THE DADE GROUP OF A FINITE GROUP

CAROLINE LASSUEUR

Abstract. The aim of this paper is to construct an equivalent of the Dade group of a p-group for an arbitrary finite group G, whose elements are equivalences classes of endo-p-permutation modules. To achieve this goal we use the theory of relative projectivity with respect to a module and that of relative endotrivial modules.

1. Introduction

The construction of the Dade group D P described by E. Dade in [Dad78a] is valid only in case the group P is a p-group. This is linked to the facts that the transitive kP -permutation modules are indecomposable, whereas for an arbitrary group G, the transitive kG-permutation modules are not indecomposable in general, and moreover that their direct summands need not be permutation modules. The classification of endo-permutation modules via the complete description of the structure of the Dade group D P was completed in 2004 by S. Bouc with [Bou06]. It had started about 25 years earlier with the first papers and results by E. Dade in [Dad78a] and [Dad78b] in 1978, and the final classification was in fact achieved through the non-effortless combined work of several (co)-authors between 1998 and 2004, including J.L. Alperin, S. Bouc, J. Carlson, N. Mazza and J. Thévenaz. Yet, for an arbitrary finite group G, no satisfying equivalent group structure to the Dade group on a class of kG-modules has been defined so far.

One way to obtain a similar notion to that of the Dade Group for arbitrary groups is to consider endo-p-permutation modules as described by J.-M. Urfer in [Urf06]. He shows that if P is a p-subgroup of a group G, this notion induces a group structure, denoted by D_{P} G , on a set of equivalence classes of indecomposable endo-p-permutation kG-modules with vertex P. (The equivalence relation being a generalisation of Dade’s compatibility relation.) However, the main drawback of this approach resides in the fact that there is not a unique indecomposable representative, up to isomorphism, for the classes in D_{P} G . More precisely, D_{P} G classifies the sources of the endo-p-permutation modules with vertex P, but not the modules themselves. Also note that if P is a Sylow p-subgroup of G, then D_{P} G D P, F_{P} G , where D P, F_{P} G is the Dade group of the fusion system F_{P} G on P defined in [LM09].

The aim of this piece of work is to show how the notion of relative endotrivial module, that we introduced in [Las11a], can generalise the Dade group in a more natural way. It is most interesting to note that crucial building pieces for the classification of endo-permutation modules are indeed the endotrivial modules, which are particular cases of endo-permutation modules. In some sense, we turn the problem upside down, and show how one can regard an endo-permutation module as an endotrivial module, of course not in the ordinary sense, but in the relative sense. This enables us to endow a well-chosen set of isomorphism classes of endo-p-permutation modules with a group structure, similar to that of the Dade group. We call this new group, the generalised Dade group of the group G, explicitly compute its structure and show how it is closely related to that of the G-stable points of the Dade group of a Sylow p-subgroup of G.

Date: January 22, 2012.

1991 Mathematics Subject Classification. 20C20.

Key words and phrases. Dade group, endo-p-permutation modules, endotrivial modules, relative endotrivial modules, relative projectivity with respect to modules.

This work is part of a doctoral thesis prepared at the Ecole Polytechnique Fédérale de Lausanne, under the supervision of Prof. Jacques Thévenaz.

2. Preliminaries and definitions

Unless otherwise mentioned, throughout this text k shall denote an algebraically closed field of prime characteristic p, G a finite group whose order is divisible by p, all modules are finitely generated, mod kG denotes the category of finitely generated left kG-modules and stmod kG the corresponding stable category. We write k for the one-dimensional trivial module. Moreover, denotes the ordinary tensor product over k, M^{}Hom_{k} M, k and Ω M the k-dual and the kernel of a projective cover of the kG-module M, respectively.

2.1. Relative projectivity with respect to a module. Projectivity relative to a kG-module was introduced by T. Okuyama [Oku91], then further developed and used in [Car96], [CP96], [CPW98], and also by the author in [Las11a]. This is a generalisation of the more classic projectivity relative to a subgroup widely used in the theory of vertices and sources. Moreover, it is also just a special case of the relative homological algebra defined for a projective class of epimorphisms or a pair of adjoint exact functors in [HS71, Chap. 10]. We recall here basic definitions and useful properties.

Definition 2.1 ([Oku91]). Let V mod kG . A finitely generated kG-module M is termed relatively V -projective, or simply V -projective, if there exists a kG-module N such that M is isomorphic to a direct summand of V _{k} N. The module M is called V -injective if and only if it is V -projective.

Proposition 2.2 (Omnibus properties, [Las11a], Prop. 2.0.2). Let U, V be kG-modules.

(a) Any direct summand of a V -projective module is V -projective and if U P roj V , then P roj U P roj V .

(b) If p ∤ dim_{k} V then P roj V mod kG . In particular P roj k mod kG .

(c) P roj U V P roj U P roj V .

(d) P roj U P roj V P roj U V P roj U P roj V .

(e) P roj V P roj V^{}P roj Ω V P roj Ω^{1} V P roj V V P roj V V .

(f) If P mod kG is projective, then P roj P P roj kG , which is equal to the whole class of projective modules in mod kG . Moreover P roj kG P roj V for any kG-module V.

Remark 2.3. The notion of relative projectivity with respect to a module encompasses the notion of projectivity relative to a subgroup, used in the theory of vertices and sources. More precisely, a kG-module M is projective relatively to the subgroup H of G if and only if M P roj k _{H}^{G} . Moreover, if H is a family of subgroups of G, then M is projective relatively to the family H if and only if M is projective relatively to the kG-module V H : _{H H} k _{H}^{G}.

In the sequel, we will state results concerned with projectivity relative to subgroups and families of subgroups in terms of modules as described here. Translating projectivity relative to a subgroup in terms of modules we have the following well-known properties (see e.g. [CR90, §19]):

- if H G, then P roj k _{H}^{G} P roj k _{^{g}H}^{G} for every g G;

- if K H G, then P roj k _{K}^{G} P roj k _{H}^{G} . Moreover, if H : H_{1}, …, H_{n} , n ℕ, is a family of subgroups of G, then, by the two preceding properties and the omnibus properties above, assuming that H_{i} _{G} H_{j} i j, 1 i, j n does not alter P roj H .

In next subsection we describe how one can use projectivity with respect to a module to construct groups of relative endotrivial modules. This essentially relies on the following theorem by Benson and Carlson.

Theorem 2.4 ([BC86], Thm. 2.1). Let k be an algebraically closed field of characteristic p (possibly p 0). Let M, N be finite-dimensional indecomposable kG-modules, then

k M N if and only if \begin{matrix}(1) M N^{}; \\ (2) p ∤ dim_{k} N . \end{matrix}

Moreover, if k is a direct summand of N^{}N then it has multiplicity one, i.e. k k is not a summand.

Definition 2.5. A kG-module V mod kG is called absolutely p-divisible if p char k divides the k-dimension of every indecomposable direct summand of V.

Proposition 2.6 ([Las11a], Prop. 2.2.2). Let V mod kG . Then, the following are equivalent: (a) The trivial kG-module k is not V -projective;

(b) V is absolutely p-divisible;

(c) P roj V mod kG .

2.2. Relative endotrivial modules. In [Las11a], we introduced and developed the notion of an endotrivial module relative to a kG-module V.

Definition 2.7. Let V be an absolutely p-divisible kG-module. A module M mod kG is termed endotrivial relative to the kG-module V or simply V -endotrivial if

End_{k} M M^{}M k V proj .

This definition is equivalent to requiring that End_{k} M is isomorphic to a trivial module in the relative stable category stmod_{V} kG .

Lemma 2.8 ([Las11a], Lem. 3.1.2, 3.2.1, 3.2.2, 3.2.3, 4.1.1). Let V mod kG be an absolutely p-divisible module. Let M, N mod kG be V -endotrivial modules. Then:

(a) dim_{k} M ^{2} 1 mod p.

(b) The modules M^{}, M N and Hom_{k} M, N are V -endotrivial.

(c) If M is indecomposable, then the vertices of M are the Sylow p-subgroups of G. Moreover, if P, S is a vertex-source pair for M, then S is a V _{P}^{G}-endotrivial module, and S has multiplicity one as a direct summand of M _{P}^{G}.

(d) There is a direct sum decomposition M M_{0} V proj where M_{0} is the unique indecomposable summand of M that is V -endotrivial.

(e) If P is Sylow p-subgroup of G, then L mod kG is V -endotrivial if and only if L _{P}^{G} is V _{P}^{G}-endotrivial.

Now, if V mod kG is an absolutely p-divisible module, one can set an equivalence relation V on the class of V -endotrivial kG-modules as follows: for M and N two V -endotrivial modules let

M _{V} N if and only if M_{0} N_{0} ,

where M_{0} and N_{0} are the unique V -endotrivial indecomposable summands of M and N, respectively, given by part (e) of Lemma 2.8. This amounts to requiring that M and N are isomorphic in stmod_{V} kG . Then let T_{V} G denote the resulting set of equivalence classes. In particular, any equivalence class in T_{V} G consists of an indecomposable V -endotrivial module M_{0} and all the modules of the form M_{0} V proj .

Proposition 2.9 ([Las11a], Prop. 3.5.1). The ordinary tensor product _{k} induces an abelian group structure on the set T_{V} G defined as follows:

M N : M _{k} N

The zero element is k and the opposite of a class M is the class M^{}.

