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Publication# Modules d'endo-p-permutation

Abstract

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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Finite group

In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. Important examples of finite groups include cyclic groups and permutation groups. The study of finite groups has been an integral part of group theory since it arose in the 19th century.

Representation theory

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations (for example, matrix addition, matrix multiplication).

Prime number

A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4.

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