RELATIVE PROJECTIVITY AND RELATIVE ENDOTRIVIAL MODULES

CAROLINE LASSUEUR

Abstract. In this paper we use projectivity relative to kG-modules to define groups of relatively endotrivial modules, which are obtained by replacing the notion of projectivity with that of relative projectivity in the definition of ordinary endotrivial modules. To achieve this goal we develop the theory of projectivity relative to modules with respect to standard group operations such as induction, restriction and inflation. As a particular example, we show how these groups can generalise the Dade group. Finally, for finite groups having a cyclic Sylow p-subgroup, we determine all the different subcategories of relatively projective modules and, using the structure of the group T G of endotrivial modules described in [MT07], the structure of all the different groups of relatively endotrivial modules.

1. Introduction

Let G be a finite group and k be an algebraically closed field of characteristic p 0. The main purpose of this paper is to build a relative notion of endotrivial modules using the notion of projectivity relative to modules introduced by T. Okuyama in [Oku91] and then developed by J. Carlson and his coauthors in [Car96], [CP96] and [CPW98]. If V is a given finitely generated kG-module, in [CPW98] it is shown that one can define an analogue of the stable module category by setting that all the V - projective modules are isomorphic to zero. This construction results in the so-called V -stable category, denoted stmod_{V} kG , which is triangulated. In particular, the notion of projectivity relative to a module encompasses the well-known notion of projectivity relative to a subgroup or to a family of subgroups.

Classically a finitely generated kG-module is endotrivial if its k-endomorphism ring is isomorphic to a trivial module in the stable module category stmod kG . Therefore, for a quick description, a finitely generated kG-module shall be termed endotrivial relatively to the module V, required to be absolutely p-divisible, if it is invertible in the relative stable module category stmod_{V} kG . This construction gives rise to a group structure T_{V} G on the collection of relatively V -endotrivial modules endowed with the ordinary tensor product _{k} over k. In particular, for V kG the group T_{V} G is the group T G of ordinary endotrivial modules.

One main reason of interest for relative endotrivial modules is that they provide a way to define a group structure on collections of representations of an arbitrary finite group G. This gives a generalization for the Dade Group D P of a finite p-group P. Endo-permutation modules are defined only for p-groups, but not for finite groups in general. One way to obtain a similar notion for arbitrary groups is to consider endo-p-permutation modules as described in [Urf06]. However, the main drawback of this approach resides in the fact that there is not a unique indecomposable representative, up to isomorphism, in the resulting group structure. Now, whereas in the theory of endo-permutation modules, ordinary endotrivial module are seen as special cases, we shall take the problem the other way around and show that any endo-permutation module can be seen as a special case of a relative endotrivial module. Indeed, a good choice of a module V leads to a natural embedding of D P in the group TV P , in which the equivalence classes do have a unique indecomposable representative, up to isomorphism.

Date: March 31, 2011.

1991 Mathematics Subject Classification. 20C20.

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

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

In section 2, we start by recalling general results concerned with projectivity relative to a module and establish a few facts describing the behaviour of this kind of relative projectivity with respect to the standard group operations such as induction, restriction and inflation. We also study relative versions of the ordinary syzygy modules as they are important building pieces for the theory of relative endotrivial modules.

Next, we shall consider groups having a normal Sylow 2-subgroup isomorphic to the Klein group, in which case we show that whatever the choice of the module V, the groups T_{V} G are all isomorphic to the group T G of ordinary endotrivial modules. Nonetheless it is worth keeping this case in mind because it is an easy-to-handle source of examples and counter-examples for the general theory.

Finally we treat groups having a cyclic Sylow p-subgroup P C_{p^{n}}, in which case we give a description by generators and relations of all the different groups of relative endotrivial modules. This description is obtained through an induction argument from the structure of the group of ordinary endotrivial modules which was recently determined in [MT07]. In order to achieve, first we need to determine all the different subcategories of relative projective modules. It turns out that in this case projectivity relative to modules is reduced to the projectivity relative to the p-subgroups of P.

We point out that throughout this paper we use notation and develop a general theory concerned with projectivity relative to kG-modules in all generality, whereas in the Klein case, cyclic case and the aforementioned generalisation of the Dade group, it would be enough to work with projectivity relative to subgroups. Nonetheless, the groups T_{V} G for a general module V are interesting for themselves as they can naturally be defined in the general theory of projectivity relative to kG-modules. Moreover, the multiplication being induced by the tensor product, the knowledge of the groups T_{V} G can give some information on the multiplicative structure of the Green ring of the group G, about which very little is known. Also note that the Klein case in which there is, up to isomorphism, only one group of relatively endotrivial modules is an oddity, however in this case there are infinitely many different subcategories of V -projective modules which do not correspond to projectivity relative to a subgroup. Furthermore, the Dade group involves projectivity relative to a family of subgroups which is not reduced to a single element, however, it can be considered as projectivity relative to a single module which is much less cumbersome to work with.

2. Relative projectivity with respect to modules

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 the modules shall be finitely generated, mod kG shall denote the category of finitely generated left kG-modules and stmod kG the corresponding stable category. Moreover, shall denote 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.

To begin with, the main purpose of this section is to recall and develop techniques and results concerned with projectivity relative to a module. This is a generalisation of the more classic projectivity relative to a subgroup widely used in the theory of vertices and sources. Its definition is 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]. Projectivity relative to a kG-module was first introduced in an unpublished manuscript by T. Okuyama [Oku91], then further developed and used by J. Carlson and several coauthors in [Car96], [CP96] and [CPW98].

Definition 2.0.1 ([Car96]). (a) A module M mod kG is termed projective relative to V or V -projective if there exists a kG-module N such that M is isomorphic to a direct summand of V _{k} N.

(b) A short exact sequence E : 0 A ^{α} B ^{β} C 0 in mod kG is termed V -split if the tensored sequence V E : 0 V A ^{V α} V B ^{V β} V C 0 splits.

The notion of a V -injective kG-module can be defined dually. However, the symmetry of the group algebra implies that the class of V -injective modules is equal to the class of V -projective modules. To settle notation, we shall denote by P roj V the subcategory of all V -projective modules in mod kG and by IP roj V the collection of V -projective modules that are indecomposable. A module U is said to be a generator for P roj V if and only if P roj U P roj V . Moreover, in computations we shall often denote by V proj a module in P roj V which does not need to be specified, and simply proj for a projective module. Also note that P roj V is a subcategory of mod kG and the properties summed up below show that it is functorially finite.

Although the notion of projectivity relative to a kG-module was first introduced by Okuyama, as he points out, the real point of origin of this notion is the following crucial observation due to Auslander and Carlson in [AC86]: if V is a kG-module, then the trace map T r : V^{}V k : f v f v is always V -split. Moreover, it splits when dim_{k} V is coprime to p.

The following omnibus proposition sums up elementary properties of relative projectivity, that we shall use extensively in the sequel of this text.

Proposition 2.0.2 (Omnibus properties). Let M, N, U, V be kG-modules.

(a) Any direct summand of a V -projective module is V -projective.

(b) If U P roj V , then P roj U P roj V .

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

(d) P roj V P roj V^{}P roj V^{}V P roj Ω^{n} V for all n ℤ.

(e) P roj U V P roj U P roj V .

(f) P roj U P roj V P roj U V P roj U P roj V .

(g) P roj _{j 1}^{n} V P roj V P roj _{j 1}^{m} V for all m, n ℕ 0 .

(h) M N is V -projective if and only if both M and N are V -projective.

(i) M P roj V if and only if End_{k} M M^{}M P roj V .

(j) M P roj V if and only if M V^{}V M.

(k) Let g G̃ G, then ^{g}P roj V P roj ^{g}V .

(l) P roj kG P roj V for all kG-modules V. Moreover, P roj kG P roj P for any non-zero projective kG-module P.

Apart from property k , all these properties appear either in [Car96, Sec. 8] or [CP96, Sec. 3.3], to which we refer for proofs. Moreover we give a proof for statement (f) which was mistyped (and not proven in [CP96, Lem. 3.3(iii)]) as P roj U P roj V P roj U V instead of P roj U P roj V P roj U V . We note that in general, P roj U P roj V P roj U V . For instance, take G : C_{9} the cyclic group of order 9, U : k _{C_{3}}^{C_{9}} and V : kG. Then, P roj V P roj U V , the set of projective modules, whereas it will be easy to compute from the results we obtain in Section 7 for cyclic p-groups that P roj U P roj V kG^{3n} n ℕ .

Proof. (g) P roj U V P roj U P roj V by the very definition of U V -projectivity. If M P roj U P roj V , by definition there are kG-modules N and L such that M U N and M V L. Since the trace map T r : M^{}M k is M-split, M M M^{}M. These three relations brought together yield

M M M^{}M U N M^{}V L U V N M^{}L .

Hence P roj U P roj V P roj U V . In addition, if M P roj U P roj V , that is M M_{U} M_{V} with M_{U} P roj V and M_{V} P roj V , then there are two modules

N_{U}, N_{V} mod kG such that M_{U} U N_{U} and M_{V} V N_{V}. It yields
M M_{U} M_{V} U N_{U} V N_{V} U V N_{U} N_{V} .
Hence the inclusion P roj U V P roj U P roj V .

(k) M P roj V if and only if M V N for some N mod kG if and only if ^{g}M ^{g} V N

gM ^{g}N if and only if ^{g}M P roj ^{g}V .
□

The notion of projectivity relative to a module encompasses the well-known notion of projectivity relative to a subgroup, as well as the notion of projectivity relative to a family of subgroups, with the advantage that it becomes somewhat less cumbersome when we look at it as projectivity relative to a single module. Using Frobenius reciprocity, it is easy to show that projectivity relative to the subgroup H of G is equivalent to projectivity relative to the kG-module k _{H}^{G} and also that a short exact sequence E : 0 A B C 0 in mod kG is H-split if and only if it is k _{H}^{G}-split. Similarly, one can show that projectivity relative to the family H of subgroups of G is equivalent to projectivity relative to the kG-module _{H H} k _{H}^{G} and the sequence E is H-split if and only if it is _{H H} k _{H}^{G}-split.

2.1. Relative projectivity and operations on groups. We now establish some notation and basic facts concerning projectivity relative to modules with respect to standard operations on groups.

Lemma 2.1.1. Let H be a subgroup of G, N a normal subgroup of G.

(a) Restriction: Let Z be a V -projective kG-module, then Z _{H}^{G} is a V _{H}^{G}-projective kH-module. We shall use the following short notation: P roj V _{H}^{G} P roj V _{H}^{G} .

(b) Induction: Let Z be a V -projective kH-module, then Z _{H}^{G} is a V _{H}^{G}-projective kG-module. We shall use the following short notation: P roj V _{H}^{G} P roj V _{H}^{G} .

(c) Inflation: let Z be a V -projective k G N -module, then Inf_{G N}^{G} Z is an Inf_{G N}^{G} V -projective kG-module. We shall use the following short notation: Inf_{G N}^{G} P roj V P roj Inf_{G N}^{G} V .

(d) Isomorphism: let φ : G G be a group homomorphism and Z be a V -projective kG-module, then IsoG ; G Z is an IsoG ; G V -projective kG-module.

Proof. (a) Z is V -projective if and only if Z V L for some kG-module L, thus

Z _{H}^{G} V L _{H}^{G} V _{H}^{G} L _{H}^{G},

i.e. Z is V _{H}^{G}-projective.

(c) and (d) can be proven in like manner.

(b) Z is V -projective if and only if Z V L for some kH-module L, hence

Z _{H}^{G} V L _{H}^{G} V _{H}^{G} L _{H}^{G}
since V _{H}^{G} L _{H}^{G}  _{HgH} ^{g}V _{H ^{g}H}^{H} L _{H ^{g}H}^{H}  _{H}^{G}. Thus Z _{H}^{G} P roj V _{H}^{G} .
□

As we shall use restriction extensively , note that the reverse inclusion for (a) does not hold in general. For instance, if G C_{3} C_{3}, let h be one of its generators, H : h and V : k _{H}^{C_{3} C_{3}} , then V _{H}^{G} k k k. It follows that P roj V _{H}^{G} P roj k^{3} P roj k mod kH , whereas using Green’s indecomposability theorem it is easy to compute that

P roj V _{H}^{G} M mod kH M a_{1}k a_{2}Ω k a_{3}kH, a_{1}, a_{2}, a_{3} 3ℤ .

Next we focus on the behaviour of relatively projective modules with respect to restrictions and inductions.

Lemma 2.1.2. Let G be a finite group and X H G be subgroups. Let U, V be kG-modules and W, Z be kX-modules. The following inclusions and equalities hold:

(a) P roj V _{H}^{G} _{X}^{H} P roj V _{X}^{G} ;
(b) P roj W _{X}^{H} _{H}^{G} P roj W _{X}^{G} ;
(c) P roj V _{H}^{G} _{H}^{G} P roj V _{H}^{G} _{H}^{G} P roj V .
If, moreover, V is H-projective, then P roj V _{H}^{G} _{H}^{G} P roj V ;
(d) If P roj V P roj U , then P roj V _{H}^{G} P roj U _{H}^{G} .
(e) If P roj W P roj Z , then P roj W _{X}^{G} P roj Z _{X}^{G} .

