i and obtain a free submodule F i of M ⊗ M mapping isomorphically onto F i under the quotient map M ⊗ M → ( M ⊗ M ) / ( K i ⊗ M ). Similarly, ( z − p − M ⊗ M is isomorphic to M ⊗ M ,hence contains a free submodule F of dimension pr r by the lemma. There-fore we have M ⊗ M = M ⊕ F where F = F ⊕ · · · ⊕ F p − is free of dimension p r r and M is a submoduleof dimension ( pr + 1)( pr + 1) − p r r = p ( r + r ) + 1.Since, for any maximal subgroup H of G , we have M j ↓ GH ∼ = k ⊕ (free) for j = 1 ,
2, the same holds for M ⊗ M and hence M ↓ GH ∼ = k ⊕ (free). We aregoing to prove that M ∼ = M ⊕ M . This will imply that M is critical. Indeed M j has no kG -free summand, because M j is critical ( j = 1 , M ⊕ M hasno kG -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 M , we consider the isomorphic module( z − p − M and our goal now is to prove that ( z − p − M ∼ = M ⊕ M .48 JON CARLSON AND JACQUES TH´EVENAZ
We work with the submodule K ⊗ M of our filtration and we first analyzeits submodule K ⊗ K ′ , where K ′ = { m ∈ M | ( z − m = 0 } is the analog of K for M . Notice that K ⊗ K ′ is a kG -module with a filtration( z − p − M ⊗ ( z − p − M ⊂ (cid:0) ( z − p − M ⊗ K ′ (cid:1) + (cid:0) K ⊗ ( z − p − M (cid:1) ⊂ K ⊗ K ′ . In the filtration, the bottom submodule is free over kG and is equal to( z − p − F by the lemma. The middle quotient of this filtration is the directsum of (( z − p − M ⊗ K ′ ) / (( z − p − M ⊗ ( z − p − M ) ∼ = ( z − p − M ⊗ k ∼ = M and ( K ⊗ ( z − p − M ) / (( z − p − M ⊗ ( z − p − M ) ∼ = k ⊗ ( z − p − M ∼ = M . This direct sum can be lifted in K ⊗ K ′ , because the submodule ( z − p − F =( z − p − M ⊗ ( z − p − M is kG -free and so the sequence0 −→ ( z − p − F −→ K ⊗ K ′ −→ ( K ⊗ K ′ ) / ( z − p − F −→ K ⊗ K ′ contains a submodule V ⊕ V with V j ∼ = M j and( V ⊕ V ) ∩ ( z − p − F = 0. It follows that ( V ⊕ V ) ∩ Soc( F ) = 0 and so( V ⊕ V ) ∩ F = 0.We now have V ⊕ V ⊕ F ⊂ K ⊗ M and therefore V ⊕ V ⊕ F intersectstrivially F ⊕ · · · ⊕ F p − because this free module has been lifted from quotientsof ( M ⊗ M ) / ( K ⊗ M ). This shows that M ⊗ M contains the submodule V ⊕ V ⊕ F ⊕ F ⊕ · · · ⊕ F p − = V ⊕ V ⊕ F . Therefore V ⊕ V is isomorphicto a submodule of M .Now we show that V ⊕ V ⊂ ( z − p − ( M ⊗ M ). Since z acts triviallyon K ′ , we have ( z − p − ( M ⊗ K ′ ) = ( z − p − M ⊗ K ′ and this contains V byconstruction of V . Similarly V ⊂ K ⊗ ( z − p − M = ( z − p − ( K ⊗ M ).Passing to the quotient by F , we deduce that V ⊕ V is isomorphic to asubmodule of( z − p − (cid:0) ( M ⊗ M ) /F (cid:1) = ( z − p − (cid:0) ( M ⊕ F ) /F (cid:1) ∼ = ( z − p − M .
In order to prove that this submodule is the whole of ( z − p − M , it suffices toprove that they have the same dimension. But V j ∼ = M j has dimension r j (for j = 1 ,
2) and we know that Dim( M ) = p ( r + r ) + 1. Therefore Dim( M ) = r + r and we are done. This shows that ( z − p − M ∼ = V ⊕ V and completesthe proof of the theorem. HE CLASSIFICATION OF TORSION ENDO-TRIVIAL MODULES
Theorem 5.9.
For every i = 1 , . . . , t , let M i be a nontrivial critical kG -module. Let ℓ i be a line in the variety of the periodic kG -module M i and assumethat ℓ i = ℓ j for i = j . Then there exists a nontrivial critical kG -module M such that V G ( M ) = S ti =1 ℓ i . Moreover , Dim( M ) ≥ t | G | / if p = 2 and Dim( M ) ≥ t | G | + 1 if p is odd.Proof . Recall that M i is periodic by Lemma 5.3, and hence V G ( M i ) isa union of lines. By Theorem 2.2, M i = L i ⊕ N i such that V G ( L i ) = ℓ i andthe variety of N i is the union of the other lines (if any; otherwise simply set L i = M i ). By Theorem 5.6, there exists a critical kG -module U i such that U i = L i .Now by Theorem 5.7 and the assumption that the lines ℓ i are distinct, weobtain a critical kG -module M such that U ⊗ U ⊗ · · · ⊗ U t = M ⊕ (free)and M = U ⊕ U ⊕ · · · ⊕ U t so that V G ( M ) = S ti =1 ℓ i .Since U i ↓ GH ∼ = k ⊕ (free) where H is a maximal subgroup of G , Dim( U i ) − | H | and therefore Dim( U i ) is a multiple of | H | /p = | G | /p . Itfollows that Dim( M ) ≥ t | G | /p 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 ) ≡ ± | G | )by Lemma 2.10 in [CaTh]. A plus sign is forced here and therefore Dim( U i ) − | 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 constructan indecomposable torsion endo-trivial module M such that V G ( M ) contains V G ( N ) for any torsion endo-trivial module N . Moreover, Dim( M ) ≥ t | G | / t | G | +1, where t is the number of components of V G ( M ). However,in view of the main theorem of this paper, it will turn out that T t ( G ) = 0 andso M ∼ = k .
6. Lower bounds on dimensions of critical modules
In this section we prove a theorem that is essential to the general cases ofour main result. Basically it says that, if an extraspecial group or an almostextraspecial group has a nontrivial critical module, then it has one of largedimension. For the proof, we need a few lemmas.50
JON CARLSON AND JACQUES TH´EVENAZ
Lemma 6.1.
Suppose that M is a nontrivial critical kG -module and let ℓ be a line in V G ( M ) . Then ℓ is not contained in any F p -rational subspace of V G ( k ) .Proof . Note that V 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) correspondsto a maximal subgroup H ⊆ G . That is, the F p -rational subspaces of V G ( k ) areprecisely the subspaces of the form res ∗ G,H ( V H ( k )). If ℓ were in res ∗ G,H ( V H ( k ))then it would have to be the case that V H ( M ↓ GH ) = { } and hence M ↓ GH would not be free as a kH -module. By Lemma 5.3, this would contradict thehypothesis that M ↓ GH ∼ = k ⊕ (free) where H is the inverse image of 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 G ( k ) = k m and suppose that ℓ is not contained in any F p -rational subspace of k m . Thenthe 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 ahomomorphism 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 theidentity then some row ( a , a , . . . , a m ) of y − I m is not zero. But then v is inthe subspace defined by the equation a x + a x + · · · + a m x m = 0. Becausethe 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 consistof 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 isdefined 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 andit follows that the group Out( G ) of outer automorphisms of G acts on the setof isomorphism classes of kG -modules. We shall also write N y for a conjugatemodule defined by an outer automorphism y ∈ Out( G ).Since G is extraspecial or almost extraspecial, we control Out( G ) in thefollowing sense. Recall that if p = 2, there is an associated quadratic form onthe F -vector space G/Z ( G ) (see Lemma 3.1). If p is odd, there is a symplecticform b on the F p -vector space G/Z ( G ) defined by [˜ x, ˜ y ] = z b ( x,y ) , where x, y ∈ G/Z ( G ), ˜ x, ˜ y ∈ G are elements of G that lift x and y , and z is a generatorof Z ( G ). HE CLASSIFICATION OF TORSION ENDO-TRIVIAL MODULES
Lemma 6.3.
Let G be an extraspecial or almost extraspecial p -group. Let Out ( G ) be the subgroup of Out( G ) consisting of outer automorphisms fixingthe center Z ( G ) pointwise. (a) If p is odd and G is extraspecial of exponent p , then Out ( G ) is iso-morphic to the symplectic group O G associated to the symplectic formcorresponding to G . (b) If p = 2, Out ( G ) is isomorphic to the orthogonal group O G associatedto the quadratic form corresponding to G .Proof . When G is extraspecial, this is one of the main results in Winter’spaper [Wi]. If G is almost extraspecial, the arguments given in Sections 3Fand 4 of [Wi] extend and yield the same result. Alternatively, this appearsexplicitly in Exercise 5 of Chapter 8 of [As]. Theorem 6.4.
