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Publication# Modules d'endo-p-permutation

Résumé

This dissertation is concerned with the study of a new family of representations of finite groups, the endo-p-permutation modules. Given a prime number p, a finite group G of order divisible by p and an algebraically closed field k of characteristic p, we say that a kG-module M is an endo-p-permutation module if its endomorphism algebra Endk(M) is a p-permutation kG-module, that is a direct summand of a permutation kG-module. This generalizes the notion, first introduced by E. Dade in 1978, of endo-permutation modules for p-groups . For P a p-group, E. Dade defined an abelian group structure on the set of isomorphism classes of indecomposable endo-permutation kP-modules with vertex P and he proved that the complete description of the structure of this group is equivalent to the classification of endo-permutation kP-modules. This group of isomorphism classes is now called the Dade group of the p-group P. The problem of describing the Dade group for an arbitrary p-group was recently solved by S. Bouc. This opens the question of studying endo-p-permutation modules, which are the natural generalization to arbitrary finite groups of endo-permutation modules. In the following text, we present the basic properties of endo-p-permutation modules and give a characterization of indecomposable endo-p-permutation modules with vertex P via properties of their sources modules. In particular, when the normalizer of P controls p-fusion, we are able to give a complete classification of sources of indecomposable endo-p-permutation modules with vertex P, using Bouc's description of the Dade group. When p is odd, we also give an alternative proof of a theorem of Dade concerned with extensions of endo-permutation modules, using our previous results. We present a consequence of this theorem of Dade on the structure of the multiplicity module associated to an indecomposable endo-p-permutation module. Finally we study some concrete examples of endo-p-permutation modules such as relative syzygies and relative Heller translates. We prove also that the Green correspondent of an indecomposable kNG(P)- endo-p-permutation module with vertex P is not in general an endo-p-permutation kG-module. The study of such representations is motivated by the important role they play in certain areas of representations theory. For instance, endo-permutation modules, and more generally endo-p-permutation modules (as is proved here), appear in the study of simple modules for p-solvable groups.

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Groupe fini

vignette|Un exemple de groupe fini est le groupe des transformations laissant invariant un flocon de neige (par exemple la symétrie par rapport à l'axe horizontal).
En mathématiques, un groupe fini es

Théorie des représentations

La théorie des représentations est une branche des mathématiques qui étudie les structures algébriques abstraites en représentant leurs éléments comme des transformations linéaires d'espaces vectori

Nombre premier

vignette|Nombres naturels de zéro à cent. Les nombres premiers sont marqués en rouge.
vignette|Le nombre 7 est premier car il admet exactement deux diviseurs positifs distincts.
Un nombre premier est

Let k be a field of characteristic p, where p is a prime number, let pp_k(G) be the Grothendieck group of p-permutation kG-modules, where G is a finite group, and let Cpp_k(G) be pp_k(G) tensored with the field of complex numbers C. In this article, we find all the composition factors of the biset functor Cpp_k restricted to the category of abelian groups.

Let k be an algebraically closed field of characteristic p, where p is a prime number or 0. Let G be a finite group and ppk(G) be the Grothendieck group of p-permutation kG-modules. If we tensor it with C, then Cppk becomes a C-linear biset functor. Recall that the simple biset functor SH,V are parametrized by pairs (H,V), where H is a finite group and V a simple COut(H)-module. If we only consider p'-groups, then Cppk = CRk is the usual representation functor and we know the simple functors which are its composition factors. If we consider only p-groups, then Cppk = CB is the Burnside functor and we also know the simple functors which are its composition factors. We want to find the composition factors of Cppk in general. In order to achieve this, we first show that the composition factors from the special cases above are also composition factors for Cppk. Then, we consider groups of little order and try to find new composition factors. This leads us to find the following new composition factors : The simple factors SCm,Cξ and SCp×Cp× Cm,Cξ, where (m,ξ) runs over the set of all pairs formed by a positive integer m prime to p and a primitive character ξ : (Z/mZ)* → C*. Their multiplicity as composition factors is 1. The simple factors SCp⋊Cl, C, where l is a number prime to p, the action of Cl on Cp is faithful and C is the trivial COut(Cp ⋊ Cl)-module. Their multiplicity as composition factors is φ(l). The simple functors SG,C, where G is a finite p-hypo-elementary B-group (for which an explicit classification is done) and C the trivial COut(G)-module. We also show that some specific simple functors appear, indexed by the groups C3 ⋊ C4, C5 ⋊ C4 and A4. On the way, we find all the composition factors of the subfunctor of permutation modules.

This dissertation is concerned with modular representation theory of finite groups, and more precisely, with the study of classes of representations, which we shall term relative endotrivial modules. Given a prime number p, a finite group G of order divisible by p, we shall say that a kG-module M is endotrivial relatively to the kG-module V if its endomorphism algebra Endk(M) is isomorphic, as a kG-module, to a direct sum of a trivial module and another module which is projective relatively to V , i.e. in short Endk(M) ≅ k ⊕ (V – projective). More accurately, in the first part of the text projectivity relative to kG-modules is used to define groups of relative endotrivial modules, which are obtained by replacing the notion of projectivity with that of relative projectivity in the definition of ordinary endotrivial modules. However, in order to achieve this goal we first need to develop the theory of projectivity relative to modules, in particular with respect to standard group operations such as induction, restriction and inflation. Then, for finite groups having a cyclic Sylow p-subgroup, using the structure of the group T(G) of endotrivial modules described in [MT07], we give a complete classification of the groups of relative endotrivial modules. We also study the case of groups that have a Sylow p-subgroup isomorphic to a Klein group C2 × C2, as well as the case of p-nilpotent groups. In a second part of the text, it is shown how our new groups of relative endotrivial modules provide a natural context to generalise the Dade group of a p-group P to an arbitrary finite group. The classification of endo-permutation modules and the complete description of the structure of the Dade group D(P) was completed in 2004 by S. Bouc with [Bou06]. This adventure had started about 25 years earlier with the first papers and results by E. Dade in [Dad78a] and [Dad78b] in 1978, and the final classification was in fact achieved through the non-effortless combined work of several (co)-authors between 1998 and 2004, including S. Bouc, J. Carlson, N. Mazza and J. Thévenaz. It is most interesting to note that crucial building pieces for this classification are indeed the endotrivial modules, which are particular cases of endo-permutation modules. Yet, for an arbitrary finite group G, no satisfying equivalent group structure to the Dade group on a class of kG-modules has been defined so far. With the goal to fill this gap, we turn the problem upside down, in some sense, and show how one can regard an endo-permutation module as an endotrivial module, of course not in the ordinary sense, but in the relative sense. This shall enable us to endow a set of isomorphism classes of endo-p-permutation modules with a group structure, similar to that of the Dade group. We shall call this new group, the generalised Dade group of the group G, explicitly compute its structure and show how it is closely related to that of the G-stable points of the Dade group of a Sylow p-subgroup of G.