Proof. (a)/(b) In both cases the inclusion was stated in Lemma 2.1.1. The reverse inclusion is a straightforward consequence of the transitivity of restrictions and inductions. E.g. V _{X}^{G} V _{H}^{G} _{X}^{H} so that V _{X}^{G} P roj V _{H}^{G} _{X}^{H} and by the omnibus properties of relative projectivity P roj V _{X}^{G} P roj V _{H}^{G} _{X}^{H}. A similar argument can be carried through for induction.

(c) The inclusion P roj V _{H}^{G} _{H}^{G} P roj V _{H}^{G} _{H}^{G} is a special case of Lemma 2.1.1, part (a). In addition, Frobenius reciprocity yields V _{H}^{G} _{H}^{G} V k _{H}^{G}, thus by 2.0.2,

P roj V _{H}^{G} _{H}^{G} P roj V P roj k _{H}^{G} P roj V .

Moreover, if V is H-projective, then P roj V P roj k _{H}^{G} by 2.0.2 (b) again. Consequently, the argument of (a) implies that:

P roj V _{H}^{G} _{H}^{G} P roj V P roj k _{H}^{G} P roj V

(d)/(e) If P roj V P roj U , then, in particular, V P roj U so that V _{H}^{G} P roj U _{H}^{G} by 2.1.1 (a), hence P roj V _{H}^{G} P roj U _{H}^{G} by 2.0.2. Swap the roles of V and U for the reverse inclusion. Property (e) is obtained likewise. □

The following lemma partly restates (a) and (b) of the two preceding ones, respectively, but focuses on a particular module rather than on a whole subcategory of relatively projective modules.

Lemma 2.1.3. Let G be a finite group and H be a subgroup of G. Let M be an H-projective module. Then, the following conditions are equivalent:

(a) M is V -projective;

(b) M _{H}^{G} is V _{H}^{G}-projective;

(c) M _{H}^{G} _{H}^{G} is V -projective.

Proof. (a) (b): is given by 2.1.1 (a).

(b) (c): Again by 2.1.1, M _{H}^{G} P roj V _{H}^{G} implies that M _{H}^{G} _{H}^{G} P roj V _{H}^{G} _{H}^{G} P roj V .

(c) (a): Finally, by H-projectivity, M M _{H}^{G} _{H}^{G} P roj V , therefore M P roj V as well. □

As a consequence, one sees the following:

Corollary 2.1.4. Let G be a finite group and P a Sylow p-subgroup of G. Let V and W be two kG-modules. Then P roj V P roj W if and only if P roj V _{P}^{G} P roj W _{P}^{G} .

Proof. The necessary condition was established in 2.1.2. For the sufficient condition, assume that P roj V _{P}^{G} P roj W _{P}^{G} . Applying 2.1.3 twice yields the following equivalences: M P roj V if and only if M _{P}^{G} P roj V _{P}^{G} P roj W _{P}^{G} if and only if M P roj W . Hence P roj V P roj W . □

Finally we establish links between V -projectivity, vertices, sources and the Green correspondence. In order to set up notation for the following sections, we recall that an admissible triple G, H; D for the Green correspondence consists of a finite group G, a p-subgroup D and a subgroup H containing N_{G} D . Using notation of [CR90, Thm. 20.6], for each such triple, define X : ^{x}D D x G H , Y : ^{x}D H x G H and A : D^{}D D^{}_{G} X . Then the Green correspondence is a bijection, that we shall denote by Γ : M N : Gr, from the set of isomorphism classes of indecomposable kGmodules M with vertex in A to the set of isomorphism classes of indecomposable kH-modules N with the same vertex in A. Furthermore, an indecomposable kG-module M with vertex in A corresponds to an indecomposable kH-module N with the same vertex if and only if M _{H}^{G} N Y proj or equivalently N _{H}^{G} M X proj .

Lemma 2.1.5. (a) Let M be an indecomposable kG-module and D, S a vertex-source pair for M. Let V be a kG-module and W be a kD-module. If S P roj W , then M P roj W _{D}^{G} , and if M P roj V , then S P roj V _{D}^{G} .

(b) Let G, H; Q be an admissible triple for the Green correspondence and V be any kG-module. Let U be an indecomposable kG-module with vertex Q and Gr U be its kH-Green correspondent. Then U P roj V if and only if Gr U P roj V _{H}^{G} .

Proof. (a) By 2.1.1 it is clear that if S P roj W , then M S _{D}^{G} P roj W _{D}^{G} . If M P roj V ,

then S M _{D}^{G} P roj V _{D}^{G} .

(b) If U P roj V , then by 2.1.1 U _{H}^{G} P roj V _{H}^{G} , therefore so does Gr U as a direct summand of U _{H}^{G}. Conversely, if Gr U P roj V _{H}^{G} , then Gr U _{H}^{G} P roj V _{H}^{G} _{H}^{G} P roj V by 2.1.2. Hence U P roj V , as a direct summand of Gr U _{H}^{G}.

□

2.2. Dimensional considerations and absolute p-divisibility. Many arguments shall use the next result by D. Benson and J. Carlson [BC86, Thm. 2.1]:

Theorem 2.2.1. 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.

In general, if M and N are finite-dimensional decomposable modules, write M _{i I} M_{i} and N _{j J} M_{j} as direct sums of indecomposable modules, then,

k M N if and only if i I, j J such that M_{i} N_{j}^{}and p ∤ dim_{k} N_{j} .

In particular, if p divides the k-dimension of all direct summands of N then k is not a summand of N^{}N End_{k} N . However, it is worth keeping in mind that the implication p ∤ dim_{k} N k N^{}N) is always true, that is even if N is decomposable, since in this case the trace map splits. Futhermore, the theorem enables us to characterize those kG-modules V relatively to which the trivial module is projective, which shall be essential later on to define the groups of relative endotrivial modules.

Proposition 2.2.2. Let V mod kG . Then, the following are equivalent:

(a) The trivial kG-module k is relatively V -projective;

(b) p char k does not divide the k-dimension of at least one of the indecomposable direct summands of V;

(c) the subcategory P roj V is equal to the whole category of finite-dimensional kG-modules mod kG .

Proof. (a) (b): By 2.0.2, k P roj V if and only if k V^{}V. Thus, V has an indecomposable direct summand whose k-dimension is not divisible by p.

(b) (c): Since V is finitely generated, write V _{j J} V_{j} as a direct sum of indecomposable modules. Then by 2.0.2,

P roj V _{j J} P roj V_{j} .

By assumption, there exists j_{0} J such that p does not divide dim_{k} V_{j_{0}} so that by proposition 2.0.2, P roj V_{j_{0}} mod kG . Therefore P roj V mod kG as well.

(c) (a): is trivial. □

In other words, the proposition shows that relative projectivity to a module V is interesting essentially if the k-dimensions of all the indecomposable direct summands of V are divisible by p char k, that is when P roj V is not equal to the whole category of finite-dimensional kG-modules mod kG . To use the terminology introduced in [BC86], in the sequel, such a kG-module V shall be called absolutely p-divisible.

As another consequence of Theorem 2.2.1 we can rephrase [Ben98, Prop. 5.8.1] to get the following characterisation for dimensions of V -projective kG-modules.

Lemma 2.2.3. Let V be an absolutely p-divisible kG-module and U P roj V . Then p divides dim_{k} U.

The behaviour of absolute p-divisibility with respect to restrictions shall turn out to be a key argument for the forthcoming study of relative endotrivial modules.

Lemma 2.2.4. Let P be a Sylow p-subgroup of G. Let V be a kG-module with vertex Q P.

(a) Then for every subgroup H P, the module V is absolutely p-divisible if and only if V _{H}^{G} is absolutely p-divisible.

(b) Furthermore, if Q P, then for every subgroup R of P such that P R Q, the module V is absolutely p-divisible if and only if V _{R}^{G} is absolutely p-divisible.

Proof. It is straightforward to see that if V is a kG-module whose restriction V _{H}^{G} to some subgroup H G is absolutely p-divisible, then V is absolutely p-divisible itself. Thus in both cases we are left with the necessary condition to prove.

(a) Let P H G be a subgroup and assume that V _{H}^{G} is not absolutely p-divisible. Then, by Lemma 2.2.2, P roj V _{H}^{G} mod kH . Besides Lemma 2.1.2 yields:

mod kH _{H}^{G} P roj V _{H}^{G} _{H}^{G} P roj V _{H}^{G} _{H}^{G} P roj V

We deduce, in particualr, that k _{H}^{G} P roj V . Finally since p ∤ dim_{k} k _{H}^{G} G : H , it follows from Lemma 2.2.3 that V is not absolutely p-divisible.

(b) Let R be a subgroup of P. By assumption, V P roj k _{Q}^{G} , so that V _{R}^{G} P roj k _{Q}^{G} _{R}^{G} and the Mackey formula yields:

k _{Q}^{G} _{R}^{G} _{g R G Q} ^{g}k _{^{g}Q R}^{^{g}Q} _{^{g}Q R}^{R} _{g R G Q} k _{^{g}Q R}^{R}

Therefore,

V _{R}^{G} _{g R G Q} P roj k _{^{g}Q R}^{R}

and so do all its direct summands. Now, the assumption that Q R implies that ^{g}Q R R for every g R G Q . Thus any direct summand of V _{R}^{G} has a vertex strictly smaller than R so that, by Lemma 3.4.1, p divides its k-dimension. Hence the result.

□

2.3. V -projective resolutions and relative syzygy modules. To end this section on properties of relative projectivity to modules, we recall some basic results linked to the corresponding relative homological algebra. The following definition is due to [Car96, Sec. 8].

Definition 2.3.1. A V -projective resolution of a module M mod kG is a nonnegative complex P_{}of V -projective modules together with a surjective kG-homomorphism P_{0} ^{ε} M such that the sequence

^{_{3}} P_{2} ^{_{2}} P_{1} ^{_{1}} P_{0} ^{ε} M 0

is totally V -split, that is such that for all i 1, the short exact sequences 0 ker _{i} P_{i} ^{_{i}} Im _{i} 0, as well as 0 ker ε P_{0} ^{ε} M 0 are V -split. The latter sequence is called a V -projective presentation of M.

Similarly, there is a notion of V -injective resolution M ^{ı} I_{}. Noticing that V^{}V M M is a V -projective presentation of M and iterating this constructions yields the existence of V -projective resolutions.

There is also a relative comparison theorem: if P_{}^{ε} M and Q_{}^{θ} M are two V -projective resolutions of the module M mod kG , then there is a chain map μ_{}: P_{}^{ε} M Q_{}^{θ} M which lifts the identity on M. In particular, the comparison theorem applied to two V -projective presentations is the relative version of Schanuel’s Lemma. Then using arguments similar to those proving the existence of minimal projective resolutions and minimal injective resolutions, it is possible to show the existence of minimal V -projective and V -injective resolutions. A corollary to this existence property provides a canonical generator for P roj V , that is, the V -projective cover V_{k} of the trivial module: P roj V P roj V_{k} . It also leads to the following definition of the modules called the relative syzygy modules or the relative Heller translates of a kG-module M.

Definition 2.3.2. Let M be a kG-module, P_{}^{ε} M be a minimal V -projective resolution of M and M ^{ı} I_{}a minimal V -injective resolution of M. Define for all n 1: Ω_{V}^{n} M : ker _{n}, Ω_{V}^{n} M : Coker ^{n 1} . Finally, define Ω_{V}^{0} to be the V -projective free part of M.

We sum the properties of the relative syzygy modules in the following omnibus proposition. They can all be found in [Car96, Sec. 8] or are more general versions of [Car96, Prop. 4.4], in which case the proofs are similar and straightforwardly obtained by replacing projectivity with relative projectivity.

Proposition 2.3.3 (Omnibus properties). Let M, N, V mod kG and m, n ℤ.

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

(b) Ω_{V}^{n} M Ω_{V}^{n} M^{}^{}.

(c) Ω_{V}^{n} M 0 if and only if M P roj V .

(d) Ω_{V}^{n} M is V -projective-free.

(e) If { \atop Ω_{V}} M denotes the kernel of a V -projective presentation of M, then { \atop Ω_{V}} M Ω_{V} M

V proj .

(f) Ω_{V}^{n} M N Ω_{V}^{n} M Ω_{V}^{n} N .

(g) Ω_{V}^{n} Ω_{V}^{m} M Ω_{V}^{n m} M .

(h) Ω_{V}^{m} M N Ω_{V}^{m} M N V proj .

(i) Ω_{V}^{m} M Ω_{V}^{n} N Ω_{V}^{m n} M N V proj .

We shall be particularly interested in the behaviour of the relative syzygy modules with respect to restriction and inflation.