Suppose that there exists a nontrivial critical kG -module N . (a) If p is odd , there exists a critical kG -module M such that Dim( M ) > | G | · | 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 ) > | G | · | O G || C | where O G is the orthogonal group associated to G and C is an odd ordercyclic subgroup of O G of maximal order.Proof . Let ℓ be a line in V G ( N ). Notice that if y ∈ O G then N y is alsoa nontrivial kG -module such that N y ↓ GH ∼ = k ⊕ (free). But then y ( ℓ ) is in thevariety V G ( N y ). If B denotes the stabilizer of ℓ in O G , we obtain a family ofmodules N y indexed by the set of cosets O G /B . So by Theorem 5.9, thereexists a critical kG -module M such that V G ( M ) = S y ∈ O G /B y ( ℓ ). Moreover,Dim M > | O G || B | · | G | p = 2 and Dim M > | 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 G ( k ) = k m . Thus by Lemma 6.2, the group B = S ∩ O G is cyclic of orderprime to p . If C is of maximal order among cyclic p ′ -subgroups of O G , wededuce the lower bound of the statement.For use in the following sections, we need to have some estimates of theorders of the orthogonal and symplectic groups and their cyclic p ′ -subgroups.52 JON CARLSON AND JACQUES TH´EVENAZ
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 . If p is odd and G is extraspecial of exponent p and order p n +1 , then O G = Sp(2 n, F p ) and | O G | = p n n Y i =1 ( p i − . If p = 2 and G ∼ = D ∗ · · · ∗ D is extraspecial of order n +1 ( type then O G = O + (2 n, F ) and | O G | = 2 · n ( n − (2 n − n − Y i =1 (2 i − . If p = 2 and G ∼ = D ∗ · · · ∗ D ∗ Q is extraspecial of order n +1 ( type then O G = O − (2 n, F ) and | O G | = 2 · n ( n − (2 n + 1) n − Y i =1 (2 i − . If p = 2 and G ∼ = D ∗ · · · ∗ D ∗ C is almost extraspecial of order n +2 ( type then O G = Sp(2 n, F ) and | O G | = 2 n n Y i =1 (2 i − . 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(2 n, F p ), respectively O G = O ± (2 n, F ), essentially by definition (see also [Wi]). In the third case,we obtain O G = O (2 n + 1 , F ) ∼ = Sp(2 n, F ) (see Theorem 11.9 of Taylor’sbook [Ta]) where orders of the four groups appear on pages 70 and 141. The listcan also be found in any of a number of text books on Chevalley groups or finitesimple groups (e.g. Gorenstein’s book [Go2]). The types of the groups of Lietype in the four cases listed are C n ( p ), D n (2), D n (2) and C n (2), respectively.In the first case the corresponding simple group is O G / {± } . In the next twocases the corresponding simple group has index 2 in O G , while the group issimple in the fourth case.For the statement about the cyclic p ′ -subgroups, note first that elementsof 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 torusis equal to | det( w − F − | = | det( F − w ) | HE CLASSIFICATION OF TORSION ENDO-TRIVIAL MODULES F ( x ) = x q is the Frobenius morphism, w is an element of the correspond-ing Weyl group, and where F and w act on the cocharacter group of a fixedmaximal torus of the corresponding algebraic group (see Proposition 3.3.5 inCarter’s book [Cart]). Since w has finite order, we obtain a product Q ni =1 ( q − ζ i )for suitable roots of unity ζ i (the eigenvalues of w ). In our case, q = p andthe rank n is the same as the integer n of the statement. Therefore if C is anycyclic p ′ -subgroup of O G , we get | C | ≤ | det( F − w ) | = (cid:12)(cid:12) n Y i =1 ( p − ζ i ) (cid:12)(cid:12) ≤ n Y i =1 ( 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 analgebraically closed field of characteristic p .We will need the following results. Recall that a nonzero element ζ ofH ( G, F p ) corresponds to a maximal subgroup of G in the sense that there is aunique maximal subgroup H of G such that res G,H ( ζ ) = 0. When p is odd, wealso need the Bockstein map β : H ( G, F p ) −→ H ( 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 , . . . , η t are nonzero elements in H ( G, F p ) and have the prop-erty that η . . . η t = 0 if p = 2 ,β ( η ) . . . 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 − t ( k ) ⊕ P has a filtration { } = L ⊆ L ⊆ · · · ⊆ L t ∼ = k ⊕ Ω − t ( k ) ⊕ P where L i /L i − ∼ = (Ω − i ( k )) ↑ GH i 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 i = H i − = K i . Then there is a projectivemodule P such that k ⊕ Ω − t ( k ) ⊕ P has a filtration { } = L ⊆ L ⊆ · · · ⊆ L t ∼ = k ⊕ Ω − t ( k ) ⊕ P where L i /L i − ∼ = (Ω − i ( k )) ↑ GH i for each i = 1 , . . . , t .Proof . This is the essence of Lemma 3.10 of [Ca2]. That lemma is statedfor Z G -modules but this does not really matter since we can tensor the whole54 JON CARLSON AND JACQUES TH´EVENAZ thing with k . Because the emphasis of our theorem is different from that ofthe results of [Ca2] we give a brief sketch of the proof here. However, all ofthe 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 modifythe arguments for odd p . Each of the cohomology elements η i corresponds toan exact sequence 0 −→ F −→ F ↑ GH i −→ F −→ . Now we splice all of these together and tensor with k to get a sequence of theform 0 −→ k −→ k ↑ GH t −→ . . . −→ k ↑ GH −→ k ↑ GH −→ k −→ , which represents the element η . . . η t = 0 in H t ( G, k ). Note that we are usingthe same notation η i for the element of H ( G, F ) and its image under thechange of rings in H ( G, k ). Now we consider the complex C obtained bytruncating the ends off of the sequence. That is, C i = k ↑ GH i +1 for i = 0 , . . . , t − C i = 0 otherwise. We see that the homology of C is a result of thetruncations. That is, H i ( C ) = k if either i = 0 or i = t − i ( C ) = 0otherwise.The next step is to collapse the complex C into a single module. This isaccomplished 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 thetrivial module k . This gives us a projective resolution of the complex C and ithas the same homology as C . Thus, in degrees above t , it is exact and is theprojective 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 itis in the wrong degree. So we take W = Ω − t ( U ). This is the module that wewant.There are now two things to note about W . First because the terms ofthe complex C are induced from the maximal subgroup H , . . . , H t , the module W has a filtration by the modules k ↑ GH i suitably translated by Ω, exactly asdescribed in the statement of the theorem. That is, the projective resolution ofthe complex C as constructed above is filtered by the projective resolutions ofthe terms of the complex, suitably translated. See the proof of Proposition 3.8of [Ca2] for this part.Next we note that the module W is isomorphic to k ⊕ Ω − t ( k ) ⊕ P for someprojective module P . This is because the original sequence that represented η . . . η t splits and hence the projective resolution of the complex is, in highdegrees, a projective resolution of the homology groups of the complex, suitablytranslated. See Proposition 3.8 of [Ca2] for this part. This proves the theoremif p = 2. HE CLASSIFICATION OF TORSION ENDO-TRIVIAL MODULES p is odd, the cohomology element η i has to be replaced by its Bock-stein β ( η i ) which corresponds to an exact sequence0 −→ F p −→ F p ↑ GK i −→ F p ↑ GK i −→ F p −→ . Again we splice all of these together and tensor with k . Using our numberingof the subgroups H i , we obtain a sequence of the form0 −→ k −→ k ↑ GH t −→ . . . −→ k ↑ GH −→ k ↑ GH −→ k −→ , which represents the element β ( η ) . . . β ( η t ) = 0 in H t ( G, k ). The complex C isobtained by truncating the ends off of the sequence and the rest of the argumentis the same, except that the integer t has to be replaced by 2 t throughout.The upper bounds wanted for the dimensions of our critical modules iscontained in the following. Theorem 7.2.
Suppose that G is a p -group which is not elementary abelian.Suppose that η , . . . , η t ∈ H ( G, F p ) are nonzero and have the property that η . . . η t = 0 if p = 2 ,β ( η ) . . . β ( η t ) = 0 if p is odd.Let r = t if p = 2 and r = 2 t if p is odd. Let H , . . . , H r be the maximal sub-groups of G as in the previous theorem. Suppose that M is an indecomposable kG -module with the property that M ↓ GH i ∼ = k ⊕ (free) for every i . Then forany s , Dim Ω s ( M ) + Dim Ω s − r +1 ( M ) ≤ r X i =1 Dim (Ω s +1 − i ( k ) ↑ GH i ) . Proof . Let P be a projective module such that W = k ⊕ Ω − r ( k ) ⊕ P hasa filtration as in the last theorem. Then tensoring W and all of the factors inthe filtration with Ω s ( M ) we get that { } = L ⊗ Ω s ( M ) ⊆ L ⊗ 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 − ⊗ Ω s ( M )) ∼ = ( L i / ( L i − ) ⊗ Ω s ( M ) ∼ = Ω − i ( k ) ↑ GH i ⊗ Ω s ( M ) ∼ = Ω − i (Ω s ( M ↓ GH i )) ↑ GH i ⊕ Q ∼ = Ω s +1 − i ( k ) ↑ GH i ⊕ Q ′ for some projective modules Q and Q ′ . Now the important thing to rememberis that kG is a self injective algebra and hence projective modules are also56 JON CARLSON AND JACQUES TH´EVENAZ injective. As a result, if a projective module is a direct summand of a sectionof a module V , then it is a direct summand of V . The consequence of this isthat (after stripping away the unnecessary projective modules Q ′ ) we can getthat, for some projective module R , the module Ω s ( M ) ⊕ Ω s +1 − r ( M ) ⊕ R hasa filtration { } = X ⊆ X ⊆ · · · ⊆ X r ∼ = Ω s ( M ) ⊕ Ω s +1 − r ( M ) ⊕ R where X i /X i − ∼ = Ω s +1 − i ( k ) ↑ GH i . The statement about dimensions followsimmediately.
