Invitation to finite groups

TL;DR

This paper provides a comprehensive introduction to finite groups, covering their structure, classification, representation theory, and applications to probability and advanced algebraic concepts.

Contribution

It offers an integrated overview of finite groups, including permutation, reflection, and unitary groups, with insights into their classification and representation theory.

Findings

Classification of complex reflection groups

Analysis of representation theory and duality

Applications to probability and analytic aspects

Abstract

This is an introduction to the finite groups, with focus on the groups of permutations and reflections, and more generally, on the finite groups of unitary matrices. We first discuss the basics of group theory, featuring the cyclic, dihedral and symmetric groups, and the structure result for finite abelian groups. Then we study the complex reflection groups, with general theory and examples, classification results, and with a look into braid groups too. We then go into the study of representation theory, and of more advanced aspects, such as Tannakian duality, Brauer theorems and Clebsch-Gordan rules. Finally, we discuss, using representation theory methods, a number of advanced analytic aspects, for the most in relation with questions coming from probability.