Lemma 2.10 ([Las11a], Prop. 3.5.3). Let V mod kG be absolutely p-divisible. If W P roj V , then the group T_{W} G can be identified with a subgroup of T_{V} G via the injective group homomorphism T_{W} G T_{V} G : M M . By abuse of notation, we write T_{W} G T_{V} G .

Since P roj kG is ordinary projectivity, the group of endotrivial modules is T G T_{kG} G . Then, by the above and part (e) of Lemma 2.2, T G T_{V} G for every absolutely p-divisible V mod kG .

One-dimensional representations: If G is a finite group, denote by X G the abelian group of all isomorphism classes of one-dimensional kG-modules endowed with the group law induced by k, which can also be identified with the group Hom G, k^{}of k-linear characters of G. This is a finite p^{}-group, isomorphic to the p^{}-part of the abelianisation G G, G of G.

Let V mod kG be an absolutely p-divisible module. Then any one-dimensional module χ is V -endotrivial, because χ^{}χ k. Therefore there is an embedding X G T_{V} G : χ χ . Thus we can identify X G with a subgroup of T_{V} G and there is always a chain of subgroups:

X G T G T_{V} G

There are also several homomorphisms between groups of relative endotrivial modules induced by a change of group.

Lemma 2.11 ([Las11a], Sect. 3.6).

1. Restriction. Let H be a subgroup of G and let V be an absolutely p-divisible kG-module, then restriction to H induces a group homomorphism, called a restriction map:

\begin{aligned}Res_{H}^{G}: & T_{V} G & & T_{V _{H}^{G}} H \\ & M & & M _{H}^{G} \end{aligned}

Moreover, if H contains the normaliser N_{G} P of a Sylow p-subgroup of G, then Res_{N_{G} P}^{G} is injective and sends the class of an indecomposable kG-module to the class of its kH-Green correspondents.

2. Inflation. Let N be a normal subgroup of a group G such that p G N . If V is an absolutely p-divisible k G N -module, then inflation induces an injective group homomorphism:

Inf_{G N}^{G}: T_{V} G N  T_{Inf_{G N}^{G} V} G
M  Inf_{G N}^{G} M

3. Isomorphism. Let φ : G_{1} G_{2} be a group isomorphism. If M is a kG_{1}-module, then it can be seen as a kG_{2}-module via φ^{1} and is denoted Iso φ M . Let V be an absolutely p-divisible kG_{1}-module. Then there is a group isomorphism:

Iso φ : T_{V} G_{1}  T_{Iso φ V} G_{2}
M  Iso φ M

Lemma 2.12. Let P be a Sylow p-subgroup of G and let H G be a subgroup containing N_{G} P . Let V mod kG be an absolutely p-divisible module.

(a) The restriction map Res_{H}^{G} : T_{V} G T_{V _{H}^{G}} H is injective.

(b) If P roj V _{H}^{G} P roj V Y , where Y ^{g}P H g G H , then the restriction map Res_{H}^{G} : T_{V} G T_{V _{H}^{G}} H is an isomorphism. Furthermore, the inverse map is induced by induction, so that

T_{V} G M _{H}^{G} M T_{V _{H}^{G}} H T_{V _{H}^{G}} H .

More accurately, on indecomposable V _{H}^{G}-endotrivial modules, the inverse map is induced by the Green correspondence, that is, if Γ M denotes the Green correspondent of an indecomposable kH-module M, then

T_{V} G Γ M M is an indecomposable V _{H}^{G} -endotrivial kH-module .

Lemma 2.13 ([Las11a], Lem. 4.4.1). Let G be a finite group with a normal Sylow p-subgroup P and let V mod kG be an absolutely p-divisible module. Then ker Res_{P}^{G} X G .

2.3. Relative syzygy modules. An important family of relative endotrivial modules is provided by the relative syzygies modules of the trivial module. We refer to [Car96, Sect. 8] for definitions of projective and injective resolutions with respect to a module V mod kG .

Definition 2.14. Let M mod kG , let P_{}, _{}^{ε} M and M ^{ı} I_{}, ^{}be minimal V - projective and V -injective resolutions of M, respectively. Define for all n 1: Ω_{V}^{n} M : ker _{n 1} , Ω_{V}^{n} M : Coker ^{n 1} . Define Ω_{V}^{0} to be the V -projective free part of M. The module Ω_{V}^{m} M , m ℕ is called the m-th V -relative syzygy module of M.

Notation. We write Ω_{V} M : Ω_{V}^{1} M and simply Ω^{n} M : Ω_{Q}^{n} M , Ω M : Ω_{Q}^{1} M if the module Q is projective. Moreover, if H is a family of subgroups of the group G, then we write Ω_{H} M instead of Ω_{V H} M . If V mod kG is absolutely p-divisible and W P roj V , we write Ω_{W} for the class of Ω_{W} k in T_{V} G and we write Ω for the class of Ω k in T_{V} G .

Lemma 2.15 ([Las11a], Lemmas 2.3.3, 2.3.4, 3.2.1). Let M, V, W mod kG .

(a) If P roj V P roj W , then Ω_{V}^{n} M Ω_{W}^{n} M for every n ℤ.

(b) Ω_{V} Ω_{W} M Ω_{V W} Ω_{V W} M and if H, K are families of subgroups of G, then this formula reads Ω_{H} Ω_{K} M Ω_{H K} Ω_{^{G}H K} M where ^{G}H K ^{g}H K H H, K K .

(c) If M is a V -endotrivial kG-module and W P roj V , then the kG-modules Ω_{W}^{n} M are V -endotrivial modules for every n ℤ.

(d) If H G and V is absolutely p-divisible, then Res_{H}^{G} Ω_{V} Ω_{V _{H}^{G}} T_{V _{H}^{G}} H .

Lemma 2.16 ([Oku91], [Las11b] Lem. 3.8.1). Let n 2 be an integer and V_{1}, …, V_{n} mod kG be pairwise non isomorphic absolutely p-divisble modules.

(a) In T_{V_{1} V_{2}} G , we have Ω_{V_{1} V_{2}} Ω_{V_{1}} Ω_{V_{2}} Ω_{V_{1} V_{2}} and Ω_{V_{1} V_{2}} Ω_{V_{1}} Ω_{V_{2}} k .

(b) More generally, in T_{V_{1} ... V_{n}} G :

Ω_{V_{1} ... V_{n}}  _{i 1}^{n} Ω_{V_{i}} _{j 2}^{n} Ω_{_{r 1}^{j 1} V_{r} V_{j}}  _{s 1}^{n} 1 ^{s 1} _{1 i_{1} ... i_{s} n} Ω_{V_{i_{1}}  V_{is}}

(c) If H : H_{1}, …, H_{n} is a family of subgroups of the group G such that the kG-module V H is absolutely p-divisible, then formula (b) reads

Ω_{H} _{i 1}^{n} Ω_{H_{i}} _{j 2}^{n} Ω_{^{G} H_{1},...,H_{j 1} H_{j}} in T_{V H} G .

2.4. Endo-permutation modules and the Dade group. If P is a p-group, then a kP -module M is called an endo-permutation module if its endomorphism algebra End_{k} M is a permutation kP -module. Furthermore, an endo-permutation module M is called capped if it possesses an indecomposable summand with vertex P.

Proposition 2.17 ([Dad78a]).

(a) The class of capped endo-permutation modules is closed under taking direct summands, duals, tensor products (over k), Heller translates, restriction to a subgroup and tensor induction to an overgroup.

(b) An endo-permutation kP -module M is capped if and only if the trivial module is a direct summand of End_{k} M .

(c) If M is capped, then any two indecomposable summands of M with vertex P are isomorphic. This unique summand, up to isomorphism, is called the cap of M and is written Cap M .

(d) An equivalence relation on the class of endo-permutation module is defined by: M N if and only if Cap M Cap N .

(e) Let D P denote the resulting set of equivalence classes. Then D P is an abelian group

for the following law:

M N M N

The zero element is the class k of the trivial kP -module, while the opposite of a class M is the class of the dual module M^{}. This group is called the Dade group of the group P.

Note that in every equivalence class in D P , there is, up to isomorphism, a unique indecomposable module, namely the cap of any module in the class. Thus D P is in bijection with the set of isomorphism classes of indecomposable endo-permutation kP -modules with vertex P which becomes a group with the law M N : Cap M N .

The classification of endo-permutation modules, through the description of the structure of the Dade group, started with [Dad78a], [Dad78b], and independently [Alp77]. It was completed in 2004 by S. Bouc in [Bou06]. Inbetween, crucial steps for this classification include the classification of the endotrivial modules of a p-group. All this was achieved through the work of [Pui90], [BT00], [CT00], [CT04], [CT05], [Bou04] and [BM04].

In [Las11a], we noted that a main reason of interest in relative endotrivial modules comes from the fact that they provide a way to define a group structure on collections of representations of an arbitrary finite group G which gives a generalisation for the Dade group of a p-group. Indeed, endo-permutation modules can always be seen as relative endotrivial modules in the following sense:

Theorem 2.18 ([Las11a], Thm 5.0.2). Let P be a p-group and let V F_{P} : _{Q P} k _{Q}^{P}. The Dade group D P can be identified with a subgroup of T_{V F_{P}} P via the canonical injective homomorphism

\begin{matrix}D P & & T_{V F_{P}} P \\ M & & Cap M . \end{matrix}

3. Projectivity relative to the family of subgroups F_{G}

Recall from the theory of vertices and sources that:

- If H is a subgroup of G and Q is a Sylow p-subgroup of H, then P roj k _{H}^{G} P roj k _{Q}^{G} .

