Groupes finis

These informal notes concern some basic themes of harmonic analysis related to representations of groups.

This article adapts the results of arXiv:1305.6557 to the ultrametric setting. This is a re-edition of the Chapter V of author thesis.

The main goal of this paper is to provide a group theoretical generalization of the well-known Euler's totient function. This determines an interesting class of finite groups.

This is a survey article on the theory of finite complex reflection groups. No proofs are given but numerous references are included.

In this Note we study the groups $G$ satisfying condition $(\mathcal{N},n)$, that is, every subset of $G$ with $n+1$ elements contains a pair $\{x,y\}$ such that the subgroup $<x,y>$ is nilpotent.

Group theory is a particularly fertile field for the design of practical algorithms. Algorithms have been developed across the various branches of the subject and they find wide application. Because of its relative maturity, computational group theory may be used to gain insight into the general structure of algebraic algorithms. This paper examines the basic ideas behind some of the more important algorithms for finitely presented groups and permutation groups, and surveys r...

These are lecture notes that arose from a representation theory course given by the first author to the remaining six authors in March 2004 within the framework of the Clay Mathematics Institute Research Academy for high school students, and its extended version given by the first author to MIT undergraduate math students in the Fall of 2008. The notes cover a number of standard topics in representation theory of groups, Lie algebras, and quivers, and contain many problems an...

For the first time we represent every finite group in the form of a graph in this book. The authors choose to call these graphs as identity graph, since the main role in obtaining the graph is played by the identity element of the group. This study is innovative because through this description one can immediately look at the graph and say the number of elements in the group G which are self-inversed. Also study of different properties, like the subgroups of a group, normal...

This is a transcript of a lecture course on Infinite Permutation Groups given by Peter M. Neumann (1940-2020) in Oxford during the academic year 1988-1989. The field of Infinite Permutation Groups only emerged as an independent field of study in the 1980's. Most of the results described in these notes were at the time of the lectures brand new and had either just recently appeared in print or had not appeared formally. A large part of the results described is either due to Pe...

Let $ H $ be a subgroup of a finite group $ G $. We say that $ H $ satisfies the $ \Pi $-property in $ G $ if for any chief factor $ L / K $ of $ G $, $ |G/K : N_{G/K}(HK/K\cap L/K )| $ is a $ \pi (HK/K\cap L/K) $-number. In this paper, we obtain some criteria for the $ p $-supersolubility or $ p $-nilpotency of a finite group and extend some known results by concerning some subgroups that satisfy the $ \Pi $-property.