8. Special cases of 2-groups of small order
In this section we consider some special cases of 2-groups that we need totreat separately, for they are not covered by the general argument of Section 10.For each of the groups we show Theorem 1.4 directly. We discuss the groups oforder 8, the almost extraspecial group D ∗ C of order 16 and the extraspecialgroup D ∗ D of order 32 (type 1).Let us start with the groups of order 8. First Q is excluded by assump-tion (and there is actually a nontrivial critical kQ -module of dimension 5;see [CaTh]). For G = D the structure of T ( D ) is known (see [CaTh]) andevery nontrivial endo-trivial kD -module is nontrivial on restriction to one ofthe two elementary abelian 2-subgroups of D . Thus the only critical moduleis the trivial one. Alternatively, we can also prove the result in the followingway. Proposition 8.1.
Let G = D . Then there exists no nontrivial critical kG -module.Proof . Let M be a critical kG -module. By Theorem 3.4, the numberof cohomology classes whose product vanishes is equal to t G = 2. ApplyingTheorem 7.2 with s = 1, we getDim Ω ( M ) + Dim M ≤ Dim Ω ( k H ) ↑ GH + Dim k ↑ GH for some maximal subgroups H and H . Since H i has order 4, Ω ( k H i ) hasdimension 3 and we obtainDim Ω ( M ) + Dim M ≤ . By Lemma 5.4, Dim Ω ( M ) = Dim M −
2. So Dim M ≤
5. This part ofthe argument is essentially the same as the one appearing in Theorem 5.3 of[CaTh].
HE CLASSIFICATION OF TORSION ENDO-TRIVIAL MODULES kG -module, thenby Theorem 6.4, there exists a nontrivial critical kG -module M of dimensionDim M > | G | · | O G || C | = 4 · , since | O G | = 2 by Proposition 6.5. This contradicts the previous upper bound.We turn now to the group G = D ∗ D of order 32. Proposition 8.2.
Let G = D ∗ D . Then there exists no nontrivialcritical kG -module.Proof . Let M be a critical kG -module. By Theorem 3.4, the number ofcohomology classes whose product vanishes is equal to t G = 3. Applying nowTheorem 7.2 with s = 1, we getDim Ω ( M ) + Dim Ω − ( M ) ≤ Dim Ω ( k ) ↑ GH + Dim k ↑ GH + Dim Ω − ( k ) ↑ GH for some maximal subgroups H , H , and H . Since H i has order 16, Ω ± ( k H i )has dimension 15 and we obtainDim Ω ( M ) + Dim Ω − ( M ) ≤
30 + 2 + 30 = 62 , so that Dim Ω ( M ) ≤ kG -module, thenby Theorem 6.4, there exists a nontrivial critical kG -module M of dimensionDim M > | G | · | O G || C | ≥ ·
729 = 128 , by Proposition 6.5. So Dim Ω ( M ) >
126 by Lemma 5.4, a contradiction.In the last case, G is the almost extraspecial group D ∗ C of order 16. Themethod 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 argumentusing the action of a Galois group. Lemma 8.3.
Let G = D ∗ C . If M is a critical kG -module , then Dim M ≤ .Proof . By Theorem 3.4, the number of cohomology classes whose productvanishes is equal to t G = 3. Applying now Theorem 7.2 with s = 1, we getDim Ω ( M ) + Dim Ω − ( M ) ≤ Dim Ω ( k ) ↑ GH + Dim k ↑ GH + Dim Ω − ( k ) ↑ GH for some maximal subgroups H , H , and H . Since H i has order 8, Ω ± ( k )has dimension 7 and we obtainDim Ω ( M ) + Dim Ω − ( M ) ≤
14 + 2 + 14 = 30 . JON CARLSON AND JACQUES TH´EVENAZ
By Lemma 5.4, Dim Ω ( M ) = Dim Ω − ( M ) = Dim M −
2. So Dim Ω ( M ) ≤
15 and Dim M ≤ Proposition 8.4.
Let G = D ∗ C . Then there exists no nontrivialcritical kG -module.Proof . Suppose that there is such a module N . We need to look at V G ( N ) ⊆ V G ( k ) ∼ = k . Suppose that p = ( α, β, γ ) is a point in V G ( N ). Bydividing by α we may assume that α = 1, so that p = (1 , β, γ ) ∈ V G ( N ).Notice that p / ∈ res ∗ G,H ( V H ( N H )) for any maximal subgroup H since N H isa free kH -module. Therefore p is not in any F -rational subspace of k , andhence β and γ cannot both be in the field with four elements (otherwise 1 , β, γ would be linearly dependent over F ). It follows that if F : k −→ k is theFrobenius map, F ( a, b, c ) = ( a , b , c ), then p , F ( p ) and F ( p ) lie on differentlines in V G ( k ).Next we need to notice that using the Frobenius homomorphism we cancreate a new module from N , by letting it act on the coefficients of the action ofthe 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 ), we let N F be the module defined by thecomposition. It is not difficult to see that N F is also critical. Moreover, F ( p )is a point in V G ( N F ). It follows that the lines through p , F ( p ), and F ( p ) are alllines in the variety of the quotient module L for some nontrivial critical module L . Thus by Theorem 5.9, kG has a nontrivial critical module of dimension atleast 25. This contradicts Lemma 8.3.
9. The groups of order p for odd p When the prime p is odd, there is one special case in the proof of The-orem 1.4 that must be handled with extra care. This involves the groups oforder p . The problem is that the general estimates of the dimensions of crit-ical modules used later are not sufficient to handle this case. The result thatwe want is the following. Proposition 9.1.
Let G = G , an extraspecial group of order p andexponent p , for p an odd prime. Then there exists no nontrivial critical kG -module. The proof proceeds in several steps. Throughout assume that a nontrivialcritical kG -module exists and use Theorem 6.4 to obtain one of large dimension,as follows. HE CLASSIFICATION OF TORSION ENDO-TRIVIAL MODULES
Lemma 9.2.
If a nontrivial critical kG -module exists , then there existsa critical kG -module M whose dimension is at least equal to ( p − p + 1 .Moreover Dim Ω( M ) ≥ ( p − p − and Dim Ω − ( M ) ≥ ( p − p − .Proof . By Theorem 6.4 there exists a critical module M whose dimensionis at least | G | | Sp(2 , F p ) | / | C | where C is a cyclic p ′ -subgroup of the symplec-tic group Sp(2 , F p ) of maximal order. Now Sp(2 , F p ) = SL(2 , F p ) has order p ( p −
1) and its cyclic p ′ -subgroup of maximal order has order p + 1. So thedimension of M must be greater than ( p − p and must be congruent to 1modulo p .Now to compute the dimension of Ω( M ), we notice from the proof ofTheorem 6.4 that the variety of the module M is the union of at least p ( p − V G ( k ) = k . So M = U ⊕ · · · ⊕ U t where, for each i , V G ( U i ) isa single line and t > p ( p − U i = r i p for some r i (we use here the fact that U i is endo-trivial and p is odd).Because U i is not a free kG -module (and, in fact, has no free submodules) andbecause a projective cover of U i has dimension p Dim U i / Rad( U i ), we musthave Dim U i / Rad( U i ) > r i . Therefore U i is minimally generated by at least r i + 1 generators and the number of generators of M is at least m = t X i =1 ( r i + 1) = t X i =1 r i ! + t . Now
M /
Rad( M ) is a quotient of M/ Rad( M ), so the minimal number of gen-erators of M is at least m . As a result, the number of copies of kG appearingin the projective cover of M must be at least m . Now the dimension of M is p ( P ti =1 r i ) + 1 and so the dimension of Ω( M ) is at least p m − Dim( M ) = p t X i =1 r i ! + t ! − p t X i =1 r i ! − tp − ≥ ( p − p − . By applying the same argument to the dual module M ∗ (which also satisfiesthe properties we need), we obtainDim Ω − ( M ) = Dim Ω − ( M ) ∗ = Dim Ω( M ∗ ) ≥ ( p − p − . This proves the lemma.
Lemma 9.3.