- If H G, then P roj k _{H}^{G} mod kG if and only if H contains a Sylow p-subgroup of G . Thus it follows from Remark 2.3 and Proposition 2.6 that a permutation module k _{R}^{G}, for a subgroup R G, is absolutely p-divisible if and only if R has a Sylow p-subgroup Q _{G} P.

Notation. Given G a finite group, fix a Sylow p-subgroup P of G and set F_{G} : Q P . Then consider the associated module V F_{G} _{Q F_{G}} k _{Q}^{G} and notice that by the above P roj V F_{G} corresponds to projectivity relative to the family of all non maximal p-subgroups of G. We emphasise that P roj V F_{G} does not depend on the choice of the Sylow p-subgroup P.

Lemma 3.1. Let H be a subgroup of G that contains a Sylow-p subgroup P of G. Then: (a) P roj V F_{G} _{H}^{G} P roj V F_{H} .

(b) V F_{H} is absolutely p-divisible.

Proof. The Mackey formula yields

V F_{G} _{H}^{G} _{Q F_{G}} k _{Q}^{G} _{H}^{G} _{Q F_{G}} _{x H G Q} k _{^{x}Q H}^{H}  _{Q P} k _{Q}^{H} X V F_{H} X

where X is a direct sum of kH-modules of the form k _{S}^{H} with S _{G} P, so that k _{S}^{H} P roj V F_{H} . Thus by Proposition 2.2 we obtain first that P roj X P roj V F_{H} and second that

P roj V F_{G} _{H}^{G} P roj V F_{H} X P roj V F_{H} .

This proves (a). Now, by Green’s indecomposability Criterion, the modules k _{Q}^{P} are indecomposable for every Q P, and moreover their dimension is divisible by p. In consequence, the module V F_{P} _{Q F_{P}} k _{Q}^{P} is absolutely p-divisible and therefore so are the modules V F_{H} for every P H G.

Indeed, .This proves (b).

□

Lemma 3.2. Let N be a normal subgroup of the group G such that p G N .Then

P roj Inf_{G N}^{G} V F_{G N} P roj V F_{G} .

Proof. Let P be a Sylow p-subgroup of G and P N N the corresponding Sylow p-subgroup of G N. By definition,

V F_{G N} _{R P N N} k _{R}^{G N} .

Moreover, if R P N N, there exists a subgroup Q such that P N Q P and R QN N. Whence

Inf_{G N}^{G} V F_{G N}  _{P N Q P} Inf_{G N}^{G} k _{QN N}^{G N}  _{P N Q P} k _{QN}^{G} .

Now, since Q is a Sylow p-subgroup of QN, P roj k _{QN}^{G} P roj k _{Q}^{G} (see above). Whence

P roj Inf_{G N}^{G} V F_{G N}  P roj _{P N Q P} k _{QN}^{G}  _{P N Q P} P roj k _{Q}^{G} P roj V F_{G} .

where the last inclusion is obtain by Proposition 2.2, parts (a) and (c), and by definition of the family F_{G}. □

4. V F_{G} -endotrivial modules

Because the module V F_{G} is absolutely p-divisible, we obtain a well-defined group T_{V F_{G}} G of V F_{G} -endotrivial modules. The following elementary properties of this group can easily be deduced from the general theory of relative endotrivial modules that is developed in [Las11a].

Proposition 4.1. Let P be a Sylow p-subgroup of G and H be a subgroup of G such that P H G . (a) There is a well-defined restriction map

\begin{aligned}Res_{H}^{G}: & T_{V F_{G}} G & & T_{V F_{H}} H \\ & M & & M _{H}^{G} . \end{aligned}

(b) If H contains N_{G} P , then the restriction map Res_{H}^{G} : T_{V F_{G}} G ^{}T_{V F_{H}} H is an isomorphism, whose inverse is induced by the Green correspondence on the indecomposable V F_{H} -endotrivial modules.

(c) ker Res_{P}^{N_{G} P} X N_{G} P .

(d) If Γ X N_{G} P denotes the subgroup of T_{V F_{G}} G made up of the classes of the kG-Green correspondents of the modules in X N_{G} P , then ker Res_{P}^{G} Γ X N_{G} P , which is a finite group isomorphic to X N_{G} P .

Proof. (a) This follows from the definition of a restriction map (section 2.2) and part (a) of

Lemma 3.1.

(b) By Lemma 3.1 P roj V F_{G} _{H}^{G} P roj V F_{H} . Therefore part (b) of Lemma 2.12 applies and yields the result. Indeed Y F_{H} (where Y is the family of subgroups in Lemma 2.12) and thus by the omnibus properties of relative projectivity, P roj V F_{H} P roj V Y .

(c) This is a straightforward application of Lemma 2.13.

(d) Since, by part (b), Res_{N_{G} P}^{G} is an isomorphism, (d) follows from (c).

□

Example 4.2. Thus far there are two obvious families of examples of V F_{G} -endotrivial modules.

(a) The kG-Green correspondents of the one-dimensional representations of the normaliser N_{G} P , provided by part (d) of Lemma 4.1.

(b) The relative syzygies Ω_{W}^{n} M with W P roj V F_{G} , n ℤ and M a V F_{G} -endotrivial module as described in part (c) of Lemma 2.15. In particular if H is a family of subgroups of G such that the associated module V H _{H H} k _{H}^{G} (see Remark 2.3) is absolutely p-divisible, then P roj V H P roj V F_{G} and therefore the relative syzygy modules Ω_{H}^{n} k of the trivial module are all V F_{G} -endotrivial modules.

It is known from [Alp01] that the relative syzygies Ω_{H}^{n} k , for families of subgroups H, are endopermutation modules when G is a p-group. In similar manner, [Urf06, Prop. 5.8] shows that they are endo-p-permutation modules when G is arbitrary. We show in section 7 that the same is true for the modules in Γ X N_{G} P . Therefore there are strong connections between V F_{G} -endotrivial modules and endo-permutation modules as well as endo-p-permutation modules.

5. Endo-p-permutation modules and the Dade group of a finite group

An endo-p-permutation kG-module is a module M mod kG whose endomorphism algebra End_{k} M is a p-permutation^{1} kG-module. I.e. if End_{k} M _{i I} N_{i} where each N_{i} is indecomposable, then for every i I, N_{i} k _{Q_{i}}^{G} for some p-subgroup Q_{i} of G. Equivalently, M is endo-p-permutation if and only if M _{Q}^{G} is an endo-permutation kQ-module for every p-subgroup Q of G. In addition, since p-permutation modules are preserved under conjugation and restriction, it is enough to check that M _{P}^{G} is an endo-permutation kP -module for P a fixed Sylow p-subgroup of G. Other elementary properties of this class of modules are the following:

Lemma 5.1. Let M mod kG be an indecomposable endo-p-permutation module with vertex P. Then:

(a) M _{P}^{G} is capped endo-permutation.

(b) p ∤ dim_{k} M.

(c) k End_{k} M with multiplicity 1.

1In English, a p-permutation module is also often termed a trivial source module.

Proof. (a) It is easy to see that M _{P}^{G} is forced to have a summand with vertex P, thus it is

capped endo-permutation. See [Urf06, Chapter 2] for details.

(b) Assume M were an indecomposable kG-module with k-dimension divisible by p, that is absolutely p-divisible. Then, as a consequence of Theorem 2.4 (see [Las11a, Lem. 2.2.4] for a fully developed argument), so would be M _{P}^{G}, which contradicts statement (a). Indeed, M _{P}^{G} being capped, it has got at least one direct summand with k-dimension not divisible by p, for according to the previous section, Cap M _{P}^{G} is an indecomposable endopermutation module, hence V F_{P} -endotrivial and thus dim_{k} Cap M _{P}^{G} 1 mod p .

(c) This is a consequence of (b) and Theorem 2.4.

□

It can be seen in [Urf07] that setting an equivalence relation on the whole class of endo-p-permutation modules with vertex P given by a generalisation of Dade’s compatibility relation (cf [Dad78a]) does not lead to a group structure induced by tensor product on the set of isomorphism classes of indecomposable endo-p-permutation modules with vertex P. The idea is then to find a subclass of this class which has more similarities with that of capped endo-permutation modules for a p-group, and secondly to obtain a group structure induced by tensor product which embeds naturally in T_{V F_{G}} G , generalising the embedding D P T_{V F_{P}} P of Theorem 2.18. In this respect we focus on endo-p-permutation modules which are, at the same time, V F_{G} -endotrivial.

Proposition 5.2. Let M mod kG be an endo-p-permutation module. The following conditions are equivalent:

(a) M is V F_{G} -endotrivial;

(b) M _{P}^{G} is V F_{P} -endotrivial;

(c) M has a unique indecomposable summand with vertex P, say M_{0} and, in addition, if S mod kP is a source for M_{0}, then the multiplicity of S as a direct summand of M _{P}^{G} is one;

(d) End_{k} M k N where N is a p-permutation kG-module, all of whose indecomposable summands have a vertex strictly contained in P.