Lemma 2.3.4. (a) Let H be a subgroup of G and M, V be kG-modules, then:

Ω_{V} M _{H}^{G} Ω_{V _{H}^{G}} M _{H}^{G} V _{H}^{G} proj

(b) Let N be a normal subgroup of G and M be a k G N -module, then :

Ω_{k _{N}^{G}} Inf_{G N}^{G} M Inf_{G N}^{G} Ω M

Proof. (a) The restriction of a minimal V -projective resolution is a V _{H}^{G}-projective resolution of M _{H}^{G}, it is not necessarily minimal though. Then, the formula follows from the comparison theorem.

(b) This formula is a version for projectivity relative to modules of a formula given in [Bou00, Cor.

4.1.2] for relative syzygies of P -sets, with P a p-group. The proof is identical.

□

2.4. Relative syzygies, vertices, sources and Green correspondence. To start with, the following characterisation of the vertices of relative syzygy modules was established in [Oku91, Cor. 9.9].

Lemma 2.4.1. Let V and W be kG-modules.

(a) Let M P roj W . Then Ω_{V}^{n} M P roj W for all n ℤ.

(b) Let M be an indecomposable non-V -projective kG-module. Then, for all n ℤ, M and Ω_{V}^{n} M have the same vertices.

In consequence, relative Heller operators commute with the Green correspondence and we can also compute their sources.

Corollary 2.4.2. Let V be a kG-module.

(a) Let G, H; Q be an admissible triple for the Green correspondence. Let U be a non V -projective indecomposable kG-module with vertex Q. If T is the kH-Green correspondent of U, then Ω_{V _{H}^{G}} T is the kH-Green correspondent of Ω_{V} U .

(b) Let M be an indecomposable non-V -projective kG-module and D, S a vertex-source pair for M. Then Ω_{V _{D}^{G}} S is a source for Ω_{V} M .

Proof. (a) First, the assumption that U is non V -projective ensures that neither Ω_{V} U , nor Ω_{V _{H}^{G}} T is zero. Indeed, by 2.1.5 U P roj V if and only if Gr U P roj V _{H}^{G} . Then, by assumption, both the modules U and T have vertex Q, thus, by the lemma, so do the modules Ω_{V} U and Ω_{V _{H}^{G}} T . Therefore, it suffices to prove that Ω_{V _{H}^{G}} T is a direct summand of Ω_{V} U _{H}^{G}. Indeed, as seen before Ω_{V _{H}^{G}} U _{H}^{G} Ω_{V} U _{H}^{G}. In addition, by the Green correspondence, T U _{H}^{G}, so that, by the properties of relative syzygies, Ω_{V _{H}^{G}} T Ω_{V _{H}^{G}} U _{H}^{G}.

(b) Let Ω_{V _{D}^{G}} S P_{V _{D}^{G}} S ↠ S be a minimal V _{D}^{G}-projective presentation of S. Then Ω_{V _{D}^{G}} S _{G}^{D} P_{V _{D}^{G}} S _{G}^{D}↠ S _{G}^{D} is a V -projective presentation of S _{G}^{D}, but it is not necessarily minimal though. Nonetheless, the relative version of Shanuel’s lemma yields:

Ω_{V _{D}^{G}} S _{G}^{D} Ω_{V} S _{G}^{D} V proj .

By assumption, S is a source of M, thus M is a direct summand of S _{G}^{D} and so Ω_{V} M is a direct summand of Ω_{V} S _{G}^{D} , which is, as seen above, in turn a direct summand of Ω_{V _{D}^{G}} S _{G}^{D}. Furthermore, according to the previous lemma, M and Ω_{V} M have a common vertex. It follows that Ω_{V _{D}^{G}} S is a source for Ω_{V} M .

□

3. The group of relative endotrivial modules

Recall that a module M mod kG is called endotrivial if its endomorphism ring is of the form End_{k} M k proj . In this section we generalize this family of modules to weaker versions by replacing ordinary projectivity with projectivity relative to modules. From now on, unless otherwise stated, V a fixed absolutely p-divisible kG-module so that the subcategory P roj V is not the whole category mod kG of kG-modules, which, as we have pointed out in the previous section, is equivalent to requiring that the trivial module k is not projective relative to V.

3.1. Relative endotrivial modules.

Definition 3.1.1. A kG-module M is termed endotrivial relative to the kG-module V or relatively V -endotrivial or simply V -endotrivial if its k-endomorphism ring is the direct sum of a trivial module and a V -projective module. That is, M is endotrivial relative to V if and only if

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

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

To begin with, here is a rudimentary but extremely useful dimensional characterisation.

Lemma 3.1.2. Let V be an absolutely p-divisible kG-module and M be a V -endotrivial module. Then: (a) dim_{k} M ^{2} 1 mod p.

(b) In case V k _{Q}^{G}, that is if we consider projectivity relative to the p-subgroup Q of G, then dim_{k} M ^{2} 1 mod P : Q where P is a Sylow p-subgroup of G containing Q.

Proof. (a) By 2.2.3 the k-dimension of any V -projective module is divisible by p, hence

dim_{k} M ^{2} dim_{k} End_{k} M dim_{k} k V proj 1 mod p .

(b) As a consequence of Green’s indecomposability theorem, the k-dimension of a module is divisible by the index of one of its vertices in the corresponding Sylow p-subgroup. (See [CR90].)

□

3.2. Constructions and stable operations for relative endotrivial modules. As the projective modules belong to any subcategory of relatively projective modules P roj V , it is clear that ordinary endotrivial modules are also endotrivial relatively to any kG-module V. In particular, so is any onedimensional kG-module. The other class of obvious examples of V -endotrivial modules is given by the kernels and cokernels of V -projective resolutions of the trivial module and in particular, the relative syzygies Ω_{V} k . More generally we have the following:

Lemma 3.2.1. (a) Let P P roj V and 0 L P N 0 be a V -split short exact

sequence. Then N is V -endotrivial if and only if L is.

(b) Let M be a V -endotrivial kG-module. Then the kG-modules Ω_{W}^{n} M are V -endotrivial modules for every kG-module W P roj V and for every n ℤ.

Proof.  (a) follows from (b). Indeed, L { \atop Ω_{V}} N Ω_{V} N  V proj and N { \atop Ω_{V}^{1}} L

Ω_{V}^{1} L V proj . However this proof can be done directly using the methods of the previous section and in particular the relative version of Shanuel’s lemma.

(b) Using the arithmetic of the relative syzygies that we developed in the previous section, we compute:

End_{k} Ω_{W}^{n} M Ω_{W}^{n} M ^{}Ω_{W}^{n} M Ω_{W}^{0} M^{}M V proj
Ω_{W}^{0} k V proj V proj
k V proj V proj k V proj

For, k P roj W P roj V mod kG and the W -projective-free part of a V -projective module is clearly V -projective.

□

Lemma 3.2.2. If M, N are V -endotrivial kG-modules, then so are the modules M^{}, M N and Hom_{k} M, N .

Proof. First, End_{k} M^{}End_{k} M . Then, using the properties of V -projectivity, compute

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

Finally, Hom_{k} M, N M^{}N which is V -endotrivial by the two preceding constructions. □

Next, we investigate the behaviour of relative endotrivial modules with respect to standard group operations.

Lemma 3.2.3. (a) If H is a subgroup of G and M a V -endotrivial kG-module, then M _{H}^{G} is a

V _{H}^{G}-endotrivial module.

(b) If N is a normal subgroup of G and M a V -endotrivial k G N -module, then Inf_{G N}^{G} M is an Inf_{G N}^{G} V -endotrivial module.

(c) Let φ : G_{1} G_{2} be a group isomorphism and M a kG_{1}-module. Then M can be seen as a kG_{2}-module, denoted by Iso_{G_{1}}^{G_{2}} M , the action of G_{2} on M being given via φ^{1} . Furthermore, if V is a kG_{1}-module and M is a V -endotrivial kG_{1}-module then Iso_{G_{1}}^{G_{2}} M becomes an Iso_{G_{1}}^{G_{2}} V - endotrivial kG_{2}-module.

Proof.

End_{k} M _{H}^{G} End_{k} M _{H}^{G} k V proj _{H}^{G} k _{H}^{G} V proj _{H}^{G} k V _{H}^{G} proj

where the last isomorphism is obtained by 2.1.1, part (a). This proves (a). The proofs for (b) and (c) are similar. □

Induction. Relative endotrivial modules are, in general, not stable under induction. This is easily seen by considering the group G : C_{3} C_{3} and its index 3 subgroup H : C_{3} 1. Then, the trivial kH-module k is endotrivial, but the induced module k _{H}^{G} can’t be endotrivial relatively to any kG-module since it is indecomposable and thus by the Benson-Carlson Theorem 2.2.1, the module k _{H}^{G} ^{}k _{H}^{G} does not have the trivial module as a direct summand. (This example extends to any indecomposable relative endotrivial kH-module M and any G H satisfying the hypothesis of Green’s indecomposability criterion, since then dim_{k} M _{H}^{G} G : H dim_{k} M is divisible by p.)

3.3. Direct sum decomposition structure of relative endotrivial modules. The first step towards the construction of an abelian group structure on the class of relative endotrivial modules is the following:

Lemma 3.3.1. Let M be a V -endotrivial kG-module and assume there is a direct sum decomposition M M_{0} M_{1}, then one of M_{0} or M_{1} is V -endotrivial and the other is V -projective. In consequence, M is V -endotrivial if and only if its V -projective-free part is indecomposable and V -endotrivial.

Proof. By assumption, we have

k V proj End_{k} M  End_{k} M_{0} Hom_{k} M_{0}, M_{1} Hom_{k} M_{1}, M_{0} End_{k} M_{1} .

As a result, the Krull-Schmidt theorem forces the trivial module k to be a direct summand of either End_{k} M_{0} , or End_{k} M_{1} . Indeed, if it were not the case, k would be a direct summand of Hom_{k} M_{0}, M_{1} or Hom_{k} M_{1}, M_{0} . But the two latter modules being dual to each other, k k would be a direct summand of End_{k} M , which is not possible because, by the assumption that V is absolutely p-divisible, k P roj V (Proposition 2.2.2). Thus we may assume that End_{k} M_{0} k V proj and End_{k} M_{1} P roj V . But, by 2.0.2 M_{1} P roj V if and only if M_{1} M_{1}^{}P roj V . In conclusion, M_{0} is V -endotrivial and M_{1} P roj V , as required. □

3.4. Vertices and sources.

Lemma 3.4.1. Let V be an absolutely p-divisible kG-module. Let M be any indecomposable V - endotrivial kG-module.

(a) The vertices of M are the Sylow p-subgroups of G.

(b) If P, S is a vertex-source pair for M, then S is a V _{P}^{G}-endotrivial module.

(c) Assume moreover that M _{P}^{G} k V _{P}^{G} proj , then the trivial kP -module is a source for M.

Proof. (a) It is well-known that the vertices of an indecomposable kG-module whose dimension is coprime to the characteristic p are the Sylow p-subgroups. Now, the dimensional characterisation of Lemma 3.1.2 yields dim_{k} M 0 mod p, hence the result.

(b) By assumption S M _{P}^{G}, so that S^{}M^{}_{P}^{G} and

S S^{}M _{P}^{G} S^{}M _{P}^{G} M^{}_{P}^{G} M M^{}_{P}^{G} k V _{P}^{G} proj .

Thus it remains to show that k S S^{}. Assume ab absurdo that it is not the case, then S S^{} has to be V _{P}^{G}-projective by the above and therefore, so is S by 2.0.2. In consequence,

M S _{P}^{G} P roj V _{P}^{G} _{P}^{G} P roj V

by Lemma 2.1.2, which contradicts the fact that for an absolutely p-divisible module V, an indecomposable V -endotrivial module is V -projective-free.

(c) Since P is a Sylow p-subgroup, M is P -projective so that

M M _{P}^{G} _{P}^{G} k V _{P}^{G} proj _{P}^{G} k _{P}^{G} V _{P}^{G} proj _{P}^{G} k _{P}^{G} V proj ,

by Lemma 2.1.2 (c). Moreover, M is V -projective-free by assumption, thus the Krull-Schmidt theorem yields that M k _{P}^{G}. In consequence, P being a vertex of M, k is a source of M.

□

3.5. Group structure. We can now copy the group structure on the ordinary endotrivial modules. Let V mod kG be an absolutely p-divisible module and 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 3.3.1. 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 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^{}.

The proof of this result is straightforward and analogous to that of the endotrivial case. We shall call T_{V} G the group of V -endotrivial modules. Also note that we use an additive notation, which is consistent with the choice made in [BT00] and related articles treating endo-permutation and endotrivial modules.

Corollary 3.5.2. If M is a self-dual, V -endotrivial kG-module, then M has order two in T_{V} G .

Proof. By assumption M M M^{}M k V proj so that 2 M  k .  □

To give a first example, this simple observation can be applied at once to the concrete case of a cyclic p-group C_{p^{n}}, n 1. Indeed, all the indecomposable kC_{p^{n}}-modules are self-dual. Therefore, whatever the choice of the absolutely p-divisible module V, we can conclude that the group T_{V} C_{p^{n}} is an elementary abelian 2-group. We shall give a complete description of all the different groups of relative endotrivial modules for cyclic p-groups in section 7.