Dim Ω p ( M ) + Dim Ω − ( M ) ≤ p ( p + p + 1) . Proof . From any one of the papers [Le1], [Ya], [BeCa] we have that thereexist η , . . . η p +1 ∈ H ( G, k ) such that β ( η ) . . . β ( η p +1 ) = 0. In Leary [Le1] therelation is given as x p x ′ − xx ′ p = 0. Now applying Theorem 7.2 with t = p + 160 JON CARLSON AND JACQUES TH´EVENAZ (hence r = 2 t = 2( p + 1)) and choosing s = 2 p in that theorem, we getDim Ω p ( M ) + Dim Ω − ( M ) ≤ p +2 X i =1 Dim(Ω p +1 − i ( k ) ↑ GH i ) , 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 , and hencethe dimensions on the right-hand side of the inequality are independent of theparticular η j . Because Dim H j ( H i , k ) = j + 1 (see Lemma 4.6), we have that(for H i = H )Dim Ω j − ( k H ) + Dim Ω j ( k H ) = p Dim H j − ( H, k ) = p (2 j ) . Induction to G multiplies the dimensions by p . Consequently the right-handside of the above inequality has the form p +2 X i =1 Dim(Ω p +1 − i ( k ) ↑ GH i )= p Dim Ω − ( k H ) + p Dim k + p p X j =1 (cid:0) Dim Ω j − ( k H ) + Dim Ω j ( k H ) (cid:1) = p (cid:0) p − p X j =1 p j (cid:1) = p + 2 p ( p )( p + 1) / p (1 + p + p )as desired.At this point we should notice that the two lemmas above are not sufficientto give us the contradiction wanted. We need some further analysis of thedimension of Ω p ( M ). For this purpose we recall that there exists an element ζ ∈ H p ( G, k ) which has the property that its restriction res
G,Z ( ζ ) is not zerowhere Z = h z i is the center of G . In Leary’s paper [Le1], the element thathe calls z will do. The element ζ can also be obtained by applying the Evensnorm map to an element in the degree 2 cohomology of a maximal elementaryabelian subgroup whose restriction to Z is not trivial.The element ζ can be represented by a unique cocycle ζ : Ω p ( k ) −→ k .Hence we have an exact sequence0 −→ L −→ Ω p ( k ) ζ −→ k −→ L is the kernel of ζ . Now by Theorem 2.2, V G ( L ) = V G ( ζ ), the varietyof the ideal generated by ζ . In particular, the restriction L ↓ GZ is free as a kZ -module. This fact can also be derived from the observation that the abovesequence is split as a sequence of kZ -modules because the restriction of ζ to Z is not zero and Ω p ( k ) ↓ GZ ∼ = k ⊕ (free). HE CLASSIFICATION OF TORSION ENDO-TRIVIAL MODULES L = L/ ( z − L ∼ = ( z − p − L . Then L is a kG -module where G = G/Z . Lemma 9.4.
The kG -module L has no projective submodules , and more-over , V G ( L ) ⊆ [ res ∗ G,E V E ( k ) where the union is over the set of all subgroups E = E/Z where E is a maximalsubgroup of G . Notice that every maximal subgroup of G is elementary abelian and theunion in the lemma is over all subgroups of order p in G . Thus the right-handside of the containment is the union of all of the F p -rational lines in V G ( k ) ∼ = k .It can be proved that the two sides are actually equal, but we do not need toknow this. Proof . If L had a kG -projective submodule then L and hence also Ω p ( k )would have projective kG -submodules. That is, if t G L = 0 then also t G L = 0.But clearly this is impossible.Now suppose that ℓ ⊆ V G ( k ) is a line that is not F p -rational. Let N bea kG -module such that V G ( N ) = ℓ (e.g. take N = kG/ ( σ −
1) where h σ i isa cyclic shifted subgroup corresponding to the line ℓ ). Then the restriction N ↓ GE is a free kE -module for any maximal subgroup E of G . So, viewing N as a kG -module by inflation, we have that V E ( N ↓ GE ) is the line determinedby the center Z , because Z acts trivially on N ↓ GE . Therefore N is periodicas a kG -module and we must have that V G ( N ) = res ∗ G,Z ( V Z ( k )), the linedetermined by the center Z . Because L is free on restriction to Z we knowthat V G ( L ) ∩ V G ( N ) = { } and hence L ⊗ N is a free kG -module. Now Z actstrivially on N and hence ( z − L ⊗ N ) = (( z − L ) ⊗ N . Thus, L ⊗ N ∼ = L ⊗ N is a free kG -module. It follows from Theorem 2.2 that V G ( L ) ∩ V G ( N ) = { } .Hence the line ℓ is not in V G ( L ) and this holds for all lines in V G ( k ) which arenot F p -rational. Thus the variety V G ( L ) must be contained in the union of the F p -rational lines. Lemma 9.5. If M is a critical kG -module , V G ( M ) ∩ V G ( L ) = { } and M ⊗ L ∼ = L ⊕ (free) .Proof . We first show that M ⊗ L is a free kG -module. That is, L is freeas a kZ -module and M ⊗ L ∼ = M ⊗ L . But from Lemmas 9.4 and 6.1 we havethat V G ( M ) ∩ V G ( L ) = { } . Hence M ⊗ L is free as a kG -module. Thus M ⊗ L is free as a kG -module.It follows that M ⊗ L has a filtration0 ⊆ (( z − p − M ) ⊗ L ⊆ · · · ⊆ (( z − M ) ⊗ L ⊆ M ′ ⊗ L ⊆ M ⊗ L JON CARLSON AND JACQUES TH´EVENAZ where M ′ = { m ∈ M | ( z − p − m = 0 } . All of the factors are isomorphic to M ⊗ L and hence are projective, except for the factor( M ′ ⊗ L ) / (( z − M ⊗ L ) ∼ = ( M ′ / ( z − M ) ⊗ L ∼ = k ⊗ L ∼ = L .
The lemma follows from the fact that free modules are also injective and henceany free composition factor is a direct summand.Now tensoring the sequence given above with M we get an exact sequence0 −→ M ⊗ L −→ M ⊗ Ω p ( k ) ⊗ ζ −→ M −→ . Any projective submodule of M ⊗ L is also a direct summand of the middleterm and can be factored out. So we have an exact sequence of the form0 −→ L −→ Ω p ( M ) ⊕ P −→ M −→ , for some projective module P . It remains to prove the following. Lemma 9.6.
In the preceding exact sequence , the projective module P iszero.Proof . Because the module L is free as a kZ -module the sequence is splitas a sequence of kZ -modules. So multiplication by z − −→ ( z − p − L −→ ( z − p − Ω p ( M ) ⊕ ( z − p − P −→ ( z − p − M −→ −→ L −→ Ω p ( M ) ⊕ P −→ M −→ , which is a sequence of kG -modules. Because V G ( L ) ∩ V G ( M ) = { } by theprevious lemma, we must have that the sequence splits. Thus, L ⊕ M ∼ = Ω p ( M ) ⊕ P .
But L ⊕ M has no projective kG -submodules by Lemma 9.4. Hence P = { } and therefore also P = { } . Proof of Proposition p ( M ) = Dim L + Dim M .By Lemma 4.4, Dim Ω p ( k ) = p ( p + 1) + 1, and so Dim L = p ( p + 1) bydefinition of L . Now by Lemma 9.2, Dim M ≥ p ( p − − ( M ) ≥ p ( p − −
1. Hence we have thatDim Ω p ( M ) + Dim Ω − ( M ) ≥ p ( p + 1) + p ( p −
1) + 1 + p ( p − − p (2 p − p + 1) . This inequality, however, is a contradiction to Lemma 9.3 since we are assumingthat p ≥ HE CLASSIFICATION OF TORSION ENDO-TRIVIAL MODULES
10. The general case in characteristic 2
We are now prepared to prove the general case by induction and completethe proof of the detection Theorem 1.4 when p = 2. Throughout, k hascharacteristic 2. Let G be an extraspecial or almost extraspecial group oforder 2 m +1 . The theorem that we are trying to prove is the following. It isequivalent to Theorem 1.4. Theorem 10.1. If G is an extraspecial or almost extraspecial -group andif G is not isomorphic to Q , then there are no nontrivial critical kG -modules. Three cases have to be treated separately, namely the groups of order atmost 16 as well as D ∗ D . But these cases have been dealt with in Section 8.Therefore we can now assume that m ≥ m > Proposition 10.2.
Let G be an extraspecial or almost extraspecial groupof order m +1 , with m = 2 n . Assume that m ≥ and m > if G is oftype . Let t G be the number of cohomology classes whose product vanishes , asdescribed in Theorem and let σ G = (cid:18) t G + m − m − (cid:19) | G | + 2 and τ G = | G | · | O G | n . If τ G > σ G then there exists no nontrivial critical kG -module.Proof . Let t = t G . In view of Theorem 3.4, there exist nonzero elements η , . . . , η t ∈ H ( G, F ) such that η . . . η t = 0 and each η i corresponds to amaximal subgroup H i . Moreover each subgroup H i is the centralizer of anoncentral involution in G and by Theorem 3.5, H i ∼ = C × U where U has thesame type as G . So H i ∼ = H 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 haveDim M ≤ Dim Ω t − ( M ) + Dim M ≤ t X i =1 Dim (Ω t − i ( k ) ↑ GH i ) . Since all the subgroups H i are isomorphic to H , we obtainDim M ≤ t − X j =0 Dim (Ω j ( k ) ↑ GH ) . JON CARLSON AND JACQUES TH´EVENAZ
Now by Corollary 3.6, which applies in view of our assumption on m (with m ,in the corollary, replaced by m − r = t G − t − X j =0 Dim (Ω j ( k ) ↑ GH ) ≤ | G | · (cid:18) m − t G − − m − (cid:19) + 2 = σ G . It follows that Dim M ≤ σ G .If there exists a nontrivial critical kG -module, then by Theorem 6.4, thereexists a nontrivial critical kG -module M of dimensionDim M > | G | · | O G || C | ≥ | G | · | O G | 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 thegroups G as above. This is a purely numerical problem which only requiresestimating the numbers τ G and σ G . We start with a lemma which will beuseful for estimating σ G . Lemma 10.3.