Proof. (a) (b): By Lemma 3.1, P roj V F_{G} _{P}^{G} P roj V F_{P} , therefore statements (a) and (b) are equivalent by part (e) of Lemma 2.8.

(a) (c): Assuming (a), M admits a decomposition M M_{0} V F_{G} proj where M_{0} is the unique indecomposable V F_{G} -endotrivial summand of M. Then dim_{k} M_{0} 0 mod p and so M_{0} is forced to have vertex P, whereas all the other summands of M have vertices strictly smaller than P by definition of P roj V F_{G} . Furthermore, by part (c) of Lemma 2.8, if S mod kP is a source for M_{0}, then S has multiplicity one in M_{0} _{P}^{G}. In consequence, since M _{P}^{G} is V F_{P} - endotrivial we have

M _{P}^{G} M_{0} _{P}^{G} V F_{P} proj S V F_{P} proj

where the Krull-Schmidt Theorem forces S to be isomorphic to the unique V F_{P} -endotrivial summand of M _{P}^{G}. Thus S has multiplicity one in M _{P}^{G} as well.

(c) (b): Write M M_{0} L with M_{0} indecomposable with vertex P and L a module all of whose indecomposable summands have a vertex strictly smaller than P. Thus L P roj V F_{G} and restricting M to P yields

M _{P}^{G} M_{0} _{P}^{G} V F_{P} proj .

Now M_{0} is endo-p-permutation as a direct summand of an endo-p-permutation module, therefore M_{0} _{P}^{G} is capped endo-permutation by Lemma 5.1. Moreover S M_{0} _{P}^{G} and because S has vertex P too, we must have S Cap M_{0} _{P}^{G} , so that the fact that the multiplicity of S is one forces all the remaining direct summands of M_{0} _{P}^{G} to have a vertex strictly smaller than P, that is to be V F_{P} -projective. Hence M _{P}^{G} is V F_{P} -endotrivial.

(a) (d): Given that M is endo-p-permutation, then End_{k} M is a p-permutation module. Thus M satisfies condition (d) if and only if it is V F_{G} -endotrivial, by definition of the family F_{G}. □

Definition 5.3. An endo-p-permutation kG-module M is said to be strongly capped if it satisfies the equivalent conditions of Proposition 5.2. Moreover, the unique summand of M with vertex P given by condition (c) is called the cap of M and denoted by Cap M .

The cap of a strongly capped endo-p-permutation module is its unique indecomposable direct summand which is itself strongly capped. Moreover, a strongly capped endo-p-permutation kGmodule has a direct sum decomposition of the form M Cap M V F_{G} proj where the V F_{G} -projective part is also an endo-p-permutation module, but not strongly capped.

Lemma 5.4. The class of strongly capped endo-p-permutation kG-modules is closed under taking duals, tensor products and restrictions to a subgroup containing a Sylow p-subgroup.

Proof. Taking duals and tensor products are stable operations for both the classes of endo-p-permutation modules and of V F_{G} -endotrivial modules, therefore they are stable for strongly capped endo-p-permutation modules. Now if H G contains a Sylow p-subgroup of G, then the restriction to H of an endo-p-permutation module is an endo-p-permutation module and the restriction to H of a V F_{G} -endotrivial module is a V F_{H} -endotrivial module by Lemma 4.1. Thus the restriction to H of a strongly capped endo-p-permutation module is strongly capped. □

Using a similar approach to that used by Dade for endo-permutation modules, one can define an equivalence relation on the class of all strongly capped endo-p-permutation modules by setting:

M N Cap M Cap N

Write M for the equivalence class of the module M and let D G denote the resulting set of equivalence classes.

Observe that this equivalence relation is the restriction to the class of strongly capped endo-p-permutation of the equivalence relation _{V F_{G}} on V F_{G} -endotrivial modules defined in Section 2.2. Thus the classes do not have the same meaning in T_{V F_{G}} G and in D G , and in general there are more representatives for a given class in T_{V F_{G}} G than in D G .

Corollary-Definition 5.5. The set D G with the composition law

M , N M N : M N ,

is an abelian group called the generalised Dade group of G, or simply the Dade group of G. Moreover, D G can be identified with a subgroup of T_{V F_{G}} G through the natural embedding

\begin{aligned}ı : & D G & & T_{V F_{G}} G \\ & M & & M . \end{aligned}

Proof. Lemma 5.4 and the uniqueness of the caps ensure that the assignment

M , N M N

is a well-defined composition law for D G . The zero element is the class k of the trivial module, while the opposite of a class M is the class M^{}of the dual module. The map ı is well-defined by the above observation on and _{V F_{G}} and it is a homomorphism because the addition is induced by _{k} on both sides. It is injective because ker ı k . Indeed, if ı M k , then M _{V F_{G}} k which is equivalent to M k because both M and k are strongly capped endo-p-permutation modules. □

We identify D G with its image ı D G and view D G as a subgroup of T_{V F_{G}} G .

Remark 5.6. Notice that any ordinary endotrivial module is strongly capped, and in particular, so is any one-dimensional kG-module. Therefore, up to identifications, the groups T G and X G can also be viewed as subgroups of D G and we have a series of subgroup inclusions:

X G T G D G T_{V F_{G}} G

The group D^{Ω} G Ω_{H} k H F_{G} is also a subgroup of D G because of the next Lemma.

Lemma 5.7. Let H F_{G}. If M is a strongly capped endo-p-permutation module, then Ω_{V H} M is a strongly capped endo-p-permutation kG-module.

Proof. Since M is assumed to be strongly capped, it is both endo-p-permutation and V F_{G} - endotrivial. In consequence, on the one hand Ω_{V H} M is V F_{G} -endotrivial by part (c) of Lemma 2.15, hence V H -endotrivial and on the second hand, it is shown in [Urf06, Proposition 5.8] that it is endo-p-permutation, hence strongly capped, as required. □

Finally, note that D G can also be identified with set of isomorphism classes of indecomposable strongly capped endo-p-permutation kG-modules endowed with the group law M N : Cap M N (where the square brackets denote the isomorphism class of a module).

6. Group operations

The operations of restriction and inflation induce group homomorphisms between the generalised Dade groups, whereas, in contrast with ordinary Dade groups, tensor induction does not.

Lemma 6.1. Let P be a Sylow p-subgroup of G and let H be a subgroup of G such that P H G. Then restriction induces a group homomorphism

\begin{aligned}Res_{H}^{G}: & D G & & D H \\ & M & & M _{H}^{G} . \end{aligned}

Furthermore, if H contains the normaliser N_{G} P of the Sylow p-subgroup P, then the map Res_{H}^{G} is injective.

Proof. As seen in 3.1, there is a restriction homomorphism for groups of relatively endotrivial modules

\begin{aligned}Res_{H}^{G}: & T_{V F_{G}} G & & T_{V F_{H}} H \\ & M & & M _{H}^{G} , \end{aligned}

which is an isomorphism if H contains N_{G} P . In consequence, it suffices to check that it maps D G to a subgroup of D H . In fact, if M D G , then, M _{H}^{G} is strongly capped by Lemma 5.4 and so M _{H}^{G} D H . Consequently, set Res_{H}^{G} : D G D H to be the restriction (of maps) to D G of the map Res_{H}^{G} : T_{V F_{G}} G T_{V F_{H}} H . It is injective if H N_{G} P . □

The injectivity of the map Res_{N_{G} P}^{G} : D G D N_{G} P allows us to identify the Dade group D G of a group G with a subgroup of the Dade group D N_{G} P .

Lemma 6.2. Let N be a normal subgroup of the group G such that G N has order divisible by p. Then inflation induces a group homomorphism

\begin{aligned}Inf_{G N}^{G}: & D G N & & D G \\ & M & & Inf_{G N}^{G} M . \end{aligned}

Proof. Consider the composite map

T_{V F_{G N}} G N ^{Inf_{G N}^{G}} T_{Inf_{G N}^{G} V F_{G N}} G ^{ı} T_{V F_{G}} G

where the first map is given by section 2.2 and the second map is given by Lemma 2.10 because

P roj Inf_{G N}^{G} V F_{G N} P roj V F_{G}

by Lemma 3.2. This composite maps D G N (viewed as a subgroup of T_{V F_{G N}} G N ) to D G . Indeed, if M D G N , then it only remains to check that Inf_{G N}^{G} M is endo-p-permutation. But if we let φ : P P N ^{}P N N denote the canonical group morphism, then

Res_{P}^{G} Inf_{G N}^{G} M Inf_{P P N}^{P} Iso φ^{1} Res_{P N N}^{G N} M

is endo-permutation because both isomorphism and inflation preserve endo-permutation modules. Hence Inf_{G N}^{G} M is endo-p-permutation, as required. It follows that there is an inflation map Inf_{G N}^{G} : D G N D G defined by restricting the map ı Inf_{G N}^{G} to D G N . □

Now, although the tensor induction of an endo-p-permutation module is an endo-p-permutation module (see [Urf07, Prop. 1.2]), the tensor induction of a strongly capped endo-p-permutation module is not necessarily strongly capped again.

