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Publication# Endo-trivial modules: a reduction to p'-central extensions

Abstract

We examine how, in prime characteristic p, the group of endotrivial modules of a finite group G and the group of endotrivial modules of a quotient of G modulo a normal subgroup of order prime to p are related. There is always an inflation map, but examples show that this map is in general not surjective. We prove that the situation is controlled by a single central extension, namely, the central extension given by a p′-representation group of the quotient of G by its largest normal p′-subgroup.

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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.

We determine all endotrivial modules in prime characteristic p, for a finite group having a cyclic Sylow p-subgroup. In other words, we describe completely the group of endotrivial modules in that case.

2007This book presents a new approach to the modular representation theory of a finite group G. Its aim is to provide a comprehensive treatment of the theory of G-algebras and to show how this theory is used to solve various problems in representation theory. Significant results have been obtained by means of this approach, which also sheds new light on modular representation theory. So it appears that a need has arisen for an expository book on the subject. I hope to meet this need and to introduce a wider audience to these new ideas.