Let t and m be integers with t ≥ and m ≥ . Then (cid:18) t + m − m (cid:19)(cid:18) t + m − m − (cid:19) < m − t . Proof . Expanding the left-hand side and eliminating the common fac-tor ( m − m − m ≥ t ≥ /m ( m −
1) by 1 /
30. Thus we get the following.2 t + m − t + m − · t + m − t + m − · . . . · t + 4 t + 2 · t + 3 t + 1 · t + 2 t · t + 1 t − · tm · t − m − < m − t < m − t . For the proof that τ G > σ G , we proceed with cases.10.1. Groups of type . Let G n = D ∗ · · · ∗ D be the central product of n copies of D , with n ≥
3. Remember that the cases n = 1 and n = 2 weretreated 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 withtwo cases.If n = 3, then t = t G = 5 by Theorem 3.4 and we have that τ = 2 · · · · > · , HE CLASSIFICATION OF TORSION ENDO-TRIVIAL MODULES σ = (cid:18) − − (cid:19) · + 2 = 35 · + 2 < · < τ . If n = 4, then t = t G = 9 by Theorem 3.4 and τ = 2 · · · · · = 2 · ,σ = (cid:18) − − (cid:19) · + 2 = 1716 · + 2 < τ . For n ≥
4, we have t G n +1 = 2 t G n by Theorem 3.4, and this allows for aninductive 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 σ n +1 − σ n − < n < τ n +1 τ n from which we get σ n +1 − < n σ n − n +1 < n τ n − < τ n +1 − τ n given by Proposition 6.5, we obtain τ n +1 τ n = | G n +1 || G n | · ( n +1) n +1 n ( n − · n +1 − n − · n − > · n · · n −
14 = 2 n +1 − n +1 > n . On the other hand, setting m = 2 n and t G n = t n = 2 n − + 2 n − (Theorem 3.4),we obtain by Lemma 10.3 σ n +1 − σ n − | G n +1 || G n | · (cid:18) t n + m − m (cid:19)(cid:18) t n + m − m − (cid:19) < · m − · t n = 2 n − (2 n − + 2 n − ) < n − n < n . Groups of type . Let G n = D ∗ · · · ∗ D ∗ Q be the central productof n − D and one of Q , 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 − (see Theorem 3.4).We start with the case n = 2, for which we need to replace τ by theslightly larger value τ ′ = | G | · | O G || C | where C is a cyclic subgroup of O G of maximal odd order. By Corollary 12.43 ofTaylor’s book [Ta], O G has a simple subgroup of index 2 isomorphicto PSL(2 , F ) (that is, A , and in fact O G is isomorphic to the symmetric66 JON CARLSON AND JACQUES TH´EVENAZ group S ). Therefore | C | = 5 and we get τ ′ = 384. On the other hand t = 5and we have σ = (cid:18) − − (cid:19) · + 2 = 322 < τ ′ . The argument of Proposition 10.2 goes through with τ ′ instead of τ .Now we prove that σ n < τ n by induction, starting with n = 3: τ = 2 · · · · = 2 · ,σ = (cid:18)
10 + 6 − − (cid:19) · + 2 = 495 · + 2 < τ . If now n ≥ σ n +1 − σ n − < n − n < τ n +1 τ n from which we conclude the proof as in the previous case. Here is the compu-tation: τ n +1 τ n = | G n +1 || G n | · ( n +1) n +1 n ( n − · n +1 + 12 n + 1 · n − > · n · · n −
14 = 2 n − n . On the other hand, we obtain by Lemma 10.3 σ n +1 − σ n − | G n +1 || G n | · (cid:18) t n + m − m (cid:19)(cid:18) t n + m − m − (cid:19) < · m − · t n = 2 n − (2 n + 2 n − ) = 2 n − + 2 n − + 2 n − + 2 n − n < n − n . Groups of type . Let G n = D ∗ · · · ∗ D ∗ C be the central productof n copies of D and one of C . Let G = G n , σ n = σ G n , τ n = τ G n , and t n = t G n = 2 n + 2 n − (see Theorem 3.4). Note that m = 2 n + 1 for type 3.We prove that σ n < τ n by induction, starting with n = 2. Remember thatthe case in which n = 1 was treated in Proposition 8.4. First we have that τ = 2 · · · = 2560 ,σ = (cid:18) − − (cid:19) · + 2 = 1282 < τ . If now n ≥ σ n +1 − σ n − < n +2 < τ n +1 τ n HE CLASSIFICATION OF TORSION ENDO-TRIVIAL MODULES τ n +1 τ n = | G n +1 || G n | · ( n +1) n · n +1) − > · n +1 · n +1 n +2 . On the other hand, we obtain by Lemma 10.3 σ n +1 − σ n − | G n +1 || G n | · (cid:18) t n + m − m (cid:19)(cid:18) t n + m − m − (cid:19) < · m − · t n = 2 n (2 n + 2 n − ) < n n +1) = 2 n +2 . This completes the proof of Theorem 10.1 and hence also the proof of Theo-rem 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 . Weassume throughout that the field k has characteristic p and that G = G n isan extraspecial group of order p n +1 and exponent p . Our aim is to prove thefollowing. Theorem 11.1. If G = G n , then there are no nontrivial critical kG -modules. If n = 1, the result follows from Section 9. Thus we can assume n ≥ σ n and τ n such that σ n is an upper bound for the dimension of any critical moduleand τ n is a lower bound for the dimension of some nontrivial critical moduleif nontrivial critical modules exist. Then we prove that σ n < τ n . First we givethe definitions.For n ≥ σ n = 2 | G n | (cid:18) t n + 2 n − n − (cid:19) where t n = 2( p + p − p n − . Let τ n be given by the rule τ n = | G n | | Sp(2 n, F p ) | c n where c n = ( p + 1) n except in the case in which p = 3 and n = 2. In that caselet c n = p + 1 = 10. Then we have the following.68 JON CARLSON AND JACQUES TH´EVENAZ
Proposition 11.2. If n ≥ and τ n > σ n , then there exists no nontrivialcritical kG n -module.Proof . Let t = t n / p + p − p n − . By Theorem 4.1, we know thatthere exist nonzero elements η , . . . , η t ∈ H ( G, F ) such that β ( η ) . . . β ( η t )= 0 and each η i corresponds to a maximal subgroup H i . Moreover each sub-group H i is the centralizer of a noncentral element of order p in G and byTheorem 4.7, H i ∼ = C p × G n − . So H i ∼ = H for each i .If M is a critical kG n -module, then by Theorem 7.2 with r = 2 t = t n and s = t n − M ≤ Dim Ω t n − ( M ) + Dim M ≤ t n X i =1 Dim (Ω t n − i ( k ) ↑ GH i ) . Since all the subgroups H i are isomorphic to H , we obtainDim M ≤ t n − X j =0 Dim (Ω j ( k ) ↑ GH ) . By Corollary 4.8, t n − X j =0 Dim (Ω j ( k ) ↑ GH ) ≤ | G | · (cid:18) t n − n − n − (cid:19) = σ n . It follows that Dim M ≤ σ n .On the other hand, if we assume that there exists a nontrivial critical kG n -module, then by Theorem 6.4, there exists a nontrivial critical kG -module M of dimension Dim M > | G | · | O G || C | where C is a cyclic p ′ -subgroup in Sp(2 n, F p ) of maximal order. In the casethat p = 3 and n = 2, we know from character tables or from direct analysison Sp(4 , F ) that C has order at most 10. In all other cases we know byProposition 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 begin-ning with the following. Lemma 11.3. τ > σ .Proof . If p = 3, then σ = 860 ,
706 while τ = 1 , , HE CLASSIFICATION OF TORSION ENDO-TRIVIAL MODULES τ ). So suppose that p ≥
5. Then σ p = 1 p p (cid:18) p + p −
1) + 13 (cid:19) = 23! (2 + 2 p − p )(2 + 2 p − p )(2 + 2 p − p ) <
13 3 · · . On the other hand, τ p = p p p ( p − p − p + 1) = p (1 − p )(1 − p )(1 + 1 p ) > p ( 45 ) ( 43 ) = 9 p > p ≥ /p < / − /p ≥ /
5. So again τ > σ . Lemma 11.4.
For n ≥ σ n +1 σ n < 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 = 49 , , ,
862 while τ = 313 , , , τ n +1 τ n = | G n +1 || G n | p ( n +1) p n ( p − . . . ( p n +2 − / ( p + 1) n +1 ( p − . . . ( p n − / ( p + 1) n = p · p n +1 · ( p n +2 − / ( p + 1) > p n +4 . The above estimate is that, since p ≥
3, we have 1 / ( p + 1) > / ( √ p ) and p n +2 − > p n +2 / √ t = t n and noting that t n +1 = pt n , we have σ n +1 σ n = 2 p n +3 p n +1 (cid:0) tp +2 n − n +1 (cid:1)(cid:0) t +2 n − n − (cid:1) = p (2 n + 1)(2 n ) ( tp + 2 n − tp + 2 n −
2) ( tp + 2 n − t + 2 n − . . . tpt ( tp − t − . Now we note that ( tp + b ) / ( t + b ) ≤ tp/t = p for all b ≥
0. Also ( tp − / ( t − < p because t ≥
3. Moreover, tp + 2 n − p + p − p n − p + 2 n −
1= 2 p n +1 (cid:18) p − p + 2 n − p n +1 (cid:19) < p n +1 (2) = 4 p n +1 . JON CARLSON AND JACQUES TH´EVENAZ
So we have that σ n +1 σ n < p (2 n + 1)(2 n ) (4 p n +1 )(4 p n +1 )( p n − )( 32 p )= 16 · / n + 1)(2 n ) p n +3 ≤ p n +3 < p n +4 . Finally σ n +1 σ n < p n +4 < τ n +1 τ n , as required. Proof of Theorem n = 1 was treatedin Proposition 9.1. We have shown that τ > σ 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 theoremfollows 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 detectiontheorem (Theorem 1.2) and the vanishing theorem (Theorem 1.1). Recall thatthey assert that if G is not cyclic, quaternion or semi-dihedral, then T ( G ) isdetected on restriction to all elementary abelian subgroups E of rank 2, andthat 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 Y H Res GH : T ( G ) −→ Y H T ( H ) is injective , where H runs through the set of all subgroups of G which areelementary abelian of rank cyclic of order p with p odd , cyclic of order and quaternion of order .Proof . First note that that there is nothing to prove if G is cyclic of order1 or 2, because T ( G ) = { } . There is also nothing to prove if G is in thedetecting family of the statement. So we can assume that G is not elementaryabelian of rank 2, C p , C , or Q . By an obvious induction argument, it sufficesto prove that Y H Res GH : T ( G ) −→ Y 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 ≥ p odd and n ≥ p = 2), thenRes C pn C pn − : T ( C p n ) −→ T ( C p n − ) is an isomorphism (both groups are isomor-phic to Z / Z generated by the class of Ω ( k )). If G is extraspecial or almost HE CLASSIFICATION OF TORSION ENDO-TRIVIAL MODULES p = 2 or p is odd and G has exponent p . If p is odd and G has exponent p (extraspecial or almost extraspecial), then the result wasproved in Section 4 of [CaTh].So we can assume that G is neither cyclic, nor elementary abelian ofrank 2, nor extraspecial, nor almost extraspecial. In that case, the result wasproved as Theorem 3.2 of [CaTh].This theorem provides a direct proof of the following result, which wasfirst 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 Dadegroup 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 induc-tive proof of Theorem 1.2. We first need to prove the result in two specialcases. Proposition 12.3.