Counterexample 6.3. Consider the 3-nilpotent group G : C_{7} C_{3} with k in characteristic 3. (If C_{7} : a and C_{3} : u , then the action of C_{3} on C_{7} is given by uau^{1} a^{2}.) Then consider the module Ω k mod kC_{3} , which is endotrivial. However the tensor induced module

Ω k \begin{matrix} \\ \end{matrix}_{C_{3}}^{G}

is neither an endotrivial module nor a strongly capped endo-3-permutation module. In fact, there exists no absolutely 3-divisible kG-module V such that the tensor induced module Ω k \begin{matrix} \\ \end{matrix}_{C_{3}}^{G} is V -endotrivial. See [Las11b, Lem. 7.6.5] for detailed computations.

7. The structure of D G

Key tools to describe the structure of the group D G are provided firstly by the following theorem proven by Dade but never published, and secondly by a characterisation of endo-p-permutation modules by Urfer.

Theorem 7.1 ([Dad82], Theorem 7.1). Let G be a finite group having a normal Sylow p-subgroup P. Let M be an endo-permutation kP -module. Then M extends to a kG-module if and only if M is G-stable.

Theorem 7.2 ([Urf07], Thm 1.5). Let G be a finite group. Let M mod kG be an indecomposable module with vertex P and source S mod kP . Then M is an endo-p-permutation module if and only if S is an endo-permutation module whose class S in the Dade group D P belongs to D P ^{G st} .

Recall that D P ^{G st} , the subgroup of G-stable points of D P , consists of the classes M D P such that Res_{^{x}P P}^{P} M Res _{^{x}P P}^{^{x}P} c_{x} M , where c_{x} denotes the conjugation by x G. In particular, D P ^{N_{G} P st} D P ^{N_{G} P}, the subgroup of fixed points of D P under the action of the normaliser N_{G} P by conjugation.

Notation. If G is a finite group with a Sylow p-subgroup P, we write X : X N_{G} P for the group of one-dimensional representations of N_{G} P , identified with a subgroup of D N_{G} P by Remark 5.6, and we write Γ X : Γ X N_{G} P for the subgroup of T_{V F_{G}} G made up of the classes of the kG-Green correspondents of the modules in X N_{G} P defined in Lemma 4.1.

Theorem 7.3. Let G be a finite group with a non-trivial Sylow p-subgroup P.Then, (a) restriction from N_{G} P to P yields an exact sequence

0 X  D N_{G} P ↠^{Res_{P}^{N_{G} P}} D P ^{N_{G} P} 0 ;

(b) restriction from G to P yields an exact sequence

0 Γ X  D G ↠^{Res_{P}^{G}} D P ^{G st} 0 .

In the following proof, we denote by R_{H}^{G} the map Res_{H}^{G} : T_{V F_{G}} G T_{V F_{H}} H and keep the notation Res_{H}^{G} : D G D H for the restriction maps at the level of the Dade groups.

Proof. First it follows from Theorem 7.2 that Im Res_{P}^{G} D P ^{G st} . For if M is an indecomposable strongly capped endo-p-permutation kG-module with source S mod kP , then Res_{P}^{G} M S . We claim that Im Res_{P}^{G} D P ^{G st} . Let S D P ^{G st} with S indecomposable. Notice that D P ^{G st} D P ^{N_{G} P}, so that by Dade’s Theorem S mod kP extends to a kN_{G} P -module S. In other words, S _{P}^{N_{G} P} S and S is a source for S. By construction S is strongly capped endo-p-permutation because its source is endo-permutation and has multiplicity 1 in its restriction. Hence S D N_{G} P and Res_{P}^{N_{G} P} S S . This proves the surjectivity of the map Res_{P}^{N_{G} P} onto D P ^{N_{G} P}. Now if Γ S is the kG-Green correspondent of S, then it has source S as well. Therefore Γ S is endo-p-permutation by Theorem 7.2. It is moreover V F_{G} -endotrivial by Lemma 4.1 because the restriction map R_{N_{G} P}^{G} is an isomorphism whose inverse is induced by Green correspondence on

indecomposable kN_{G} P -modules. Thus Γ S  D G and Res_{P}^{G} Γ S  S D P ^{G st} , as

required. Next we claim that the kernel of the restriction map Res_{P}^{G} : D G D P is Γ X . It was established in Lemma 4.1 that ker R_{P}^{N_{G} P} X. Therefore

ker Res_{P}^{N_{G} P} ker R_{P}^{N_{G} P} D N_{G} P  X D N_{G} P  X

because X D N_{G} P as noticed in Remark 5.6. Furthermore,

ker Res_{P}^{G} Res_{N_{G} P}^{G} ^{1} ker Res_{P}^{N_{G} P}  Res_{N_{G} P}^{G} ^{1} X
R_{N_{G} P}^{G} ^{1} X D G Γ X D G

and it remains to show that Γ X D G , i.e. that the indecomposable representatives of the classes in Γ X are endo-p-permutation modules. Indeed, if χ X, then its kG-Green correspondent Γ χ has the same source as χ, that is the trivial module k mod kP . Therefore Γ χ k _{P}^{G}, or in other words, it is a p-permutation module and thus an endo-p-permutation module. Hence ker Res_{P}^{G} Γ X . □

Corollary 7.4. The generalised Dade group D G of a finite group G is finitely generated.

Proof. The group Γ X X is finite. The group D P ^{G st} is finitely generated as a subgroup of D P , which is finitely generated by [Pui90]. Thus the exact sequence

0 Γ X  D G ↠^{Res_{P}^{G}} D P ^{G st} 0 .

of Theorem 7.3 implies that D G is finitely generated too. □

8. The generalised Dade group and control of p-fusion

The Dade group D G may always be identified, via restriction, with a subgroup of the Dade group D N_{G} P of the normaliser of a Sylow p-subgroup P of G. Then one may naturally ask when these groups are equal. The control of p-fusion in G by a subgroup H gives a partial answer to this question.

Proposition 8.1. Let H be a subgroup of G such that N_{G} P H G. Then D G D H if and only if D P ^{G st} D P ^{H st} .

Proof. Since H G, D P ^{G st} D P ^{H st} . Thus, there is a commutative diagram with exact rows given by Theorem 7.3

\xymatrix{0 \ar[r] & Γ_{G} X \ar[d]^{} \ar[r] & D G \ar[r]^{^{Res{G \atop P}}} \ar@{^{(}->}[d]^{_{Res{G \atop H}}} & D P ^{G st} \ar[r] \ar@{^{(}->}[d]^{i} & 0 \\ 0 \ar[r] & Γ_{H} X \ar[r] & D H \ar[r]^{_{Res{H \atop P}}} & D P ^{H st} \ar[r] & 0}

where i denotes the inclusion of D P ^{G st} in D P ^{H st} as subgroup, and where Γ_{G} X ker Res_{P}^{G} and Γ_{H} X ker Res_{P}^{H} . By part (d) of Proposition 4.1, Γ_{G} X X Γ_{H} X . Then, by the five-lemma the map Res_{H}^{G} is surjective if and only if the map i is. Thus, up to identification, D G D H if and only if D P ^{G st} D P ^{H st} . □

Links between control of p-fusion and the G-stable elements of the Dade group of a p-group were already established in [Urf07]:

Proposition 8.2 ([Urf07], Prop. 1.9). Let P be a p-subgroup of G and assume that p-fusion in G is controlled by H G. Then D P ^{G st} D P ^{H st} .

Corollary 8.3. Assume that the p-fusion of G is controlled by a subgroup H G.

(a) If G H N_{G} P , then D G D H .

(b) If N_{G} P H P, then D G D N_{G} P .

Proof. (a) is a straightforward consequence of Propositions 8.1 and 8.2.

(b) If N_{G} P H P, and H controls p-fusion then so does N_{G} P and part (a) yields the

result.

□

Example 8.4. Let G be a finite group and P be a Sylow p-subgroup of G. In the following cases, it follows from the above corollary that D G D N_{G} P :

(a) in case P is abelian, because then the normaliser N_{G} P controls p-fusion in G by Burnside’s Theorem;

(b) in case G is a p-nilpotent group, because then P controls p-fusion;

(c) in case p is odd and G is a group with P metacyclic, because then N_{G} P controls p-fusion in G, since such p-groups are resistant to fusion.

Example 8.5. An example in which D G D N_{G} P is provided by G : GL_{3} 𝔽_{3} and its extraspecial Sylow 3-subgroup P of order 27 which consists of the upper unitriangular matrices. The subgroup of P generated by the matrix

x : { \atop }1 0 1 0 1 1 0 0 1 { \atop }

is cyclic of order 3 and it is proven in [Urf07, Section 4] that the class in D P of the relative syzygy module Ω_{k _{x}^{P}} k is N_{G} P -stable but not G-stable. Thus D P ^{G st} D P ^{N_{G} P} and it follows from Proposition 8.1 that D G D N_{G} P .

9. The p-nilpotent case

In this section, consider G is a p-nilpotent group. In other words, G is a semidirect product G N P, with N O_{p^{}}G and P a Sylow p-subgroup of G. Thus G NP, N P 1 , and we let φ : P P N P NP N G N be the canonical isomorphism. For the structure of the groups T_{V} G of V -endotrivial modules for an arbitrary absolutely p-divisible kG-module V, we refer to [Las11b, Chap. 6].

Theorem 9.1. Let G N P be a p-nilpotent group. The restriction map Res_{P}^{G} : D G D P is split surjective. In consequence there is a group isomorphism

D G X N_{G} P D P .