The following lemma points out relations of inclusion between groups of relative endotrivial modules.

Lemma 3.5.3. Let U, V mod kG be absolutely p-divisible modules. Assume moreover that P roj V P roj U , then:

(a) every V -endotrivial kG-module is U-endotrivial;

(b) if M and N are V -endotrivial modules such that M _{V} N, then M _{U} N as well. In consequence, T_{V} G can be identified with a subgroup of T_{U} G via the injective group homomorphism ı : T_{V} G T_{U} G : M _{V} M _{U}.

Proof. (a) Let M be a V -endotrivial module, then End_{k} M k V proj k U proj , i.e. M is U-endotrivial.

(b) There exists two indecomposable V -endotrivial modules M_{0} and N_{0} such that M M_{0} V proj and N N_{0} V proj and M _{V} N implies that M_{0} N_{0}. By (i), M_{0} and N_{0} are U-endotrivial, so that M _{U} N. In consequence ı is a well-defined group homomorphism. The injectivity follows from the uniqueness of the summand M_{0}.

□

The study of relative endotrivial modules for the Klein Group C_{2} C_{2} will show that it is possible to have a strict inclusion P roj V P roj U but an isomorphism T_{V} G T_{U} G . Nevertheless, a strict inclusion P roj V P roj U implies that the class of V -endotrivial modules is strictly contained in the class of U-endotrivial modules. Indeed, let M P roj U P roj V . Then, on the one hand, L : k M is U-endotrivial, since

End_{k} M k M M^{}M M^{}k U proj ,

but on the other hand it is not V -endotrivial, otherwise M would be V -projective. Besides, this argument also shows that there are more modules belonging to the class k in T_{U} G than in T_{V} G .

3.6. Standard homomorphisms between groups of relative endotrivial modules. In order to make further links between different groups of relative endotrivial modules, we are now going to define group homomorphisms and actions which are induced by group operations.

Restriction. Let M be a kG-module, H a subgroup of G and V an absolutely p-divisible kG-module such that V _{H}^{G} is also absolutely p-divisible. Then the groups T_{V} G and T_{V _{H}^{G}} H are well-defined and the module M _{H}^{G} is a V _{H}^{G}-endotrivial kH-module. Therefore, restriction to a subgroup induces a well-defined group homomorphism:

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

Indeed, Res_{H}^{G} is a group homomorphism since restriction and _{k} commute. Furthermore, by Lemma 2.1.2, the map Res_{H}^{G} is independent of the choice of the generator V for P roj V .

Inflation. Let N be a normal subgroup of a group G, M be a k G N -module. Then M can be seen as a kG-module, denoted Inf_{G N}^{G} M , by letting N act by identity. Furthermore, if V is an absolutely p-divisible k G N -module, then the groups T_{V} G N and T_{Inf_{G N}^{G} V} G are well-defined. In addition,

if M is V -endotrivial, then Inf_{G N}^{G} M is Inf_{G N}^{G} V -endotrivial, therefore, 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

Isomorphism. Let φ : G_{1} G_{2} be a group isomorphism and M a kG_{1}-module. Then M can be seen as a kG_{2}-module, denoted by Iso_{G_{1}}^{G_{2}} M , the action of G_{2} on M being given via φ^{1} . Furthermore, if V is an absolutely p-divisible kG_{1}-module, then Iso_{G_{1}}^{G_{2}} V is an absolutely p-divisible kG_{2}-module and if M is a V -endotrivial kG_{1}-module then M becomes a Iso_{G_{1}}^{G_{2}} V -endotrivial kG_{2}-module. This operation induces a group isomorphism:

Iso_{G_{1}}^{G_{2}}: T_{V} G_{1}  T_{Iso_{G_{1}}^{G_{2}} V} G_{2}
M  Iso_{G_{1}}^{G_{2}} M

A concrete example of such an isomorphism between groups of relative endotrivial modules is provided below by conjugation.

Remark 3.6.1. It should be noted that the three cases of restriction, inflation, and isomorphism can be unified in the single case of restriction along a group homomorphism G_{1} G_{2}. Nonetheless, we do not do it in these terms because the case of restriction, in which we need to require that the module V _{H}^{G} is absolutely p-divisible, shows that an arbitrary group homomorphism, and in particular an inclusion of subgroups, would not necessarily induce a well-defined group homomorphism between the corresponding groups of relative endotrivial modules.

Conjugation. Let H G be a normal subgroup and V be an absolutely p-divisible G-invariant kH-module. Then, for all g G, ^{g}P roj V P roj V and ^{g}H H. Therefore, conjugation induces a well-defined action of G (or rather G H), on the group T_{V} H given by:

\begin{matrix}G T_{V} H & & T_{V} H \\ g, M & & ^{g}M \end{matrix}

In case the subgroup H and the module V are not assumed to be normal nor G-invariant, then the above assignment does not yield a group action, nevertheless, for any element g G, the conjugation isomorphism γ_{g} : H ^{g}H induces a group isomorphism

\begin{aligned}γ_{g}: & T_{V} H & & T_{^{g}V} ^{g}H \\ & M & & ^{g}M \end{aligned}

In particular, if H G, then T_{V} H T_{^{g}V} H . Also, if V is G-invariant then T_{V} H T_{V} ^{g}H .

4. Properties of restriction maps

The purpose of this section is to relate groups of relative endotrivial modules for a group G to those for a Sylow p-subgroup P of G or a subgroup H containing P. In particular, links between endotrivial modules for G and the normalizer N_{G} P of the Sylow subgroup can be obtained by Green correspondence. Most of the result presented in this section are generalisations of results concerning ordinary endotrivial modules which can be found in [MT07], [CMN06] and [Maz07].

4.1. Restrictions to Sylow p-subgroups. To begin with, we describe restrictions to a Sylow psubgroup. The following easy properties generalise [CMN06, Prop. 2.6].

Lemma 4.1.1. Let P be a Sylow p-subgroup of G and H a subgroup of G containing P.

(a) Let M be a V -endotrivial kG-module. Then M is a direct summand of a V _{H}^{G}-endotrivial module induced from H to G, namely the module M _{H}^{G} _{H}^{G}.

(b) Assume V _{H}^{G} is absolutely p-divisible and let M be a kG-module such that M _{H}^{G} is V _{H}^{G}- endotrivial, then M is V -endotrivial.

Proof. (a) Since H P, by H-projectivity M M _{H}^{G} _{H}^{G} where M _{H}^{G} is V _{H}^{G}-endotrivial by 3.2.3. (b) As M _{H}^{G} is V _{H}^{G}-endotrivial and V _{H}^{G} absolutely p-divisible,

dim_{k} M ^{2} dim_{k} M _{H}^{G} ^{2} 1 mod p .

In consequence both the trace map and its restriction to H split, so that M^{}M k ker T r and M _{H}^{G} ^{}M _{H}^{G} k ker T r _{H}^{G}, where ker T r _{H}^{G} has to be V _{H}^{G}-projective by the assumption that M _{H}^{G} is V _{H}^{G}-endotrivial. Besides, by H-projectivity, ker T r ker T r _{H}^{G} _{H}^{G} P roj V by 2.1.3. Therefore ker T r is a V -projective module as well and M^{}M k V proj as required.

□

We now treat the special case of a normal Sylow p-subgroup. The next proposition and its corollary partly generalise [CMN06, Prop. 2.6, (d)] and [Maz07, Cor. 2.7].

Proposition 4.1.2. Let P be a normal Sylow p-subgroup of G. Let V be an absolutely p-divisible kG-module. Then, an indecomposable kG-module M is V -endotrivial if and only if its restriction to P is an indecomposable V _{P}^{G}-endotrivial module.

Proof. Assume that M is an indecomposable V -endotrivial module. Let M _{P}^{G} N_{1} N_{s}, s ℕ, be a decompostion into indecomposable summands. Since P is a vertex of M (see 3.4.1), one may assume, without loss of generality, that N_{1} is a source for M, so that M N_{1} _{P}^{G} as well. Thus, given that P is normal in G, the Mackey formula yields

M _{P}^{G} N_{1} _{P}^{G} _{P}^{G} _{g G P} ^{g}N_{1} .

Now, on the one hand M _{P}^{G} is V _{P}^{G}-endotrivial, which is more accurately the direct sum of an indecomposable V _{P}^{G}-endotrivial module, whose k-dimension is coprime to p, and a V _{P}^{G}-projective module, all of whose indecomposable summands have k-dimension divisible by p. On the other hand the G-conjugates ^{g}N_{1} of N_{1} are all indecomposable with k-dimension equal to that of N_{1}. Therefore, this forces M _{P}^{G} to be indecomposable (V _{P}^{G}-endotrivial). Conversely, let M be such that M _{P}^{G} is an indecomposable V _{P}^{G}-endotrivial module. Firstly the fact that M _{P}^{G} is indecomposable forces M to be indecomposable as well, and secondly it follows from part (c) of Lemma 4.1.1 that M is V -endotrivial. □

As a consequence, when the Sylow p-subgroup P is normal in the group G, then the V -endotrivial modules are detected upon restriction to P. Since the restriction of a V -endotrivial module is Ginvariant, at the level of groups of relatively endotrivial modules, there is an inclusion

Im Res_{P}^{G} T_{V _{P}^{G}} P ^{N_{G} P P} .

A natural question is to ask when this inclusion is indeed an equality, that is when the restriction map is actually surjective onto the N_{G} P P -fixed points of T_{V _{P}^{G}} P . We shall see further in the last section that, for instance, it is always the case for groups with cyclic Sylow p-subgroups.

Corollary 4.1.3. Let P be a normal Sylow p-subgroup of G. Let V be an absolutely p-divisible kGmodule, M be an indecomposable V -endotrivial module and X_{}, _{}be a V -projective resolution of M. Then:

(a) X_{}, _{}is minimal if and only if X_{}_{P}^{G}, _{}_{P}^{G} is a minimal V _{P}^{G}-projective resolution of

M _{P}^{G};
(b) in particular, Ω_{V}^{n} M _{P}^{G} Ω_{V _{P}^{G}}^{n} M _{P}^{G} for all integer n.

Proof. Given that X_{}, _{}is a minimal V -projective resolution, for each integer n 0 there is a V -split short exact sequence

\xymatrix{0 \ar[r] & Ω_{V}^{n 1} M \ar[r] & X_{n} \ar[r]^{_{n}} & Ω_{V}^{n} M \ar[r] & 0} .

Restricting it from G to P yields a V _{P}^{G}-projective presentation of Ω_{V _{P}^{G}}^{n} M _{P}^{G}:

\xymatrix{0 \ar[r] & Ω_{V}^{n 1} M _{P}^{G} \ar[r] & X_{n} _{P}^{G} \ar[r]^{_{n} _{P}^{G}} & Ω_{V}^{n} M _{P}^{G} \ar[r] \ar[r] & 0}

although, it is not necessarily minimal. However, by 2.3.3,

Ω_{V}^{n 1} M _{P}^{G} Ω_{V}^{n 1} M _{P}^{G} V _{P}^{G} proj and Ω_{V}^{n} M _{P}^{G} Ω_{V}^{n} M _{P}^{G} V _{P}^{G} proj .

Besides, by the proposition, both these modules are indecomposable so that the V _{P}^{G}-projective factors are zero. Therefore the above short exact sequence is indeed

\xymatrix{0 \ar[r] & Ω_{V}^{n 1} M _{P}^{G} \ar[r] & X_{n} _{P}^{G} \ar[r]^{_{n} _{P}^{G}} & Ω_{V}^{n} M _{P}^{G} \ar[r] & 0} .

Hence the minimality of X_{}_{P}^{G}, _{}_{P}^{G} . The converse is trivial. □

4.2. Restriction to the normalizer of a Sylow p-subgroup. The goal is now to figure out the behaviour of restriction maps from a group G to the normalizer of a Sylow p-subgroup P or a subgroup H containing N_{G} P . It follows from Sections 2.2 and 3.6 that for every absolutely p-divisible kGmodule V there is a well-defined restriction map

Res_{H}^{G} : T_{V} G T_{V _{H}^{G}} H .

The following statement generalises [CMN06, Prop. 2.6.(a)].

Lemma 4.2.1. Let P be a Sylow p-subgroup of G, H a subgroup of G containing N_{G} P and V an absolutely p-divisible kG-module. Then the restriction map Res_{H}^{G} : T_{V} G T_{V _{H}^{G}} H is injective.