Suppose that G ∼ = Q × C or G ∼ = D ∗ C . Then T ( G ) is detected on restriction to all elementary abelian subgroups E of rank .Proof . Suppose that M is a nontrivial endo-trivial module such that M ↓ GE ∼ = k ⊕ (free) for every elementary abelian subgroup E of rank 2. Assumethat M has minimal dimension among such modules. On restriction to a max-imal subgroup of the form C × C , we must have that M ↓ GC × C ∼ = k ⊕ (free),because T ( C × C ) −→ T ( E ) is an isomorphism for E = C × C ⊂ C × C . Itfollows that Dim( M ) ≡ M ↓ GC ∼ = k ⊕ (free) forany cyclic subgroup C , because C is contained in a maximal subgroup of theform C × C .Since M is nontrivial, it must be detected on some restriction (Theo-rem 12.1). So there exists a quaternion subgroup H ∼ = Q in G such that M ↓ GH is nontrivial. Then M ↓ GH ∼ = Ω ( k H ) ⊕ (free), because Ω ( k H ) is the onlyindecomposable endo-trivial kH -module other than k H itself whose dimensionis congruent to 1 modulo 8 (see [CaTh, § z be the generator of the center of H (which is also central in G ). Weconsider the variety V G ( M ) ⊆ V G ( k ) ∼ = k where, as in Section 5, G = G/ h z i and M ∼ = ( z − M . On restriction to H , we have V H ( M ) = V H ( Ω ( k H ) )and Ω ( k H ) is a periodic kH -module by Lemma 5.3. Since Ω ( k H ) is invariantunder Galois automorphisms, so is Ω ( k H ), and therefore V H ( M ) is a union of72 JON CARLSON AND JACQUES TH´EVENAZ lines permuted by Galois automorphisms. But these lines are not F -rational(by Lemma 6.1 applied to the kH -module Ω ( k H ), which is critical), hence notfixed by Galois automorphisms. It follows that there are at least two lines in V H ( M ) (and in fact exactly two, which are F -rational, because this is the onlypossibility for the 4-dimensional module Ω ( k H ) ). Now V G ( M ) also containsat least two lines since it contains res ∗ G,H ( V H ( M )). So M ∼ = M ⊕ M where V G ( M ) is one of the two lines. Now following the procedure of Theorem 5.6we can construct a nontrivial endo-trivial kG -module N such that N = M .Moreover N is trivial on restriction to every elementary abelian subgroup.But Dim( N ) < 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 n ⋊ C for some n ≥ and some action of C on Q n . Then one of the following propertiesholds :(a) G contains a semi-dihedral subgroup S such that S ⊇ Q ⊆ Q n . (b) G contains a subgroup Q ∗ C with Q ⊆ Q n . (c) G contains a subgroup Q × C with Q ⊆ Q n .Proof . Let u be a generator of C . We use induction on n and firstconsider the case n = 3. If the action of u on Q /Z ( Q ) is nontrivial, then wecan choose two generators x and y of Q such that uxu − = y . In that case G is semi-dihedral and we are in case (a). If now u acts trivially on Q /Z ( Q ),then u fixes each of the three cyclic subgroups of order 4 of Q . If u actstrivially on Q , then G = Q × C and we are in case (c). Otherwise it easy tosee that u must invert two of the cyclic subgroups of order 4 and fix pointwisethe third one, say h x i . But then the actions of u and x coincide, so that ux − acts trivially and G = Q ∗ C , which is case (b).Assume now that n ≥
4. Let x and y be generators of Q n with x n − = 1, y = x n − and yxy − = x − . All elements of the form x b y have order 4(where 0 ≤ b ≤ n − ). Conjugation by u must satisfy uxu − = x a for someodd integer a and uyu − = x b y for some b . Since u = 1, we must have thefollowing congruences modulo 2 n − : a ≡ ± , n − ± a + 1) b ≡ . If a ≡ − b is odd, we can replace x by x b and we get a standard presenta-tion of the semi-dihedral group SD n +1 , so we are in case (a). Otherwise b mustbe even, because this is forced by the condition ( a + 1) b ≡ a
6≡ −
1. There-fore conjugation by u stabilizes the subgroup Q n − generated by x and y .The result now follows by induction applied to the group Q n − ⋊ h u i . HE CLASSIFICATION OF TORSION ENDO-TRIVIAL MODULES
Theorem 12.5.
Suppose that G is a p -group which is not cyclic , quater-nion , or semi-dihedral. Then T ( G ) is detected on restriction to all elementaryabelian subgroups E of rank .Proof . We use induction on the order of G . First recall that the result isknown if G is abelian or dihedral (see [CaTh]); so we assume that G is neitherabelian nor dihedral.Let M be an endo-trivial module such that Res GE [ M ] = 0 for every elemen-tary abelian subgroup E of rank 2, where [ M ] denotes the class of M in T ( G ).It suffices to prove that Res GH [ M ] = 0 for every maximal subgroup H of G ,because then [ M ] = 0 by Theorem 12.1. For every maximal subgroup H whichis not cyclic, quaternion, or semi-dihedral, M ↓ GH satisfies the same assumptionas M , so that Res GH [ M ] = 0 by induction. Now, we are left with the caseswhere 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 the-ory (see Theorem 4.4 in Chapter 5 of [Go1]), G is either abelian, or isomorphicto a group P to be described below, or in addition when p = 2, isomorphic to D n +1 , Q n +1 , or SD n +1 . The cases of the cyclic group C p n +1 , the quaterniongroup Q n +1 , or the semi-dihedral group SD n +1 , are excluded by our hypoth-esis. The cases of an abelian group or a dihedral group D n +1 have alreadybeen dealt with. So we are left with the case G = P = H ⋊ C p , with respect tothe action uxu − = x p n − , where x is a generator of H and u is a generatorof C p . This case occurs if n ≥ p is odd and n ≥ p = 2. Now G also contains a maximal subgroup K = h x p i × h u i ∼ = C p n − × C p and we alreadyknow that Res GK [ M ] = 0. ThereforeRes GC pn − [ M ] = Res KC pn − Res GK [ M ] = 0 . But we also have Res GC pn − = Res HC pn − Res GH andRes HC pn − : T ( H ) −→ T ( C p n − )is an isomorphism since both T ( H ) and T ( C p n − ) are cyclic of order 2 generatedby the class of Ω ( k ) (because n ≥ n ≥ p = 2). It follows thatRes GH [ M ] = 0.Assume now that H ∼ = SD n is semi-dihedral. We know that the torsionsubgroup T t ( H ) is cyclic of order 2 generated by the class of an endo-trivialmodule whose dimension is congruent to 1 modulo 2 n − (see [CaTh, § GH , because all endo-trivial modulesfor G have dimension congruent to ± n , by Lemma 2.10 in [CaTh].It follows that the image of Res GH is contained in h [Ω H ( k )] i ∼ = Z , because T ( H ) = T t ( H ) ⊕ h [Ω H ( k )] i . But now the restriction mapRes HE : h [Ω H ( k )] i −→ T ( E ) = h [Ω E ( k )] i JON CARLSON AND JACQUES TH´EVENAZ is an isomorphism where E is an elementary abelian subgroup of rank 2. SinceRes HE Res GH [ M ] = 0, we must have Res GH [ M ] = 0 as required. Note that thesame argument shows that Res GS [ M ] = 0 for any semi-dihedral subgroup S of G .Assume finally that H ∼ = Q n is quaternion. Since G is neither cyclic norquaternion, its 2-rank cannot be 1 (see Chapter 5 of [Go1]) and so there existsan element of order 2 outside H . Therefore G ∼ = Q n ⋊ C for some action of C on Q n . By Lemma 12.4, G contains a subgroup R which is isomorphic to Q ∗ C , Q × C , or semi-dihedral, and such that R ⊇ Q ⊆ H . In the firsttwo cases we have Res GR [ M ] = 0 by Proposition 12.3 and in the third we haveRes GR [ M ] = 0 by the argument above. It follows thatRes HQ Res GH [ M ] = Res GQ [ M ] = Res RQ Res GR [ M ] = 0 . We know that T ( H ) ∼ = Z / Z ⊕ Z / Z , where Z / Z is generated by the classof Ω H ( k ) and Z / Z is generated by the class of an endo-trivial module ofdimension 2 n − + 1 (see [CaTh, § GH , because all endo-trivial modules for G have dimension congruent to ± n . Thus the image of Res GH is contained in Z / Z = h [Ω H ( k )] i .But now the restriction mapRes HQ : h [Ω H ( k )] i −→ h [Ω Q ( k )] i is an isomorphism. Since Res HQ Res GH [ M ] = 0, we must have Res GH [ M ] = 0 asrequired.We immediately deduce the vanishing theorem (Theorem 1.1 of the intro-duction). Corollary 12.6. If G is not cyclic , quaternion or semi-dihedral , thenthe torsion subgroup of T ( G ) is trivial.Proof . By the theorem, we know that T ( G ) is embedded in a product ofcopies 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 max-imal elementary subgroup has rank at least . Then T ( G ) ∼ = Z , generated bythe class of the module Ω ( k ) .Proof . The assumption implies that G cannot be cyclic, quaternion orsemi-dihedral. Therefore, by the theorem, T ( G ) is detected on restrictionto elementary abelian subgroups of rank 2. The rest of the proof followsAlperin [Al2] and we recall the argument (also used in [BoTh]). The par-tially ordered set of all elementary abelian subgroups of rank at least 2 is con-nected, in view of the assumption and by a well-known result of the theory of HE CLASSIFICATION OF TORSION ENDO-TRIVIAL MODULES 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 thatall 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 ) ofall 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 thatthe torsion-free rank of D ( G ) has been determined in [BoTh] so that we areparticularly interested in the torsion subgroup D t ( G ). We first state an easyconsequence of Theorem 12.1. Theorem 13.1.