Proof. Since G is p-nilpotent, the Sylow p-subgroup P controls p-fusion in G, thus Proposition 8.2 yields D P ^{G st} D P ^{P st} D P . Now [Las11b, Thm. 6.2.2] states that the restriction map Res_{P}^{G} : T_{V F_{G}} G T_{V F_{P}} P is split surjective and moreover that a section is provided by the map

T_{V F_{P}} P ^{Iso φ} T_{V F_{G N}} G N ^{Inf_{G N}^{G}} T_{V F_{G}} G .

By Section 6, both these maps can be restricted to the Dade groups so that Inf_{G N}^{G} Iso φ : D P D G is a section for Res_{P}^{G} : D G D P . In consequence, in view of Theorem 7.3, D G decomposes as a direct sum

D G Γ X D P ^{G st} Γ X D P .

Finally, Γ X X X N_{G} P by Lemma 4.1. The result follows. □

10. The cyclic case

Consider G is a finite group with a non-trivial cyclic Sylow p-subgroup P C_{p^{n}}, n 1. In this case, the classification provided in [Las11a, Sect. 8] for the groups of relative endotrivial modules of G allows us to determine the generalised Dade group D G with ease.

Proposition 10.1. Let G be a finite group with a non-trivial cyclic Sylow p-subgroup P C_{p^{n}}, n 1, and for 0 r n 1 let Z_{r} denote the unique cyclic subgroup of order p^{r} of P. Then

D G T_{V F_{G}} G T_{k _{Z_{n 1}}^{G}} G Γ X N_{G} P , Ω_{k _{Zs}^{G}} 0 s n 1 .

Proof. Since P is abelian, N_{G} P : N controls p-fusion by Burnside’s Theorem. Therefore, by Corollary 8.3, D G D N . Next we claim that D N T_{V F_{N}} N . By definition V F_{N} _{s 0}^{n 1} k _{Z_{s}}^{N} so that

P roj V F_{N} _{s 0}^{n 1} P roj k _{Z_{s}}^{N} P roj k _{Z_{n 1}}^{N}

because P roj k _{Z_{s}}^{N} P roj k _{Z_{n 1}}^{N} for every s n 1 as pointed out in remark 2.3. Therefore

T_{V F_{N}} N T_{k _{Z_{n 1}}^{N}} N .

In addition, by [Las11a, Thm. 8.2.6], we have

T_{k _{Z_{n 1}}^{N}} N X N , Ω_{k _{Zs}^{N}} 0 s n 1 .

Now, X N D N by Remark 5.6 and the relative syzygy modules Ω_{k _{Zs}^{N}} k are endo-p-permutation modules by Lemma 5.7. Whence D N T_{V F_{N}} N . Finally, T_{V F_{G}} G T_{V F_{N}} N via restriction, by Lemma 4.1. Consequently, there is a commutative diagram:

[figure: 0; T_{V F_{G}} G; ^{Res{G \atop N}}; ; T_{V F_{N}} N; 0; 0; D G; _{Res{G \atop N}}; ; D N; 0]

where the left-hand side vertical arrow is the inclusion as a subgroup of D G in T_{V F_{G}} G . Thus, the equality D G T_{V F_{G}} G follows by commutativity of the diagram. □

11. The Klein case

Consider G is a finite group with a Sylow 2-subgroup P C_{2} C_{2} and assume that the characteristic of the field k is 2.

Theorem 11.1 ([Las11a], Thm. 6.0.4). Let G be a finite group with a normal Sylow 2-subgroup P C_{2} C_{2}. Let V be any absolutely 2-divisible kG-module. Then there is a group isomorphism φ : T_{V} G T G : M M_{0} where M M_{0} V proj with M_{0} the unique indecomposable and V -endotrivial summand of M.

Proposition 11.2. Let G be a finite group with a Sylow 2-subgroup P C_{2} C_{2}.

(a) For any absolutely 2-divisible kG-module V, the group T_{V} G identifies with a subgroup of T_{V F_{G}} G T N_{G} P .

(b) Moreover D G T_{V F_{G}} G .

Proof. Set N : N_{G} P .

(a) If V mod kG is absolutely 2-divisible, then, by Lemma 2.11, the restriction map Res_{N}^{G} : T_{V} G T_{V _{N}^{G}} N is injective and sends the class of an indecomposable V -endo-trivial kG-module to the class of its kN-Green correspondent. By Lemma 4.1, the map Res_{N}^{G} : T_{V F_{G}} G T_{V F_{N}} N is an isomorphism whose inverse is induced by Green correspondence on the indecomposable V F_{N} -endotrivial modules. Furthermore, by Theorem 11.1, T_{V F_{N}} N T N T_{V _{N}^{G}} N . Therefore, the situation is as described in the following diagram:

[figure: T_{V F_{G}} G; _{Res{G \atop N}}; ; T_{V} G; _{Res{G \atop N}}; T_{V F_{N}} N; ; T N; ; T_{V _{N}^{G}} N]

Thus, there is an injective group homomorphism T_{V} G  T_{V F_{G}} G : L  L ,

where L denotes an indecomposable V -endotrivial module.

(b) The series of embeddings T N D N T_{V F_{N}} N and Theorem 11.1, which identifies T N with T_{V F_{N}} N , allow us to conclude that D N T_{V F_{N}} N . Then, to prove that D G T_{V F_{G}} G , use the same argument as in the proof of Proposition 10.1 in the cyclic case, because P is also abelian and thus N_{G} P controls p-fusion in G.

□

12. The group D^{Ω} G

If P is a p-group, with p odd, then one of the main results of the classification of endo-permutation modules asserts that D P D^{Ω} P Ω_{k _{Q}^{P}} Q P (see [Bou06]). In this section we ask whether or not a similar result holds for the generalised Dade group.

Recall from 5.6 and 5.7 that the group D^{Ω} G Ω_{H} k H F_{G} is a subgroup of D G .

Lemma 12.1. The group D^{Ω} G is generated by the relative syzygies Ω_{k _{Q}^{G}}, where Q runs over the proper subgroups of P, that is D^{Ω} G Ω_{k _{Q}^{G}} Q F_{G} .

Proof. If H F_{G} is a family of subgroups, set n_{H} : max H H H . We claim that Ω_{H} Ω_{k _{Q}^{G}} Q F_{G} for every H F_{G} and the proof proceeds by induction on the natural number n_{H}. First, if n_{H} 1, then P roj H is projectivity relative to the trivial subgroup 1_{G} , which is projectivity in the usual sense. Hence

Ω_{H} Ω_{k _{1_{G}}^{G}} Ω_{k _{Q}^{G}} Q F_{G} .

Then, let H : H_{1}, …, H_{n} , n ℕ be a subfamily of F_{G} such that n_{H} 2 and assume as induction hypothesis that Ω_{F} Ω_{k _{Q}^{G}} Q F_{G} for every subfamily F F_{G} such that 1 n_{F} n_{H}. Furthermore, according to Remark 2.3, we may assume that H_{i} _{G} H_{j} i j, 1 i, j n. Then, according to Lemma 2.16 we can write

Ω_{H} _{i 1}^{n} Ω_{H_{i}} _{j 2}^{n} Ω_{^{G} H_{1},...,H_{j 1} H_{j}} in T_{V H} G .

The sum _{i 1}^{n} Ω_{H_{i}} Ω_{k _{Q}^{G}} Q F_{G} . Moreover, for every 2 j n, the family of subgroups G H_{1}, …, H_{j 1} H_{j} is made up of the subgroups of the form ^{g}H_{i} H_{j} with g G and 1 i j 1, which all satisfy ^{g}H_{i} H_{j} H_{j} by the above assumption. In consequence, the sum _{j 2}^{n} Ω_{^{G} H_{1},...,H_{j 1} H_{j}} belongs to Ω_{k _{Q}^{G}} Q F_{G} by induction hypothesis, and the result follows. □

Remark 12.2. If H is a subfamily of F_{G}, then it follows from the preceding proof that Ω_{H} Ω_{k _{Q}^{G}} Q H for some H H .

Question: In case G P is a p-group and p is odd, then D P D^{Ω} P (see [Bou06]). Does a similar result hold in general for D G when G is an arbitrary finite group?

Because the one-dimensional representations are always in D G this result obviously has to be adapted when G is not a p-group. Nonetheless, we show that in the following cases, D G is D^{Ω} G modulo the Green correspondents Γ X of one-dimensional representations of N_{G} P (with P a Sylow p-subgroup of G):

(a) when G has a cyclic Sylow p-subgroup;

(b) when p is odd and P is normal in G;

(c) when N_{G} P controls p-fusion in P;

(d) it is also true for G GL_{3} 𝔽_{p} with p odd. The question of determining if this result holds in general is left open.

(a) The cyclic case. In case the group G has a cyclic Sylow p-subgroup P, then it was proven in Proposition 10.1 that

D G T_{V F_{G}} G T_{k _{Z_{n 1}}^{G}} G Γ X N_{G} P , Ω_{k _{Zs}^{G}} 0 s n 1 .

Hence D G is indeed D^{Ω} G modulo Γ X .