Proof. Let M be an indecomposable V -endotrivial kG-module. By Lemma 3.4.1 P is a vertex of M. Then, on the one hand, the Green correspondence for the triple G, H; P yields:

M _{H}^{G} Gr M X

where Gr M is an indecomposable kH-module with vertex P and X P roj Y with Y ^{x}P H x G H . But ^{x}P H ^{x}P for all x G H, otherwise ^{x}P would be a Sylow p-subgroup of H which is not possible, since then there would be h H such that ^{hx}P P, that is hx N_{G} P H and x H. Therefore all the direct summands of X have a vertex strictly smaller than P. On the other hand, M _{H}^{G} is a V _{H}^{G}-endotrivial module, that is:

M _{H}^{G} M_{0} V _{H}^{G} proj

with M_{0} an indecomposable V _{H}^{G}-endotrivial module, thus with vertex P by 3.4.1. In consequence, the Krull-Schmidt theorem implies that M_{0} Gr M , the kH-Green correspondent of M, whose uniqueness yields the injectivity of Res_{H}^{G}. □

4.3. Cases in which restriction maps are isomorphisms. Knowing that the restriction map Res_{H}^{G} : T_{V} G T_{V _{H}^{G}} H is injective for every subgroup H containing the normalizer N_{G} P of a Sylow p-subgroup P, the next question that arises is to understand when this map is an isomorphism. The last section on groups with cyclic Sylow p-subgroup shall provide us with examples in which the answer depends on the module V to which relative projectivity is considered. Notwithstanding, one can show that in case the subgroup H is strongly p-embedded in G, then Res_{H}^{G} is always an isomorphism, however the choice of the module V. This result generalises the similar result for ordinary endotrivial modules that can be found, for instance, in [MT07, Lem. 2.7]. Furthermore, the proof of this result provides us with the following more general sufficient condition on the module V for the restriction map to be an isomorphism.

Lemma 4.3.1. Let P be a Sylow p-subgroup of G and H G a subgroup containing the normalizer N_{G} P of P. Let V be an absolutely p-divisible kG-module. If P roj V _{H}^{G} P roj Y , where Y is the family of subgroups ^{g}P H g G H involved in the Green correspondence, 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 .

Proof. By 4.2.1, the map Res_{H}^{G} is one-to-one, therefore it only remains to show that it is onto as well. Let L be an indecomposable V _{H}^{G}-endotrivial module. The Mackey formula yields the isomorphism.

L _{H}^{G} _{H}^{G} L _{\begin{matrix}g H G H \\ g G H \end{matrix}} ^{g}L _{^{g}H H}^{^{g}H} _{^{g}H H}^{H} : L L^{},

where, by the proof of the Green correspondence, L^{}P roj Y , so that

L _{H}^{G} _{H}^{G} L Y proj L V _{H}^{G} proj

by assumption. In other words, L _{H}^{G} _{H}^{G} is V _{H}^{G}-endotrivial and consequently L _{H}^{G} is V -endotrivial by 4.1.1. Therefore, Res_{H}^{G} L _{H}^{G} L _{H}^{G} _{H}^{G} L . Hence the surjectivity of Res_{H}^{G}. Moreover, the proof of the injectivity of Res_{H}^{G} shows that the unique indecomposable V -endotrivial summand of L _{H}^{G} has to be isomorphic to the kG-Green correspondent of L. It follows from the proof of the injectivity (Lemma 4.2.1) that the inverse map is induced by Green correspondence on the indecomposable modules. To see that, alternatively, it is induced by induction, let M T_{V _{H}^{G}} H an write M M_{0} V _{H}^{G} proj with M_{0} an indecomposable V _{H}^{G}-endotrivial module. Then,

M _{H}^{G} M_{0} _{H}^{G} V _{H}^{G} _{H}^{G} Γ M_{0} X proj V proj

where X is the familly of subgroups involved in the Green correspondence, as described in section 2, and P roj V _{H}^{G} _{H}^{G} P roj V by Lemma 2.1.2. As just mentioned above, Γ M_{0} is V -endotrivial, therefore it remains to check that P roj X P roj V . But this is a consequence of the hypothesis that P roj V _{H}^{G} P roj Y . Indeed, at the level of kH-modules, P roj Y P roj X by definition of the families X and Y, thus P roj V _{H}^{G} P roj X . Inducing to G yields in mod kG the required inclusions

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

Corollary 4.3.2. If the subgroup H is strongly p-embedded in G, then Res_{H}^{G} : T_{V} G T_{V _{H}^{G}} H is an isomorphism.

Proof. If H is strongly p-embedded in G, then for any g G H the subgroup ^{g}H H has order coprime to p, thus Y 1 . Therefore Res_{H}^{G} is an isomorphism, regardless of the module V, since then P roj V _{H}^{G} P roj Y P roj for any kG-module V.

□

For instance, if the Sylow p-subgroup P is a trivial intersection subgroup (TI), then N_{G} P is strongly p-embedded in G. Moreover, any strongly p-embedded subgroup contains the normalizer of some Sylow p-subgroup of G.

Besides, the first explicit example that springs to mind for a module satisfying the hypotheses of the lemma is the absolutely p-divisible module

V : _{Q F} k _{Q}^{G}

where F : Q P is the family of all proper p-subgroups of the Sylow p-subgroup P. Indeed, for any p-subgroup Q P G it results from the Mackey formula that k _{Q}^{H} k _{Q}^{G} _{H}^{G}, thus

P roj V _{H}^{G} P roj F P roj \overline{F} P roj Y .

Finally, it is also worth emphasizing that in general the kG-Green correspondent Γ L of an indecomposable V _{H}^{G}-endotrivial module L might or might not be a V -endotrivial module. Again, the section on groups with cyclic Sylow p-subgroups shall provide us with a handful of examples illustrating this phenomenon. Nonetheless, whenever the lemma applies Γ L is always V -endotrivial.

4.4. On the kernels of the restriction maps. If G is a finite group, we shall follow the notation of [MT07] and 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. It is a p^{}-group, isomorphic to the p^{}-part of the abelianization G G, G of G.

Any one-dimensional module χ is endotrivial (χ^{}_{k} χ k since in this case the trace map splits). Therefore, for all absolutely p-divisible kG-module V, there is an embedding

\begin{matrix}X G & & T_{V} G \\ χ & & χ . \end{matrix}

mapping a one-dimensional module to its class in T_{V} G . Formalism would require to denote by X_{V} G the image of X G in T_{V} G , where the law is written additively, nonetheless, in order to keep light notation, when no confusion is to be made we shall simply use X G instead of X_{V} G .

The next lemma gives conditions on the module V under which the kernel of the restriction map Res_{Q}^{G} : T_{V} G T_{V _{Q}^{G}} P is exactly X G . This generalises [MT07, Lem. 2.6]. The proof is the same, it is only analysed more deeply in order to state the results in terms of V -projectivity, which is less restricting than ordinary projectivity. This criterion shall be especially useful for the forthcoming case of groups having a cyclic Sylow p-subgroup.

Lemma 4.4.1. Let G be a finite group and P be a Sylow p-subgroup of G. Let V be an absolutely p-divisible kG-module.

(a) Let Q be any p-subgroup of G such that the restriction map Res_{Q}^{G} : T_{V} G T_{V _{Q}^{G}} Q is

well-defined, that is such that V _{Q}^{G} is absolutely p-divisible. Then X G ker Res_{Q}^{G} .

(b) If all the direct summands of V _{P}^{G} have a vertex strictly included in ^{x}P P, up to conjugation, for all x G N_{G} P , then X G ker Res_{P}^{G} . In particular, if P is normal in G, then X G ker Res_{P}^{G} .

Proof. (a) This is clear since the only one-dimensional kQ-module is the trivial module.

(b) It remains to show the reverse inclusion. So let M be an indecomposable V -endotrivial kGmodule such that M ker Res_{P}^{G} , i.e. M _{P}^{G} k V _{P}^{G} proj . Thus, by P -projectivity, we have:

M M _{P}^{G} _{P}^{G} k _{P}^{G} V _{P}^{G} proj _{P}^{G} k _{P}^{G} V proj

where by 2.1.1 and 2.1.2 P roj V _{P}^{G} _{P}^{G} P roj V _{P}^{G} _{P}^{G} P roj V . Now, since by assumption M is indecomposable and V -endotrivial, that is V -projective-free, M must be a direct summand of k _{P}^{G}, therefore restricting to P and applying the Mackey formula yields:

M _{P}^{G} k _{P}^{G} _{P}^{G} k^{N_{G} P :P} _{\begin{matrix}x P G P \\ x N_{G} P \end{matrix}} k _{^{x}P P}^{P}

Each summand k _{^{x}P P}^{P} has a vertex equal to ^{x}P P. Write V _{P}^{G} _{i 1}^{m} V_{i}, m ℕ, as a sum of indecomposable modules and for all i \overline{1, n} and let Q_{i} be a vertex of V_{i}, then

P roj V _{P}^{G}  _{i 1}^{m} P roj V_{i}  _{i 1}^{m} P roj k _{Q_{i}}^{G} .

Assume then that k _{^{x}P P}^{P} P roj V _{P}^{G} , thus k _{^{x}P P}^{P} P roj k _{Q_{i}}^{G} for some 1 i m. However, Q_{i} _{G} ^{x}P P by assumption, contradicting the fact that ^{x}P P is a vertex. Therefore, none of the summands k _{^{x}P P}^{P} belongs to P roj V _{P}^{G} , which forces M _{P}^{G} to be a direct summand of k^{N_{G} P :P}. Using once more that M _{P}^{G} is V _{P}^{G}-endotrivial allows us to deduce that M _{P}^{G} k, for V _{P}^{G} being absolutely p-divisible, k P roj V _{P}^{G} . Hence M X G .

□

Notice that in case V kG, that is if we consider ordinary endotrivial modules, then condition (b) is equivalent to requiring that ^{x}P P is non trivial for all x G, as is stated in [MT07, Lem. 2.6].

5. Relative endotrivial modules as a generalisation for the Dade group

We come in this section to a chief reason of interest in relative endotrivial modules: it provides a way to define a group structure on collections of representations of an arbitrary finite group G which gives a generalization for the Dade group D P of a finite p-group P.

Endo-permutation modules are defined only for p-groups, but not for finite groups in general. One way to obtain a similar notion for arbitrary groups is to consider endo-p-permutation modules as described in [Urf06]. 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. 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 .

So let us see in which way our notion of relative endotrivial modules can generalize the Dade group. Let us fix P a finite p-group. The first observation to make is that an indecomposable capped endopermutation kP -module M (i.e. with vertex P) is in fact a relative endotrivial module as well, that is relatively to some intrinsincally defined kP -module V_{M}. Indeed, it is an elementary fact about capped endo-permutation modules that the trivial kP -module k has to be a direct summand of End_{k} M ([Dad78a, Prop. 3.7]), while by 2.2.1 it is clear that the multiplicity of k is exactly one. It yields the characterization:

End_{k} M  permutation module k k _{Q_{1}}^{P}  k _{Q_{s}}^{P}

for some subgroups Q_{1}, …, Q_{s} P, s ℕ. Therefore one can set V_{M} : _{i 1}^{s} k _{Q_{i}}^{P} (clearly absolutely p-divisible!) so that, by very definition, M becomes a V_{M}-endotrivial module. Besides, adding to V_{M} direct summands which are permutation modules, with no trivial summand, allows to build modules W : V_{M} perm such that M is also W -endotrivial. Such a construction always results in an absolutely p-divisible module since a non trivial indecomposable permutation kP -module is of the form k _{Q}^{P} for some Q P. Therefore, one can easily define a universal V, relatively to which any indecomposable capped endo-permutation module is endotrivial, namely

V : _{Q P} k _{Q}^{P} .

This construction leads to the following natural embedding of D P in TV P , in which the equivalence classes do have a unique indecomposable representative, up to isomorphism.

Theorem 5.0.2. The Dade group D P can be identified with a subgroup of TV P via the canonical injective homomorphism

\begin{matrix}D P & & TV P \\ M & & Cap M . \end{matrix}

Proof. According to the above construction, any indecomposable capped endo-permutation module M is a V -endotrivial module. Since both in D P and in TV P there is a unique indecomposable representative for the classes, the map D P TV P : M Cap M is a well-defined, injective group morphism. □

Open Problem 5.0.3. The question that arises naturally at this stage is the question of wether or not D P is actually isomorphic to TV P . We shall see in the next sections that if P C_{p^{n}}, a cyclic p-group, or if p 2 and P C_{2} C_{2}, then the answer is positive.

Finally we note that the embedding of D P in TV P shall provide us with the right setting to define an analog of the Dade group for an arbitrary finite group G with char k G . If P is a Sylow p-subgroup of G, let { \atop V_{G}} : _{Q P} k _{Q}^{G}. The idea is then to find a subgroup of T{ \atop V_{G}} G , that we shall denote by D G and that has many properties in common with the Dade group of a p-group. In particular, the indecomposable representatives of the classes shall be endo-p-permutation modules and if G is a p-group then D G coincides with the Dade group. This new group is studied in the Ph.D. Thesis of the author [Las12].

6. C_{2} C_{2}: the normal case

For a first example, we consider groups G having a normal Sylow 2-subgroup isomorphic to the Klein group C_{2} C_{2}, which we shall rather denote by V_{4} for ease of notation. Furthermore, k shall denote an algebraically closed field of characteristic 2. In this case, we show that relative endotrivial modules are not of much interest, since any group of relative endotrivial modules turns out to be isomorphic to the group T G of ordinary endotrivial modules, whose structure is made explicit in [Maz07]. Nonetheless this case is still worth considering because it is a nice source of examples and counter-examples for general behaviours of the groups of relative endotrivial modules.