Let G be a finite p -group. (a) The product of all restriction-deflation maps Y K/H
Def
KK/H
Res GK : D ( G ) −→ Y K/H D ( K/H ) is injective , where K/H runs through the set of all sections of G whichare elementary abelian of rank cyclic of order p with p odd , cyclic oforder or quaternion of order . (b) For the torsion subgroup , the product of all restriction-deflation maps Y K/H
Def
KK/H
Res GK : D t ( G ) −→ Y K/H D t ( K/H ) is injective , where K/H runs through the set of all sections of G whichare cyclic of order p if p is odd , quaternion of order or cyclic of order if p = 2 .Proof . The argument is exactly the same as the one given in Theorem 1.6of [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 nontrivial torsion element in D ( G ) has order . In otherwords , for any indecomposable endo-permutation kG -module M with ver-tex G , the class of M is a torsion element if and only if M is self-dual. (b) If p = 2, any nontrivial torsion element in D ( G ) has order or .Proof . The nontrivial elements of D ( C p ) have order 2, while those of D ( Q ) and D ( C ) have order 2 or 4. Moreover, an element of order 2 corre-sponds to a self-dual module by definition of the group law.76 JON CARLSON AND JACQUES TH´EVENAZ If p is odd, the detection theorem above allows for a complete descriptionof the torsion subgroup of D ( G ) (Theorem 1.5 of the introduction), by thepartial results already obtained in [BoTh]. Theorem 13.3. If p is odd and G is a finite p -group , the torsion subgroupof D ( G ) is isomorphic to ( Z / Z ) s , where s is the number of conjugacy classesof nontrivial cyclic subgroups of G . Note that explicit generators are described in [BoTh].
Proof . Theorem 6.2 in [BoTh] asserts that a certain quotient D t ( G ) ofthe torsion subgroup D t ( G ) is isomorphic to ( Z / Z ) s , where s is as above.So we only have to prove that D t ( G ) = D t ( G ). But by definition, D t ( G ) = D t ( G ) / Ker( ψ ), where ψ is the product of all restriction-deflation maps ψ = Y K/H
Def
KK/H
Res GK : D t ( G ) −→ Y 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 ofsection needed on the right-hand side. However, we will also change the targetby including all groups having torsion endo-trivial modules, namely cyclic,quaternion, and semi-dihedral groups.If S = h x i is cyclic of order p n , then D ( S ) = D t ( S ) ∼ = n Y i =1 T t ( S/ h x p i i ) , and we let π S : D t ( S ) → T t ( S ) denote the projection onto the factor indexedby i = n . The situation is easier if S is a quaternion or semi-dihedral group,since D t ( S ) = T t ( S ) by [CaTh, § 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 classof all subgroups H of G such that N G ( H ) /H is cyclic. If p = 2, let X bethe class of all subgroups H of G such that N G ( H ) /H is cyclic of order ≥ quaternion of order ≥ or semi-dihedral of order ≥ . Let [ X /G ] be a system HE CLASSIFICATION OF TORSION ENDO-TRIVIAL MODULES of representatives of conjugacy classes of subgroups in X . Then the map Y H ∈ [ X /G ] π N G ( H ) /H Def N G ( H ) N G ( H ) /H Res GN G ( H ) : D t ( G ) −→ Y 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 GN G ( H ) /H ( a ) = 0 for every H ∈ X , where we write for sim-plicity DefRes GK/H = Def
KK/H
Res GK for every section K/H . By Theorem 13.1above, it suffices to prove that DefRes
GK/H ( a ) = 0 for every section K/H iso-morphic to C p , C or Q . We are going to show that DefRes GN G ( H ) /H ( a ) = 0 andthe result will follow from this since DefRes GK/H = Res N G ( H ) /HK/H DefRes GN G ( H ) /H .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 nothingto prove because L = G , π G/H = id, andDefRes
GG/H ( a ) = π G/H
DefRes
GG/H ( a ) = 0 , by assumption if p is odd and by the fact that D ( G/H ) = { } if p = 2. Let F be a subgroup such that H < F ≤ L . By induction, DefRes GN G ( F ) /F ( a ) = 0 andconsequently DefRes GN L ( F ) /F ( a ) = 0. This holds for every such F and thereforeDefRes GL/H ( a ) ∈ \ H L/HN L ( F ) /F ) = T ( L/H ) . The last equality is a well-known characterization of T ( L/H ) as a subgroupof D ( L/H ) (see Lemma 2.1 in [CaTh] and note that this characterization isalso at the heart of the proof of Theorem 13.1). Since a was chosen to be atorsion element in D ( G ), we have proved that DefRes GL/H ( a ) ∈ T t ( L/H ).If L/H is not cyclic, quaternion, or semi-dihedral, then T t ( L/H ) = { } by Theorem 1.1 and so DefRes GL/H ( a ) = 0. The same holds if L/H is cyclic oforder 2. If L/H is quaternion or semi-dihedral, then π L/H is the identity mapand π L/H DefRes GL/H ( a ) = 0 by assumption, so DefRes GL/H ( a ) = 0. If L/H iscyclic of order ≥ 3, then π L/H : D t ( L/H ) → T t ( L/H ) restricts to the identityon T t ( L/H ). Since π L/H DefRes GL/H ( a ) = 0 by assumption, we obtain againDefRes GL/H ( a ) = 0.In order to illustrate the efficiency of Theorem 13.4 compared to Theo-rem 13.1, suppose that G is abelian. Then there are numerous sections of G isomorphic to C p or C and the map in Theorem 13.1 is an injection in amuch larger group, whereas the map in Theorem 13.4 hits exactly every cyclicquotient of G and is an isomorphism (Dade’s theorem).78 JON CARLSON AND JACQUES TH´EVENAZ If G is a dihedral 2-group, there are many sections of G isomorphic to C ,but N G ( H ) /H is never cyclic of order ≥ 4, quaternion, or semi-dihedral, sothat X is empty and D t ( G ) = { } , a result also obtained in [CaTh, § D t ( G )was not previously known. Proposition 13.5. Suppose that G is an extraspecial -group of type that is , a central product of copies of D . Then D t ( G ) = { } .Proof . We claim that X is empty and so D t ( G ) = { } . If H is a sub-group of G containing Z ( G ), then H is a normal subgroup, G/H is elementaryabelian, and H / ∈ X . If H does not contain Z ( G ), then for any g ∈ N G ( H ), wehave that [ g, h ] ∈ H ∩ Z ( G ) = { } . 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 ) = { } . Using the quadratic form on G/Z ( G ), it is nothard to prove that if n is the number of copies of D in the central product andif H = ( C ) k , then C G ( H ) = H × L where L is a central product of n − k copiesof D (possibly n − k = 0 and L = Z ( G )). Therefore N G ( H ) /H is extraspecialand 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 to the case of an odd prime, for which the solutionof 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 oftype 2 and the almost extraspecial groups (type 3). For simplicity, we shallonly deal with the smallest of the groups, namely D ∗ Q and D ∗ C , butour results can easily be generalized to the other groups of types 2 and 3.If follows from Theorem 13.4 that the product of all restriction-deflationmaps Y H ∈ [ X /G ] Def N G ( H ) N G ( H ) /H Res GN G ( H ) : D t ( G ) −→ Y H ∈ [ X /G ] D t ( N G ( H ) /H )is injective. In the opposite direction, there is the sum of all maps obtained bycomposing inflation maps Inf N G ( H ) N G ( H ) /H and tensor induction Ten GN G ( H ) , namely X H ∈ [ X /G ] Ten GN G ( H ) Inf N G ( H ) N G ( H ) /H : M H ∈ [ X /G ] D t ( N G ( H ) /H ) −→ D t ( G ) . We let D t ( G ) be the image of this map. The question of the surjectivity ofthis map does not seem to be easy and this is why we have to introduce thesubgroup D t ( G ). In similar situations for odd primes, or for the Dade group HE CLASSIFICATION OF TORSION ENDO-TRIVIAL MODULES Q , we can prove the surjectivity of the map (see Sections 4 and 6of [BoTh]), so it seems natural to conjecture that D t ( G ) = D t ( G ). In our twoexamples, we shall be able to compute D t ( 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 ( 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 ofthe form Ten GK Inf KK/H ( x ), but for simplicity we only consider two very specialcases. 