(b) The normal odd case: In order to prove (b), we first recall that a set of generators for D P ^{G st} is provided in [Urf06]:

Proposition 12.3 ([Urf06], Cor. 3.7). Suppose that p is an odd prime and P is a Sylow p-subgroup of the group G. Then the abelian group D P ^{N_{G} P} is spanned by the elements

f_{Q} : _{g N_{G} P P N_{G} P,Q}Ω_{k _{^{g}Q}^{P}}

where N_{G} P, Q g N_{G} P ^{g}Q Q and Q runs over F_{G}.

In what follows, we consider that P G, so that N_{G} P, Q N_{G} Q for every subgroup Q P. We still need another technical result on projectivity relative to p-subgroups.

Lemma 12.4. Let G be a group with a normal Sylow p-subgroup P and R be a proper subgroup of P. Then

P roj k _{R}^{G} _{P}^{G} P roj _{x G P N_{G} R}k _{^{x}R}^{P} .

Proof. The Mackey formula yields P roj k _{Q}^{G} _{P}^{G} P roj _{x G P} k _{^{x}Q}^{P} . Now, in order to obtain the equality of the statement, recall from Proposition 2.2 that if V, W mod kG and P roj V P roj W then P roj V W P roj V . Therefore, in

P roj _{x G P}k _{^{x}Q}^{P} _{x G P}P roj k _{^{x}Q}^{P}

it is enough to keep only one copy of the summands generating the same relative projectivity. Thus, compute that for x, y G, P roj k _{^{x}Q}^{P} P roj k _{^{y}Q}^{P} if and only if there exists p P such that ^{px}Q ^{y}Q if and only if y^{1} x P N_{G} Q (since P G if and only if y x mod P N_{G} Q . Whence P roj k _{Q}^{G} _{P}^{G} P roj _{x G P N_{G} Q} k _{^{x}Q}^{P} . □

Proposition 12.5. Let p be an odd prime and P be a normal Sylow p-subgroup of G. Then the restriction map Res_{P}^{G} : D^{Ω} G ↠ D P ^{G} is surjective. More accurately, if Q P, then any generator f_{Q} of D P ^{G} described in Proposition 12.3 can be expressed as

f_{Q} _{g G P N_{G} Q}Ω_{k _{^{x}Q}^{P}} Res_{P}^{G} Ω_{k _{Q}^{G}} X

where X f_{R} D P ^{G} R P and R Q .

Proof. The proof proceeds by induction on the order of the subgroup Q. Case Q 1: by part (d) of Lemma 2.15, Res_{P}^{G} Ω_{k _{1}^{G}} Ω_{k _{1}^{G}} f_{1}. Hence f_{1} Res_{P}^{G} D^{Ω} G .

Induction step: Let Q P such that Q 1 and assume as induction hypothesis that for every subgroup S P such that S Q , the generator f_{S} _{x G P N_{G} S} Ω_{k _{^{x}S}^{G}} of D P ^{G} belongs to Res_{P}^{G} D^{Ω} G . Again by part (d) of Lemma 2.15, in D P we have

Res_{P}^{G} Ω_{k _{Q}^{G}} Ω_{k _{Q}^{G} _{P}^{G}} Ω_{V}

where V : _{x G P} k _{^{x}Q}^{P}, so that the second equality follows from the Mackey formula. Taking the vision of P -sets, Ω_{V} Ω_{Y} where Y is the P -set defined by Y : _{x G P} P ^{x}Q. Then [Bou00, Lem. 5.2.3] yields the formula

Ω_{Y} _{\begin{matrix}U,V s_{P} \\ U _{P} V \\ Y^{V} \end{matrix}}μ_{P} U, V Ω_{P U}

where s_{P} is a set of representatives of conjugacy classes, under the action of P, of subgroups in P and μ_{P} is the Möbius function of the poset s_{P} , _{P} . Translating this in terms of kP -modules yields:

Ω_{V} _{\begin{matrix}U s_{P} \\ U _{G}Q \end{matrix}} _{\begin{matrix}V s_{P} \\ U _{P} V _{G}Q \end{matrix}} μ_{P} U, V Ω_{k _{U}^{P}}  _{\begin{matrix}U s_{P} \\ U _{G}Q \end{matrix}}Ω_{k _{U}^{P}}  _{\begin{matrix}U s_{P} \\ U _{G}Q \end{matrix}} _{\begin{matrix}V s_{P} \\ U _{P} V _{G}Q \end{matrix}} μ_{P} U, V Ω_{k _{U}^{P}}

_{x G P N_{G} Q}Ω_{k _{^{x}Q}^{P}} _{\begin{matrix}U G s_{P} \\ U _{G}Q \end{matrix}} _{\begin{matrix}V s_{P} \\ U _{P} V _{G}Q \end{matrix}} μ_{P} U, V _{x G P N_{G} U}Ω_{k _{^{x}U}^{P}}

f_{Q} _{\begin{matrix}U G s_{P} \\ U _{G}Q \end{matrix}} _{\begin{matrix}V s_{P} \\ U _{P} V _{G}Q \end{matrix}} μ_{P} U, V f_{U}

where G s_{P} denotes a set of representatives of conjugacy classes of classes of subgroups in s_{P} under the left action of G. Then set

X : _{\begin{matrix}U G s_{P} \\ U _{G}Q \end{matrix}} _{\begin{matrix}V s_{P} \\ U _{P} V _{G}Q \end{matrix}} μ_{P} U, V f_{U} f_{R} D P ^{G} R P, R Q .

By induction hypothesis X Res_{P}^{G} D^{Ω} G . It follows that f_{Q} Res_{P}^{G} Ω_{k _{Q}^{G}} X Res_{P}^{G} D^{Ω} G , as required. □

Theorem 12.6. Let p be an odd prime and G a finite group having a normal Sylow p-subgroup. Then

D G X G D^{Ω} G .

Proof. By Theorem 7.3 the restriction map Res_{P}^{G} : D G D P ^{G} induces an isomorphism D G X G D P ^{G}. Moreover the previous proposition states that restriction of Res_{P}^{G} to D^{Ω} G is surjective onto D P ^{G}. Hence the result. □

Notice that the sum D G X G D^{Ω} G of Theorem 12.6 need not be direct. A counterexample is provided by taking G to be a group with a normal Sylow p-subgroup isomorphic to a cyclic p-group C_{p^{n}} with p, n 3. Indeed, Theorem 10.1 states that D G T_{k _{Z_{n 1}}^{G}} G , and it

can be seen from the description by generators and relations of this group given in [Las11a, Thm. 8.2.6] that there exists a class ν X G , ν k , such that 2Ω_{k _{Z_{n 1}}^{G}} ν and thus ν belongs

to both X G and D^{Ω} G .

(c) D^{Ω} and control of fusion.

Lemma 12.7. Let H be a subgroup of G containing the Sylow p-subgroup P of G and assume that H controls p-fusion in G. Then the restriction map Res_{H}^{G} : D^{Ω} G D^{Ω} H is surjective. Moreover, if H contains N_{G} P , then restriction induces an isomorphism D^{Ω} G D^{Ω} H .

Proof. The proof is similar to that of Proposition 12.5. We claim that for every subgroup Q P, Res_{H}^{G} Ω_{k _{Q}^{G}} Ω_{k _{Q}^{H}} X with X Ω_{k _{R}^{H}} D^{Ω} H R P, R Q . We proceed by induction on the order of the subgroup Q. Case Q 1 : Ω_{k _{Q}^{G}} Ω so that by Lemma 2.15, part (d), Res_{H}^{G} Ω Ω Ω_{k _{1}^{H}} Res_{H}^{G} D^{Ω} G .

Induction step: Let Q P be a subgroup such that Q 2 and assume that Res_{H}^{G} Ω_{k _{S}^{G}} has the required form for every subgroup S P such that S Q . Compute, by part (d) of Lemma 2.15, that

Res_{H}^{G} Ω_{k _{Q}^{G}} Ω_{k _{Q}^{G} _{H}^{G}} Ω_{V} ,

where by the Mackey Formula one can set V : _{x H G Q} k _{^{x}Q H}^{H} . Then decompose

V _{x H G Q} k _{^{x}Q H}^{H}  _{\begin{matrix}x H G Q \\ ^{x}Q H \end{matrix}} k _{^{x}Q}^{H}  _{\begin{matrix}x H G Q \\ ^{x}Q H \end{matrix}} k _{^{x}Q H}^{H} .

Write V_{1} : _{x H G Q ,^{x}Q H} k _{^{x}Q}^{H} and V_{2} : _{x H G Q ,^{x}Q H} k _{^{x}Q H}^{H} . Then, by part (a) of Lemma 2.16, Ω_{V} Ω_{V_{1}} Ω_{V_{2}} Ω_{V_{1} V_{2}} . Now, firstly, since H controls fusion, for every x H G Q such that ^{x}Q H, there exists h H, such that ^{x}Q ^{h}Q. In consequence P roj V_{1} P roj k _{Q}^{H} by Proposition 2.2 (c) and (e), and thus by part (a) of Lemma 2.15 we have Ω_{V_{1}} Ω_{k _{Q}^{H}}. Secondly, P roj V_{2} corresponds to projectivity relative to the family of subgroups H : ^{x}Q H x H G Q , ^{x}Q H , all of whose elements have order strictly smaller than Q . Therefore Remark 12.2 states that

Ω_{V_{2}} Ω_{H} Ω_{k _{S}^{G}} S P, S Q .