Theorem 6.0.4. Let G be a finite group with a normal Sylow 2-subgroup isomorphic to the Klein group V_{4}. Let V be any absolutely 2-divisible kG-module. Then there is a group isomorphism φ : T_{V} G T G : M _{V} M_{0} , where M M_{0} V proj with M_{0} indecomposable and V -endotrivial. In particular, if G V_{4}, then T_{V} G Ω k _{V} ℤ.

Proof. To begin with, consider the case G V_{4} itself. The Klein group is a 2-group, therefore the indecomposable modules that bear chances to be V -endotrivial must have odd k-dimension. By the classification of indecomposable kV_{4}-modules, the odd-dimensional indecomposable modules are precisely the modules Ω^{n} k , n ℤ, which are all endotrivial modules in the usual sense. In consequence, on the one hand, Ω k _{V} Ω k T G ℤ and on the other hand, T_{V} G T G via φ. Hence

T_{V} G Ω k _{V} ℤ

although the classes in T_{V} G may contain more modules than the classes in T G . Now, let G be an arbitrary group with a normal Sylow 2-subgroup isomorphic to V_{4}. By 4.1.2, a kG-module M is indecomposable V -endotrivial if and only if its restriction M _{V_{4}}^{G} is indecomposable and V _{V_{4}}^{G}- endotrivial. But we have just shown that any such kV_{4}-module is in fact an ordinary endotrivial module hence, by the same criterion, M is actually endotrivial. In consequence, φ is a well-defined group homomorphism. The uniqueness of the summand M_{0} then yields the bijection. □

Remark 6.0.5. For G V_{4}, D G T G (see [Dad78b]), hence the positive answer to Problem 5.0.3 in this case. Also note in the general case that the structure of T G is described more accurately in [Maz07, Thm. 2.6] as follows:

T G X G Ω k X G ℤ

By Corollary 4.1.3, the indecomposable endotrivial kG-modules consist of all the extensions to G of the kV_{4}-modules Ω^{n} k , n ℤ, which are given by the family of modules Ω^{n} k k_{ω} such that n ℤ and k_{ω} is a one-dimensional kG-module.

Although there is, up to isomorphism, only one group of relatively endotrivial kG-modules, there are infinitely many different subcategories of V -projective modules which, in particular, do not correspond to projectivity relative to a subgroup.

Lemma 6.0.6. Let λ ℙ^{1} k and n 1 be an integer. Let M_{2n} λ be the unique 2n-dimensional indecomposable kV_{4}-module with projective variety λ . Then the indecomposable modules projective relative to M_{2n} λ are:

(a) IP roj M_{2n} λ M_{2m} λ 1 m n kV_{4} if λ 0, 1, ;

(b) IP roj M_{2} λ M_{2} λ , M_{4} λ , kV_{4} if λ 0, 1, ;
(c) IP roj M_{2n} λ M_{2m} λ 1 m n kV_{4} if λ 0, 1, and n 2.

Proof. This is a consequence of the Green ring structure on kV_{4} computed by Conlon [Con65]. □

7. The cyclic case: relative endotrivial modules for cyclic p-groups.

Let G : C_{p^{n}} be a cyclic p-group of order p^{n}, n 1, generated by g and k an algebraically closed field of characteristic p. Then kC_{p^{n}} k X X 1 ^{p^{n}} as k-algebras and M_{i} : k X X 1 ^{i} is the unique indecomposable kC_{p^{n}}-module of dimension i, up to isomorphism. Moreover, for 1 i p^{n} this provides a complete list of indecomposable kC_{p^{n}}-modules, up to isomorphism. In particular, M_{1} k, the trivial module, and M_{p^{n}} kC_{p^{n}} is the indecomposable projective module. (See [Thé95, Exercises 5.4, 17.2 and 28.3] for details.) Besides, for all 1 i p^{n}, a simple comparison of dimensions yields Ω M_{i} M_{p^{n} i} . Also note that the indecomposable absolutely p-divisible modules are the M_{i}’s with p dividing their dimension i. Finally, according to notation used in [MT07], for all integers 0 r n, we shall denote by Z_{r} the unique cyclic subgroup of P of order p^{r}, with Z_{0} 1, Z_{1} : Z and Z_{n} P. Thus there are isomorphisms M_{p^{r}} k _{Z_{n r}}^{G} . In this section we shall classify the relative endotrivial modules relatively to any absolutely p-divisible kG-module V.

7.1. Determination of all types of relative projectivities to modules. The aim is first to find out all the absolutely p-divisible modules V for which the subcategories P roj V are strictly different and secondly to describe explicitly all the modules they contain.

Lemma 7.1.1. For every integer 1 r n, IP roj M_{p^{r}} M_{αp^{r}} α ℕ, 1 α p^{n r} .

Proof. Let 1 r n and 1 α p^{n r} be integers. Consider the subgroup Z_{n r} C_{p^{n}} of index p^{r}. By 2.0.2 (c), mod kZ_{n r} P roj k . In particular, the kZ_{n r} -module M_{α} P roj k thus, by Lemma 2.1.1, we get

M_{α} _{Z_{n r}}^{C_{p^{n}}} P roj k _{Z_{n r}}^{C_{p^{n}}} .

In addition, by Green’s indecomposability theorem, both M_{α} _{Z_{n r}}^{C_{p^{n}}} and k _{Z_{n r}}^{C_{p^{n}}} are indecomposable. Because for every 1 i p^{n}, there is a unique indecomposble kC_{p^{n}}-module with k-dimension i, it is necessary that M_{α} _{Z_{n r}}^{C_{p^{n}}} M_{αp^{r}} C_{p^{n}} and k _{Z_{n r}}^{C_{p^{n}}} M_{p^{r}}. In other words M_{αp^{r}} P roj M_{p^{r}} . This yields the inclusion

M_{αp^{r}} 1 α p^{n r} IP roj M_{p^{r}} .

On the other hand, projectivity relative to the module M_{p^{r}} is exactly the same thing as projectivity relative to the subgroup Z_{n r} of C_{p^{n}}. Therefore, if M is projective relatively to Z_{n r} , then by 3.1.2 the index p^{r} C_{p^{n}} : Z_{n r} divides dim_{k} M , which proves the second inclusion. □

Corollary 7.1.2. For every integer 1 r n, the collection of kC_{p^{n}}-modules projective relatively to the kC_{p^{n}}-module M_{p^{r}} is given as follows:

P roj M_{p^{r}} _{I finite} M_{α_{i}p^{r}} α_{i} ℕ and 1 α_{i} p^{n r} i I

Lemma 7.1.3. Let M_{i} be an indecomposable kC_{p^{n}}-module such that p^{r}, with 1 r n 1, is the largest power of p dividing dim_{k} M_{i} i. Write i : α_{i}p^{r} with 1 α_{i} p 1 an integer. Then P roj M_{i} P roj M_{p^{r}} .

Proof. By Lemma 7.1.1, M_{i} M_{α_{i}p^{r}} IP roj M_{p^{r}} . In consequence, P roj M_{i} P roj M_{p^{r}} . In order to show the reverse inclusion, consider again the subgroup Z_{n r} . Since p ∤ α_{i}, by 2.0.2 (c), P roj M_{α_{i}} mod kZ_{n r} . In particular, the trivial kZ_{n r} -module k P roj M_{α_{i}} , hence

M_{p^{r}} k _{Z_{n r}}^{C_{p^{n}}} P roj M_{α_{i}} _{Z_{n r}}^{C_{p^{n}}} P roj M_{α_{i}p^{r}}

by Green’s indecomposability theorem again. Thus P roj M_{i} P roj M_{p^{r}} . □

We shall now show that in mod kC_{p^{n}} projectivity relative to modules is indeed reduced to projectivity relative to subgroups. In other words:

Proposition 7.1.4. Let V be an absolutely p-divisible kC_{p^{n}}-module. Then P roj V P roj M_{p^{r}} P roj k _{Z_{n r}}^{C_{p^{n}}} for some subgroup Z_{n r} of C_{p^{n}} with r 1.

Proof. If V is indecomposable then V M_{i} for some i divisible by p, and the result has been shown in the preceding lemma. If V is decomposable, write V : _{i 1}^{s} M_{i}, s ℕ. Factor every 1 i s, as i : α_{i}p^{r_{i}} with 1 α_{i} p 1 and 1 r_{i} n 1. Let m : min_{i} r_{i} . Then, using 2.0.2, compute:

P roj V P roj _{i 1}^{s} M_{i}  _{i 1}^{s} P roj M_{i}  _{i 1}^{s} P roj M_{p^{r_{i}}}  _{i 1}^{s} P roj M_{p^{m}} P roj M_{p^{m}}

where clearly P roj M_{p^{r_{i}}} P roj M_{p^{m}} either by a classical argument on projectivity relative to subgroups or by Lemma 7.1.1. □

In particular, note that for G C_{p} a cyclic group of prime order, there is no relative projectivity to modules other than ordinary projectivity. More generally, we note that there is a unique chain of strict inclusions of subcategories of relatively projective kC_{p^{n}}-modules given as follows:

P roj P roj M_{p^{n}} P roj M_{p^{n 1}} P roj M_{p^{2}} P roj M_{p} .

7.2. Structure of the groups of relatively endotrivial modules. There are exactly n different proper subcategories of relatively projective modules in mod kC_{p^{n}} , given by P roj M_{p^{r}} for 0 r n and therefore also n different groups of relatively endotrivial modules: T_{M_{p^{r}}} C_{p^{n}} for 0 r n. Besides, since there is a unique indecomposable kC_{p^{n}}-module for each k-dimension between 1 and p^{n}, it is clear that every such module is self-dual. Therefore, by Corollary 3.5.2, any group T_{M_{p^{r}}} C_{p^{n}} is an elementary abelian 2-group (or trivial). It remains to figure out their respective ranks. First, we need the following technical result.

Lemma 7.2.1. Let C_{p^{n}} be a cyclic p-group with n 2 and let 1 r n be an integer. Then, there is no indecomposable kC_{p^{n}}-module, whose k-dimension lies between p^{n 1} and p^{n} p^{n 1} , which is M_{p^{r}}-endotrivial.

The main idea of the proof is based on the following restriction formula (see [Thé95, Exercise 28.3 (a)]):

(1)  M_{i} _{Z}^{C_{p^{n}}} sM_{a 1} p^{n 1} s M_{a}

with i ap^{n 1} s, 0 s p^{n 1} and 0 a p, for all 1 i p^{n}.

Proof. The case p 2 is trivial since 2^{n 1} 2^{n} 2^{n 1} , therefore, we may assume that p is odd. Furthermore, an M_{p^{r}}-endotrivial module is necessarily M_{p}-endotrivial since P roj M_{p^{r}} P roj M_{p} , hence we may also asume that r 1. The indecomposable modules, candidates to be M_{p}-endotrivial are the indecomposable modules of the form M_{βp 1} for some 1 β p^{n 1} . We claim that, if p^{n} βp 1 p^{n} p^{n 1} , then M_{βp 1} is not M_{p}-endotrivial. First note that the symmetry given by the Heller operator Ω allows us to consider only the case βp 1. The proof proceeds ab absurdo: we assume that M_{βp 1} is M_{p}-endotrivial and compute End_{k} M_{βp 1} _{Z}^{C_{p}^{n}}. Since p^{n} βp 1 p^{n} p^{n 1} , we have p^{n 2} β p^{n 1} and we can write β : γp^{n 2} σ with γ and σ integers such that 1 γ p 1 and 0 σ p^{n 2} . So that

βp 1 γp^{n 2} σ p 1 γp^{n 1} σp 1.

Now, M_{βp 1} is M_{p}-endotrivial, thus End_{k} M_{βp 1} k M_{p} proj and

End_{k} M_{βp 1} _{Z}^{C_{p^{n}}} k M_{p} _{Z}^{C_{p^{n}}} proj .

Let us count the number of trivial summands on both sides of this isomorphism. On the right-hand side, there is one modulo p by formula (1) (this easily follows from the fact that M_{p}-projective modules have dimension divisible by p by 2.2.3). On the left-hand side we get by formula (1):

End_{k} M_{βp 1}  _{Z}^{C_{p^{n}}} M_{βp 1} _{Z}^{C_{p^{n}}} M_{βp 1} _{Z}^{C_{p^{n}}}
σp 1 M_{γ 1} p^{n 1} σp 1 M_{γ} ^{2}
σp 1 ^{2} M_{γ 1} ^{2} 2 p^{n 1} σp 1 σp 1 M_{γ 1} M_{γ}
p^{n 1} σp 1 ^{2} M_{γ} ^{2}

Since 1 γ p 1, p ∤ dim_{k} M_{γ} and p ∤ dim_{k} M_{γ 1} , but by 2.2.1 there is exactly one trivial summand k in M_{γ} M_{γ} as well as in M_{γ 1} M_{γ 1} and, moreover, k is not a direct summand of M_{γ 1} M_{γ}. Therefore, altogether there are σp 1 ^{2} p^{n 1} σp 1 ^{2} 2 mod p trivial summands

in End_{k} M_{βp 1}  _{Z}^{C_{p^{n}}}, which is a contradiction. Hence the result.  □

For simplicity of notation, we shall, from now on, denote by Ω_{M_{p^{s}}} the class of the relative syzygy module Ω_{M_{p^{s}}} k in T_{M_{p^{r}}} C_{p^{n}} and simply use Ω : Ω_{M_{p^{n}}}. The classification theorem is the following.