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 inSection 3 of [BoTh]. Lemma 14.1. Let G be a p -group and let K and H be subgroups of G suchthat 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 furtherthat 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 PP/R Res GP Ten GK Inf KK/H = γ | R : R ∩ H | Iso P/R ( P ∩ K ) / ( R ∩ H ) (Iso K/H ( P ∩ K ) / ( R ∩ H ) ) − , 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 KK/L Inf KK/H = Inf K/LK/HL Def K/HK/HL . Proof . (a) Since there is a single double coset, the Mackey formula impliesthat Def PP/R Res GP Ten GK Inf KK/H = Def PP/R Ten PP ∩ K Res KP ∩ K Inf KK/H . Now Proposition 3.10 in [BoTh] asserts thatDef PP/R Ten PQ = γ | R : Q ∩ R | Ten P/RQR/R Iso QR/RQ/Q ∩ R Def QQ/Q ∩ R . 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 thereforeDef PP/R Ten PP ∩ K = γ | R : R ∩ H | Iso P/RP ∩ K/R ∩ H Def P ∩ KP ∩ K/R ∩ H . JON CARLSON AND JACQUES TH´EVENAZ Composing on the right with Res KP ∩ K Inf KK/H , it is easy to see thatDef P ∩ KP ∩ K/R ∩ H Res KP ∩ K Inf KK/H = (Iso K/HP ∩ K/R ∩ H ) − , using either the definitions of the maps or the methods of Corollary 3.9 in [BoTh].It follows thatDef PP/R Ten PP ∩ K Res KP ∩ K Inf KK/H = γ | R : R ∩ H | Iso P/RP ∩ K/R ∩ H (Iso K/HP ∩ K/R ∩ H ) − , and the result follows.(b) This follows either from the definitions of the maps or from the meth-ods of Corollary 3.9 in [BoTh].Now we can start with our first example D ∗ C . Let S , S , S be rep-resentatives of the three conjugacy classes of noncentral subgroups of order 2(the two classes in D and the product of a generator of C with an elementof order 4 in D ). Proposition 14.2. Let G = D ∗ C be the almost extraspecial group oforder . Then D t ( G ) is cyclic of order generated by the class of the module Ten GS × C Inf S × C S × C /S (Ω S × C /S ( k )) .Proof . We have that N G ( S i ) = S i × C and so N G ( S i ) /S i ∼ = C and S i isin the class X of Theorem 13.4. These are the only subgroups in X (becauseevery other nontrivial subgroup H contains the Frattini subgroup and G/H iselementary abelian). Therefore Theorem 13.4 yields an injective map Y i =1 Def S i × C S i × C /S i Res GS i × C : D t ( G ) −→ Y i =1 D t ( S i × C /S i ) ∼ = ( Z / Z ) , each factor D t ( S i × C /S i ) ∼ = D t ( C ) being cyclic of order 2 generated bythe class of Ω S i × C /S i ( k ). Now by definition D t ( G ) is generated by the threeelements Ten GS i × C Inf S i × C S i × C /S i (Ω S i × C /S i ( k )) (1 ≤ i ≤ . We claim that they are all equal and have order 2. This will complete theproof of the proposition.In order to prove the claim, we show that the image of any of these threeelements by the injective map above is equal to the “diagonal element”(Ω S × C /S ( k ) , Ω S × C /S ( k ) , Ω S × C /S ( k ) ) . This follows from a straightforward application of Lemma 14.1. If i = j , weobtain Def S j × C S j × C /S j Res GS j × C Ten GS i × C Inf S i × C S i × C /S i (Ω S i × C /S i ( k ))= γ | S j :1 | Iso S j × C /S j C / (Iso S i × C /S i C / ) − (Ω S i × C /S i ( k ))= γ (Ω S j × C /S j ( k )) = Ω S j × C /S j ( k ) , HE CLASSIFICATION OF TORSION ENDO-TRIVIAL MODULES ( k ) is defined over the prime field F and hence is fixed bythe Frobenius map γ . In the case i = j , we have for any k [ S i × C ]-module M ,Res GS i × C Ten GS i × C ( M ) = M ⊗ g M , where g is a representative of the nontrivial class of G/S i × C and g M denotesthe conjugate module. Therefore, ignoring inflation for simplicity, we obtainDef S i × C S i × C /S i Res GS i × C Ten GS i × C Inf S i × C S i × C /S i (Ω S i × C /S i ( k ))= Def S i × C S i × C /S i (cid:0) Ω S i × C /S i ( k ) ⊗ g (Ω S i × C /S i ( k )) (cid:1) = Def S i × C S i × C /S i (cid:0) Ω S i × C /S i ( k ) (cid:1) ⊗ Def S i × C S i × C /S i (cid:0) Ω S i × C /gS i g − ( k ) (cid:1) = Ω S i × C /S i ( k ) ⊗ Def S i × C S i × C /S i (Ω S i × C /gS i g − ( k )) . But the second factor is trivial because, by part (b) of Lemma 14.1 with K = S i × C , we haveDef KK/S i Inf KK/gS i g − = Inf K/S i K/S i ( gS i g − ) Def K/gS i g − K/S i ( gS i g − ) and a deflation of the class of Ω ( k ) is trivial (see Lemma 1.3 of [BoTh]).In this example, we see that D t ( G ) embeds in three copies of Z / Z andthat D t ( G ) ∼ = Z / Z . So in order to prove the conjectural equality D t ( G ) = D t ( G ), we would have to improve Theorem 13.4 by showing the injectivity ofthe restriction-deflation map to a single section S i × C /S i . In this specificexample, we have been able to do this by a rather delicate argument not givenhere.The methods are similar with our second example D ∗ Q , but anothercomplication occurs. Recall that D t ( Q ) is generated by the class of Ω Q ( k ),which has order 4, and the class of a certain 5-dimensional module M , whichhas order 2 (see [CaTh, § M is defined over the field F (sowe assume here that k contains F ) and M is not invariant under the Galoisautomorphism γ . Actually γ ( M ) ∼ = Ω ( M ), another 5-dimensional module,and Ω ( k ), M , Ω ( M ) are the three elements of order 2 in D t ( Q ) ∼ = Z / Z ⊕ Z / Z .Let S , . . . , S be representatives of the five conjugacy classes of noncentralsubgroups of order 2 (the two classes in D and the product of an element oforder 4 in D with one of the three possible elements of order 4 in Q ). Proposition 14.3. Let G = D ∗ Q be the extraspecial group of order 32 ( type Then D t ( G ) ∼ = Z / Z ⊕ Z / Z generated by the class of the module Ten GN G ( S ) Inf N G ( S ) N G ( S ) /S (Ω N G ( S ) /S ( k )) ( order JON CARLSON AND JACQUES TH´EVENAZ and by the class Ten GN G ( S ) Inf N G ( S ) N G ( S ) /S ( M N G ( S ) /S ) ( order where M N G ( S ) /S is the module M viewed as a module for the group N G ( S ) /S , which is isomorphic to Q .Proof . We have that N G ( S i ) = S i × C (for some subgroup C isomorphicto Q ) and so N G ( S i ) /S i ∼ = Q and S i is in the class X of Theorem 13.4.These are the only subgroups in X , because every other nontrivial subgroup H contains the Frattini subgroup and G/H is elementary abelian. Therefore,by Theorem 13.4, the map Y i =1 Def N G ( S i ) N G ( S i ) /S i Res GN G ( S i ) : D t ( G ) −→ Y i =1 D t ( N G ( S i ) /S i ) ∼ = ( D t ( Q )) is injective. Now by definition D t ( G ) is generated by the classes of the modulesTen GN G ( S i ) Inf N G ( S i ) N G ( S i ) /S i ( X ) (1 ≤ i ≤ , where X is either Ω ( k ) or M (viewed in D t ( N G ( S i ) /S i ) ).If X = Ω ( k ), we always obtain the same element, independently of i ,mapping to the diagonal element consisting of Ω ( k ) in each component underthe injective map above. The proof of this follows exactly the same argumentas the one used in the proof of Proposition 14.2, with the following minormodification. For every pair S i , S j with i = j , the group generated by S i and S j is isomorphic to D . Its centralizer C is isomorphic to Q , 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 theargument is similar to that used in Proposition 14.2.If now X = M , we again use Lemma 14.1, but the computation changesbecause of the presence of the Galois automorphism γ which does not fixthe class of M . Moreover, for each i , we need to fix a choice of isomorphism N G ( S i ) /S i ∼ = Q in order to be able to make a consistent computation. We skipthe details and only give the result. It turns out that, under the injective mapabove, the image of Ten GN G ( S i ) Inf N G ( S i ) N G ( S i ) /S i ( M ) is the 5-tuple ( M, M, M, M, M ),again independent of i . 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