Thirdly, by (b) of Lemma 2.15, Ω_{V_{1} V_{2}} Ω_{H ^{H} Q} . Since H consists of subgroups all of order strictly smaller than Q , so does the family H ^{H} Q . Thus, the same argument as above yields

Ω_{V_{1} V_{2}} Ω_{H ^{H} Q} Ω_{k _{S}^{G}} S P, S Q .

Therefore set X : Ω_{V_{2}} Ω_{V_{1} V_{2}} so that

Ω_{k _{Q}^{H}} Res_{H}^{G} Ω_{k _{Q}^{G}} X
with X  Ω_{k _{R}^{H}} D^{Ω} H R P, R  Q , as required. Then, by induction hypothesis,

X Res_{H}^{G} D^{Ω} G and thus so does Ω_{k _{Q}^{H}}. In conclusion all the generators of D^{Ω} H are in

Res_{H}^{G} D^{Ω} G and the surjectivity of Res_{H}^{G} : D^{Ω} G D^{Ω} H follows. Finally, if N_{G} P H G, the map Res_{H}^{G} : D G D H is injective by Lemma 6.1. Hence the isomorphism follows. □

Corollary 12.8. Let p be an odd prime. If P is a Sylow p-subgroup of G and N_{G} P controls p-fusion in G, then the Dade group decomposes as

D G D^{Ω} G Γ X .

Proof. Theorem 7.3 provides us with the exact sequence

0 Γ X  D G ↠^{Res_{P}^{G}} D P ^{G st} 0 .

Thus it suffices to prove that the map Res_{P}^{G} : D^{Ω} G D P ^{G st} is surjective. Indeed, since N_{G} P controls p-fusion, D P ^{N_{G} P} D P ^{G st} by Proposition 8.2. Therefore, the map Res_{P}^{G} : D^{Ω} G D P ^{G st} is equal to the composition

D^{Ω} G ↠^{Res_{N_{G} P}^{G}} D^{Ω} N_{G} P ↠^{Res_{P}^{N_{G} P}} D P ^{N_{G} P} D P ^{G st}

where Res_{N_{G} P}^{G} is surjective by the previous lemma and Res_{P}^{N_{G} P} is surjective by Proposition 12.5. Hence the result. □

(d) The example of GL_{3} 𝔽_{p} . Let G GL_{3} 𝔽_{p} with p an odd prime. This group has an extraspecial Sylow p-subgroup P of order p^{3} consisting of the upper unitriangular matrices and generated by the three matrices

x : { \atop }1 0 1 0 1 1 0 0 1 { \atop }, y : { \atop }1 1 1 0 1 0 0 0 1 { \atop }and z : { \atop }1 0 1 0 1 0 0 0 1 { \atop }.

Since fusion in P is the same under the action of GL_{3} 𝔽_{p} or under the action of P SL_{3} 𝔽_{p} , D P ^{G st} D P ^{P SL_{3} 𝔽_{p} st} . This last group was computed in [LM09, Example 6.6], by the following general method:

D P ^{G st} D P ^{N_{G} P} _{E P E p-essential} D P ^{N_{G} E st} .

(This is actually a consequence of Alperin’s fusion Theorem.) In the current case GL_{3} 𝔽_{p} has exactly two p-essential subgroups, namely E_{1} : x, z and E_{2} : y, z . Moreover

N_{G} E_{1}  { \atop }\begin{matrix}GL_{2} 𝔽_{p} & \begin{matrix} \\  \end{matrix} \\ 0 0 & 𝔽_{p}^{} \end{matrix}{ \atop }: H_{p} and N_{G} E_{2}  { \atop }\begin{matrix}𝔽_{p}^{} &  \\ {0 \atop 0} & GL_{2} 𝔽_{p} \end{matrix} { \atop }: K_{p} .

which are the two maximal standard parabolic subgroups in GL_{3} 𝔽_{p} and both of which contain N_{G} P . Hence

D P ^{G st} D P ^{H_{p} st} D P ^{K_{p} st} .

Proposition 12.9. Let p be an odd prime. Let G : GL_{3} 𝔽_{p} and let H_{p} and K_{p} be as above. Then the three restriction maps Res_{P}^{G} : D^{Ω} G D P ^{G st} , Res_{P}^{H_{p}} : D^{Ω} H_{p} D P ^{H_{p} st} , and Res_{P}^{K_{p}} : D^{Ω} H_{p} D P ^{K_{p} st} are surjective.

Proof. Detailed computations for the proof of this proposition can be found in [Las11b, Sect. 7.11]. The method is to find sets of generators for the groups D P ^{H_{p} st} and D P ^{K_{p} st} and thus for D P ^{G st} D P ^{H_{p} st} D P ^{K_{p} st} , and then show that all these generators are in the image of the corresponding restriction map.

□

Note that, in particular, the computations in [Las11b, Sect. 7.11] prove that the set of generators for D P ^{P SL_{3} 𝔽_{p} st} computed in [LM09, Example 6.6] misses one generator to be complete.

Finally, as above, the surjectivity of the three restriction maps Res_{P}^{G}, Res_{P}^{H_{p}}, and Res_{P}^{H_{p}} of Proposition 12.9 imply that

D G D^{Ω} G Γ X N_{G} P ;
D H_{p} D^{Ω} H_{p} Γ X N_{H_{p}} P ;
D K_{p} D^{Ω} K_{p} Γ X N_{K_{p}} P .

References

[Alp77] J. L. Alperin. Invertible modules for groups. Notices Amer. Mazh. Soc., 27:A–64, 1977.

[Alp01] J. L. Alperin. A construction of endo-permutation modules. J. Group Theory, 4(1):3–10, 2001.

[BC86] D. J. Benson and J. F. Carlson. Nilpotent elements in the Green ring. J. Algebra, 104(2):329–350, 1986.

[BM04] S. Bouc and N. Mazza. The Dade group of (almost) extraspecial p-groups. J. Pure Appl. Algebra, 192(1-3):21–51, 2004.

[Bou00] S. Bouc. Tensor induction of relative syzygies. J. Reine Angew. Math., 523:113–171, 2000.

[Bou04] S. Bouc. A remark on the Dade group and the Burnside group. J. Algebra, 279(1):180– 190, 2004.

[Bou06] S. Bouc. The Dade group of a p-group. Invent. Math., 164(1):189–231, 2006.

[BT00] S. Bouc and J. Thévenaz. The group of endo-permutation modules. Invent. Math., 139(2):275–349, 2000.

[Car96] J. F. Carlson. Modules and group algebras. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 1996. Notes by Ruedi Suter.

[CP96] J. F. Carlson and C. Peng. Relative projectivity and ideals in cohomology rings. J.

Algebra, 183(3):929–948, 1996.

[CPW98] J. F. Carlson, C. Peng, and W. W. Wheeler. Transfer maps and virtual projectivity. J.

Algebra, 204(1):286–311, 1998.

[CR90] C. W. Curtis and I. Reiner. Methods of representation theory. Vol. I. Wiley Classics Library. John Wiley & Sons Inc., New York, 1990. With applications to finite groups and orders, Reprint of the 1981 original, A Wiley-Interscience Publication.

[CT00] J. F. Carlson and J. Thévenaz. Torsion endo-trivial modules. Algebr. Represent. Theory, 3(4):303–335, 2000. Special issue dedicated to Klaus Roggenkamp on the occasion of his 60th birthday.

[CT04] J. F. Carlson and J. Thévenaz. The classification of endo-trivial modules. Invent. Math., 158(2):389–411, 2004.

[CT05] J. F. Carlson and J. Thévenaz. The classification of torsion endo-trivial modules. Ann.

of Math. (2), 162(2):823–883, 2005.

[Dad78a] E. C. Dade. Endo-permutation modules over p-groups I. Ann. of Math, 107:459–494, 1978.

[Dad78b] E. C. Dade. Endo-permutation modules over p-groups II. Ann. of Math, 108:317–346, 1978.

[Dad82] E. C. Dade. Extending endo-permutation modules. Preprint 1982. (unpublished).

[HS71] P. J. Hilton and U. Stammbach. A course in homological algebra. Springer-Verlag, New York, 1971. Graduate Texts in Mathematics, Vol. 4.

[Las11a] C. Lassueur. Relative projectivity and relative endotrivial modules. J. Algebra, 337:285– 317, 2011.

[Las11b] C. Lassueur. Relative Projectivity and Relative Endotrivial Modules. PhD thesis, EPFL, Lausanne, 2011.

[LM09] M. Linckelmann and N. Mazza. The Dade group of a fusion system. J. Group Theory, 12(1):55–74, 2009.

[Oku91] T. Okuyama. A generalization of projective covers of modules over group algebras. 1991. Unpublished manuscript.

[Pui90] L. Puig. Affirmative answer to a question of Feit. J. Algebra, 131(2):513–526, 1990.

[Urf06] J.-M. Urfer. Modules d’endo-p-permutation. PhD thesis, EPFL, Lausanne, 2006.

[Urf07] J.-M. Urfer. Endo-p-permutation modules. J. Algebra, 316(1):206–223, 2007.

THE DADE GROUP OF A FINITE GROUP 23

EPFL/SB/MATHGEOM/CTG, Section de Mathématiques, Station 8, CH-1015 Lausanne, Switzerland

E-mail address: caroline.lassueur@a3.epfl.ch