Theorem 7.2.2. Let G : C_{p^{n}} with n 1 be a cyclic p-group and M_{p^{r}} with 1 r n be an absolutely p-divisible kC_{p^{n}}-module.

(a) If p is odd, or if p 2 and r 2, then

T_{M_{p^{r}}} C_{p^{n}} Ω_{M_{p^{s}}} r s n _{j 1}^{n r 1} C_{2} .

(b) If p 2 and r 1, then

T_{M_{2}} C_{2^{n}} Ω_{M_{p^{s}}} 1 s n _{j 1}^{n 1} C_{2} .

To begin with, the following lemma on the structure of T_{M_{p^{r}}} C_{p^{n}} shall enable us to prove the theorem by induction on the integer n.

Lemma 7.2.3. Assume G C_{p^{n}} with n 2 and write C_{p^{n 1}} C_{p^{n}} Z. Then for every integer 1 r n,

T_{M_{p^{r}}} C_{p^{n}} Inf_{C_{p^{n 1}}}^{C_{p^{n}}} T_{M_{p^{r}}} C_{p^{n 1}} Ω T_{M_{p^{r}}} C_{p^{n 1}} C_{2} .

Proof. Inflation induces an injective group homomorphism Inf_{C_{p^{n 1}}}^{C_{p^{n}}} : T_{M_{p^{r}}} C_{p^{n 1}} T_{M_{p^{r}}} C_{p^{n}} . The

indecomposable representatives for the classes in the image subgroup Inf_{C_{p^{n 1}}}^{C_{p^{n}}} T_{M_{p^{r}}} C_{p^{n 1}} are kC_{p^{n}}- modules whose k-dimension is less than or equal to p^{n 1} . Moreover, as inflation commutes with direct sums, it is clear that M_{i} Inf_{C_{p^{n 1}}}^{C_{p^{n}}} M_{i} is M_{p^{r}}-endotrivial if and only if M_{i}, seen as a kC_{p^{n 1}} -module, is an M_{p^{r}}-endotrivial kC_{p^{n 1}} -module. As seen in Lemma 7.2.1 there is no indecomposable M_{p^{r}}-endotrivial module with k-dimension between p^{n 1} and p^{n} p^{n 1} . Furthermore, for all p^{n} p^{n 1} i p^{n}, M_{i} Ω M_{p^{n} i} is M_{p^{r}}-endotrivial if and only if M_{p^{n} i} is and for such a module in T_{M_{p^{r}}} C_{p^{n}} we have M_{i} Ω M_{p^{n} i} Ω M_{p^{n} i} . Whence the direct product

T_{M_{p^{r}}} C_{p^{n}} Inf_{C_{p^{n 1}}}^{C_{p^{n}}} T_{M_{p^{r}}} C_{p^{n 1}} Ω .
□

Proof of Theorem 7.2.2. (a) The proof proceeds by induction on n. First, the cyclic p-group of smallest order for which projectivity relative to an indecomposable module of dimension p^{r} can be considered is C_{p^{r}}, in which case P roj M_{p^{r}} P roj as M_{p^{r}} kC_{p^{r}}. Therefore T_{M_{p^{r}}} C_{p^{r}} T C_{p^{r}} Ω C_{2} by the classification made in [Dad78b]. Then by the lemma and the induction hypothesis we get:

T_{M_{p^{r}}} C_{p^{n}} Inf_{C_{p^{n 1}}}^{C_{p^{n}}} T_{M_{p^{r}}} C_{p^{n 1}}  Ω

Inf_{C_{p^{n 1}}}^{C_{p^{n}}} Ω_{M_{p^{s}}} r s n 1 Ω

Ω_{M_{p^{s}}} r s n 1  Ω  Ω_{M_{p^{s}}} r s n

_{j 1}^{n r 1} C_{2}

since by Corollary 3.5.2 any element of T_{M_{p^{r}}} C_{p^{n}} has order 2.

(b) If r 1, then T_{M_{2}} C_{2} T C_{2} k 1 . Hence the missing factor C_{2} in the product. Notice that Ω_{M_{2}} k k, which is the reason why Ω_{M_{2}} is not a generator for T_{M_{2}} C_{2^{n}} . Nonetheless the set of generators is obtained in like manner as it was in part (a). □

Corollary 7.2.4. Let C_{p^{n}} with n 1 be a cyclic p-group. Then the Dade group D C_{p^{n}} T_{M_{p}} C_{p^{n}} .

Proof. By the description of the Dade group for cyclic p-groups made in [Dad78b], any indecomposable M_{p}-endotrivial kC_{p^{n}}-module is an endo-permutation module, therefore the injective homomorphism

D C_{p^{n}} T_{M_{p}} C_{p^{n}} : M  Cap M of Theorem 5.0.2 is an isomorphism.  □

Remark 7.2.5. Even though we showed that for cyclic p-groups projectivity relative to modules is reduced to projectivity relative to subgroups, we kept notation using modules rather than subgroups because it was more manageable firstly in the description of relatively projective modules, secondly in computations and thirdly in arguments involving inflation. Nevertheless in the next section, treating the case of groups having a cyclic Sylow p-subgroup, it will be easier to think in terms of subgroups. In this system of notation the groups of relatively endotrivial modules are generated as follows:

T_{M_{p^{r}}} C_{p^{n}} T_{k _{Z_{n r}}^{C_{p^{n}}}} C_{p^{n}} Ω_{k _{Zs}^{C_{p^{n}}}} 0 s n r if M_{p^{r}} M_{2} and

T_{M_{2}} C_{2^{n}} T_{k _{Z_{n 1}}^{C_{p^{n}}}} C_{2^{n}} Ω_{k _{Zs}^{C_{2^{n}}}} 0 s n 1 .

8. The cyclic case: groups with cyclic Sylow p-subgroups.

In this section G is a finite group having a non-trivial cyclic Sylow p-subgroup P C_{p^{n}}, n 1. Recall from the previous section that Z_{r} denotes the unique cyclic subgroup of P of order p^{r} with 0 r n. Moreover, for 0 r n 1, one has the following chain of inclusions of subgroups of G: Z_{r} Z_{r 1} P N_{G} P N_{G} Z_{r 1} N_{G} Z_{r} G.

8.1. Determination of the different types of V -projectivities. To start with we show that the only types of relative projectivity that occur are again the projectivities relative to subgroups.

Proposition 8.1.1. Let G be a finite group with a cyclic Sylow p-subgroup P C_{p^{n}} with n 1.

(a) Let V be any kG-module. Then P roj V P roj k _{Q}^{G} for some subgroup Q of P. In particular, V is absolutely p-divisible if and only if Q is a proper subgroup of P.

(b) There is a unique chain of proper inclusions of subcategories of relatively projective kG-modules:

P roj P roj k _{Z_{1}}^{G} P roj k _{Z_{2}}^{G} P roj k _{Z_{n 1}}^{G} P roj k _{P}^{G} mod kG

Proof. (a) Recall that, by Lemma 2.1.4, subcategories of relatively projective modules are determined upon restriction to P in the sense that for all U, W mod kG , P roj U _{P}^{G} P roj W _{P}^{G} if and only if P roj U P roj W . First of all, by 7.1.4 there exists a subgroup Q of P such that P roj V _{P}^{G} P roj k _{Q}^{P} . (Q P in case V _{P}^{G} is not absolutely p-divisible.)

Therefore, by the above remark, in order to show that P roj V P roj k _{Q}^{G} , it is enough to check that P roj k _{Q}^{G} _{P}^{G} P roj k _{Q}^{P} . Indeed, applying the Mackey formula yields

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

where the subgroups ^{x}Q P form a chain of subgroups of Q ^{1}Q P, since P is cyclic. Hence P roj k _{^{x}Q P}^{P} P roj k _{Q}^{P} for all x P G Q so that

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

Now, the module V is absolutely p-divisible if and only if P roj V mod kG P roj k , if and only if P roj V _{P}^{G} P roj k _{Q}^{P} P roj k _{P}^{G} P roj k . Thus by the characterization given in 7.1.4, V is absolutely p-divisible if and only if Q is a proper subgroup of P.

(b) For G P, we have shown in the previous section that there is a unique chain of inclusions of subcategories of relatively projective kP -modules given by

P roj P roj k _{Z_{1}}^{P} P roj k _{Z_{2}}^{P} P roj k _{Z_{n 1}}^{P} P roj k _{P}^{P} mod kP .

But we proved in (a) that P roj k _{Q}^{G} _{P}^{G} P roj k _{Q}^{P} for all subgroup Q P, therefore another application of Lemma 2.1.4 yields the result.

□

As a corollary of the proof, we obtain that for all 0 r n, projectivity relatively to the p-subgroup Z_{r} of G restricted to a subgroup H of G such that either P H or Z_{r} H P remains projectivity relative to Z_{r}, i.e. P roj k _{Z_{r}}^{G} _{H}^{G} P roj k _{Z_{r}}^{H} . For, we showed in the proof of the proposition that P roj k _{Z_{r}}^{G} _{P}^{G} P roj k _{Z_{r}}^{P} , but the argument remains true if P is replaced with a subgroup H as given above.

Groups of relatively endotrivial modules are defined only for absolutely p-divisible modules V, in consequence and in view of Proposition 8.1.1, we shall assume for the remainder of the section that V k _{Z_{r}}^{G} for some proper subgroup Z_{r} of P. The remainder of the section is devoted to the determination of the structure of the groups T_{k _{Zr}^{G}} G with 0 r n.

Remark 8.1.2. Proposition 8.1.1 also gives full control of absolute p-divisibility with respect to restrictions: the restriction of an absolutely p-divisible kG-module V remains absolutely p-divisible whenever either P H or Z_{r} H P. Indeed, we have P roj V P roj k _{Z_{r}}^{G} for some Z_{r} P then, by Lemma 2.1.2 and the above remarks

P roj V _{H}^{G} P roj k _{Z_{r}}^{G} _{H}^{G} P roj k _{Z_{r}}^{H} modkH .

Hence V _{H}^{G} is an absolutely p-divisible kH-module. In consequence, for all subgroups H as above, the restriction maps Res_{H}^{G} from the group T_{k _{Zr}^{G}} G are well-defined and all have the form

Res_{H}^{G} : T_{k _{Zr}^{G}} G T_{k _{Zr}^{H}} H .

8.2. Properties of the restriction maps and the structure theorem. To begin with we develop a few more properties of the restriction maps. We shall then use them to deduce the structure of the groups of relatively endotrivial modules T_{k _{Zr}^{G}} G from our knowledge of the structure of T_{k _{Zr}^{P}} P .

In order to ease up notation we simply use the symbole Ω_{V} to denote the class Ω_{V} k in T_{V} G and k _{Q} instead of k _{Q}^{H} in indices when it is clear to which subgroup H G induction goes. We avoid to use a simpler notation like Ω_{Q} because it has been widely used to denote the class of the ordinary syzygy Ω k in mod kQ in articles concerned with endotrivial and endo-permutation modules.

Lemma 8.2.1. Let H G be a subgroup such that either P H or Z_{r} H P and Res_{H}^{G} : T_{k _{Zr}} G T_{k _{Zr}} H be a restriction map. Then:

(a) Res_{H}^{G} Ω_{k _{Zs}} Ω_{k _{Zs}} for all Z_{s} Z_{r} so that Ω_{k _{Zs}} 0 s r Im Res_{H}^{G} .
(b) If Z_{r} H P, then Res_{H}^{G} is surjective.

Proof. (a) By 2.3.4, Ω_{k _{Zs}} k _{H}^{G} Ω_{k _{Zs}} k k _{Z_{s}}^{H} proj . Hence Res_{H}^{G} Ω_{k _{Zs}} Ω_{k _{Zs}}.

(b) Follows from (a) since by 7.2.2 the group T_{k _{Zr}} H is generated by the set of all relative syzygies

Ω_{k _{Zs}} k such that Z_{s} Z_{r}.

□

Corollary 8.2.2. Let P be a cyclic p-group and Z_{r} a proper subgroup of P. Then, the restriction maps Res_{H}^{P} : T_{k _{Zr}} P T_{k _{Zr}} H are isomorphisms for all Z_{r} H P.

Proof. By the lemma Res_{H}^{P} is surjective and, by 7.2.2, T_{k _{Zr}} P  T_{k _{Zr}} H .  □

Using the criterion described in Lemma 4.3.1, we can show that the group T_{k _{Zr}} G is indeed entirely determined by restriction to N_{G} Z_{r 1} .

Proposition 8.2.3. Let G be a finite group with a non-trivial cyclic Sylow p-subgroup P and Z_{r} be a proper subgroup of P. Then, the restriction map

Res_{N_{G} Z_{r 1}}^{G} : T_{k _{Zr}} G T_{k _{Zr}} N_{G} Z_{r 1}

is an isomorphism, with inverse map induced by Green correspondence or alternatively by induction:

T_{k _{Zr}} G  Γ M M is an indecomposable k _{Z_{r}} -endotrivial kN_{G} Z_{r 1} -module
M _{N_{G} Z_{r 1}}^{G} M T_{k _{Zr}} N_{G} Z_{r 1}

Proof. The isomorphism and both the descriptions of T_{k _{Zr}} G using Green correspondence and induction follow from the criterion given in Lemma 4.3.1. Thus, it suffices to check that

P roj k _{Z_{r}}^{N_{G} Z_{r 1}} P roj Y ,

where Y ^{g}P N_{G} Z_{r 1} g G N_{G} Z_{r 1} . For all g G N_{G} Z_{r 1} , the subgroup ^{g}P N_{G} Z_{r 1} is a p-subgroup of ^{g}P, hence of the form ^{g}Z_{l} for some l ℕ_{n} since P C_{p^{n}} is cyclic. Besides, ^{g}Z_{l} N_{G} Z_{r 1} as well, thus contained in some Sylow p-subgroup of N_{G} Z_{r 1} , say ^{h}P with h N_{G} Z_{r 1} , so that by uniqueness of the subgroup of order p^{l} in ^{h}P, we have ^{g}Z_{l} ^{h}Z_{l}. Hence h^{1} g normalizes Z_{l} and g hN_{G} Z_{l} N_{G} Z_{r 1} N_{G} Z_{l} N_{G} Z_{r 1} since g does not normalize Z_{r 1} . This forces N_{G} Z_{l} to contain strictly N_{G} Z_{r 1} , because the subgroups N_{G} Z_{i} are totally ordered by inclusion, hence Z_{l} Z_{r}. As a consequence, P roj ^{g}P N_{G} Z_{r 1} P roj Z_{l} P roj k _{Z_{r}}^{N_{G} Z_{r 1}} and as required:

P roj Y _{g G N_{G} Z_{r 1}} P roj ^{g}P N_{G} Z_{r 1} P roj k _{Z_{r}}^{N_{G} Z_{r 1}}

□

In view of the proposition we can restrict our attention to the groups T_{k _{Zr}} N_{G} Z_{r 1} . First of all,

computing the kernel of the restriction map Res_{P}^{N_{G} Z_{r 1}} : T_{k _{Zr}} N_{G} Z_{r 1} T_{k _{Zr}} P provides us with a set of generators.

Lemma 8.2.4. (a) There is an exact sequence

0 X N_{G} Z_{r 1}  T_{k _{Zr}} N_{G} Z_{r 1}  ^{Res_{P}^{N_{G} Z_{r 1}}} T_{k _{Zr}} P 0 .

(b) The group T_{k _{Zr}} N_{G} Z_{r 1} is a finite abelian group generated by X N_{G} Z_{r 1} and the r 1

relative syzygy modules Ω Ω_{k _{1}}, Ω_{k _{Z}} , …, Ω_{k _{Zr}} .

Proof. (a) The map Res_{P}^{N_{G} Z_{r 1}} is surjective by 8.2.1. In addition, V : k _{Z_{r}}^{N_{G} Z_{r 1}} fulfills the hypotheses of Lemma 4.4.1. Indeed, recall from the proof of Proposition 8.1.1 that

k _{Z_{r}}^{N_{G} Z_{r 1}} _{P}^{N_{G} Z_{r 1}} _{g P N_{G} Z_{r 1} Z_{r}}k _{^{g}Z_{r} P}^{P}

where each indecomposable summand k _{^{g}Z_{r} P}^{P} has a vertex equal to ^{g}Z_{r} P Z_{r}, which is strictly contained in ^{x}P P for all x N_{G} Z_{r 1} N_{G} P . Indeed, any such x normalizes Z_{r 1} ,

thus Z_{r} Z_{r 1} ^{x}P P. Therefore 4.4.1 yields ker Res_{P}^{N_{G} Z_{r 1}} X N_{G} Z_{r 1} .

(b) By Theorem 7.2.2, T_{k _{Zr}} P Ω_{k _{Zs}} 0 s r . Now, by 8.2.1, Ω_{k _{Zs}} is a preimage

by Res_{P}^{N_{G} Z_{r 1}} for the generator Ω_{k _{Zs}} of T_{k _{Zr}} P for all 0 s r . Thus X N_{G} Z_{r 1}

Ω_{k _{Zs}} 0 s r is a set of generators for T_{k _{Zr}} N_{G} Z_{r 1} . The finiteness of T_{k _{Zr}} N_{G} Z_{r 1}

follows from both that of T_{k _{Zr}} N_{G} P and of X N_{H} Z_{r 1} .
□

Knowing that the generators Ω, Ω_{k _{Z_{1}}} , …, Ω_{k _{Zr}} of T_{k _{Zr}} N_{G} Z_{r 1} are preimages for the generators Ω_{k _{Zs}} , 0 s r, of T_{k _{Zr}} P which all have order 2, it remains to identify 2Ω_{k _{Zs}}, for all 0 s r, with an element of the kernel, that is, a one-dimensional representation of N_{G} Z_{r 1} . These identifications will follow from an induction argument and use the structure of the group of endotrivial modules T G described in [MT07, Thm. 3.2]. This result makes use of a distinguished element of X N_{G} Z , which we need to describe and understand before use.

For Z the unique subgroup of P of order p, let H : N_{G} Z be its normalizer in G. As H acts by conjugation on Z, the quotient H C_{G} Z embeds as a subgroup of Aut Z ℤ pℤ ^{}, thus given c H, for all u Z we have

cu u^{ν c} for some ν c ℤ pℤ ^{}

where in addition ν c can be considered as an element of k^{}via the canonical embedding ℤ pℤ k. In consequence, the composition H H C_{G} Z Aut Z ℤ pℤ^{}k^{}defines a linear character of H. For simplicity, ν is then identified with a one-dimensional module in X H . In fact, a similar construction can be applied to any subgroup of G which normalizes Z. Furthermore, a Frattini argument applied to H and its normal subgroup C_{G} Z yields the decomposition H N_{H} P C_{G} Z N_{G} P C_{G} Z , therefore as C_{G} Z acts trivially on Z, ν is entirely defined by its value on N_{G} P . In other words, ν can be viewed as a kN_{G} P -module which can be extended in a kH-module for all subgroup H such that H H N_{G} P , and for ease of notation, we also denote these modules by ν, so that:

Res{ \atop H_{1}} ; { \atop H_{2}} ν ν whenever H { \atop H_{1}} { \atop H_{2}} N_{G} P .

In particular, we are interested in the subgroup H_{r 1} : N_{G} Z_{r 1} . Our aim will be to apply an induction argument to its quotient H_{r 1} Z_{r}. In this respect, note that P Z_{r} is a cyclic Sylow p-subgroup of H_{r 1} Z_{r}, Z_{r 1} Z_{r} its unique cyclic p-subgroup of order p, moreover H_{r 1} Z_{r} N_{H_{r 1} Z_{r}} Z_{r 1} Z_{r} and N_{H_{r 1} Z_{r}} P Z_{r} N_{G} P Z_{r}. Moreover, a Frattini argument yields more precisely

H_{r 1} Z_{r} N_{G} P Z_{r} C_{H_{r 1} Z_{r}} Z_{r 1} Z_{r} .

Therefore, there is also a corresponding kN_{G} P Z_{r}-module ν ν_{N_{G} P Z_{r}} which extends to H_{r 1} Z_{r}. Finally, the following technical result computes the inflation of ν_{N_{G} P Z_{r}} to a kN_{G} P -module.

Lemma 8.2.5. With the notation above we have Inf_{N_{G} P Z_{r}}^{N_{G} P} ν_{N_{G} P Z_{r}} ν_{N_{G} P}, that is, by abuse

of notation, Inf_{N_{G} P Z_{r}}^{N_{G} P} ν ν.

Proof. Let P : u u^{p^{n}} 1 . Then Z u^{p^{n 1}} , Z_{r} u^{p^{n r}} and Z_{r 1} Z_{r} u^{p^{n r 1}} Z_{r} . Let d N_{G} P , we have ^{d}u u^{j} for some integer j such that 1 j p^{n}. Then ^{d} u^{x} u^{x} ^{j} for all 1 x n. Therefore ^{d} u^{p^{n 1}} u^{p^{n 1}} ^{j} so that ν d j mod p. Likewise ^{dZ_{r}} u^{p^{n r 1}} Z_{r}

du Z_{r} ^{p^{n r 1}}  u^{j}Z_{r} ^{p^{n r 1}}  u^{p^{n r 1}} Z_{r} ^{j}, hence Inf_{N_{G} P Z_{r}}^{N_{G} P} ν d j mod p. Hence the result.
□

Theorem 8.2.6. Let G be a finite group with a non trivial cyclic Sylow p-subgroup P C_{p^{n}}. For all 0 r n, let Z_{r} be the unique proper p-subgroup of P of order p^{r} and H_{r 1} be its normalizer in G. Then

\begin{aligned}T_{k _{Zr}} H_{r 1} & X H_{r 1} , Ω_{k _{Zs}} 0 s r \\ & X H_{r 1}  Ω_{k _{Zs}} 0 s r  ν 2Ω_{k _{Zs}} , 0 s r . \end{aligned}

Proof. We need to identify each class 2Ω_{k _{Zs}} with an element of X H_{r 1} . We claim that 2Ω_{k _{Zs}} ν for all 0 s r. The proof proceeds by induction on r. The case r 0 holds by [MT07, Thm 3.2], because projectivity relative to Z_{0} 1 is ordinary projectivity, thus r 0 is the ordinary endotrivial case. So we may assume that r 0 and as T_{k _{Z_{r 1}}} H_{r 1} can be seen as a subgroup of T_{k _{Zr}} H_{r 1} , by induction hypothesis, we may assume that the relations 2Ω_{k _{Zs}} ν hold for all 0 s r 1. Thus it remains to show that 2Ω_{k _{Zr}} ν .

Factoring out H_{r 1} N_{G} Z_{r 1} by its normal subgroup Z_{r} enables us to apply the induction hypothesis again to the group T_{k _{Zr Zr}} H_{r 1} Z_{r} T H_{r 1} Z_{r} , for which [MT07, Thm. 3.2] provides the relation

2Ω ν in T H_{r 1} Z_{r} , that is, 2Ω_{k _{Zr Zr}} ν in T_{k _{Zr Zr}} H_{r 1} Z_{r} .

The following commutative square yields the desired relation for T_{k _{Zr}} H_{r 1} :

\xymatrix{T_{k _{Zr}} H_{r 1} \ar@{^{(}->}[r]_{}^{Res} & T_{k _{Zr}} N_{G} P \\ T_{k _{Zr Zr}} H_{r 1} Z_{r} \ar@{^{(}->}[r]_{Res} \ar[u]_{_{Inf{Hr 1 \atop Hr 1 Zr}}} & T_{k _{Zr Zr}} N_{G} P Z_{r} \ar[u]_{_{Inf{NG P \atop NG P Zr}}}}

By Lemma 2.3.4, Inf_{H_{r 1} Z_{r}}^{H_{r 1}} Ω_{k _{Zr Zr}} Ω_{k _{Zr}} , therefore, inflationing our relation to T_{k _{Zr}} H_{r 1} yields

2Ω_{k _{Zr}} Inf_{H_{r 1} Z_{r}}^{H_{r 1}} ν in T_{k _{Zr}} H_{r 1} .

By the previous lemma Inf_{N_{G} P Z_{r}}^{N_{G} P} ν ν , so that the result follows from the injectivity of Res_{N_{G} P}^{H_{r 1}} (Lemma 4.2.1). □

Remark 8.2.7. Since 2Ω_{k _{Zs}} ν 2Ω for all 0 s r the generators Ω_{k _{Z}} , …, Ω_{k _{Zr}} can be replaced with the generators Ω Ω_{k _{Z}} , …, Ω Ω_{k _{Zr}} , all of which have order 2. Thus the abelian group T_{k _{Zr}} N_{G} Z_{r 1} contains a direct sum of r copies of ℤ 2ℤ.

Finally, using the isomorphism of Proposition 8.2.3, the description by generators and relations of T_{k _{Zr}} N_{G} Z_{r 1} extends to T_{k _{Zr}} G which is a finite abelian group generated by the relative syzygy modules Ω Ω_{k _{1}^{G}}, Ω_{k _{Z}^{G}}, …, Ω_{k _{Zr}^{G}} and an isomorphic copy of X N_{G} Z_{r 1} , made up of all the classes of the Green correspondents of the one-dimensional kN_{G} Z_{r 1} -modules, with the relations 2Ω_{k _{Zs}^{G}} Γ ν for all 0 s r.

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

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