Abstract

We present some recent advances in the study of the problem of recognizing finite groups by the degrees of their irreducible complex representations. We especially focus on simple groups and more generally quasi-simple groups.

Introduction

Representation theory of finite groups was originally developed to analyze groups in terms of linear transformations or matrices. A representation of degree n (where n is a positive integer) over a field F of a group is a way to represent elements in the group by n × n invertible matrices with entries in F in such a way that the rule of group operation corresponds to matrix multiplication. Degree certainly is the most important piece of information in a representation, and therefore the degrees of irreducible representations are a key tool to study the structure of finite groups.

This is an expository paper in which we survey some recent advances on the problem of recognizing finite groups by the degrees of their (complex) representations, especially for simple groups and more generally quasi-simple groups. For a finite group G, we denote the set of degrees of irreducible representations of G by cd(G) and call it the degree set of G. The multiplicity of each degree is the number of irreducible representations of that degree, and when these numbers are taken into account, we will similarly have the degree multiset of G, denoted by cd* (G).

A fundamental question in group representation theory is whether one can recover a group or some of its properties from the degrees of its irreducible representations. In the late 1980s, I.M. Isaacs [19] proved that if cd*(G) = cd*(H) and p is a prime, then G has a normal p-complement if and only if H has a normal p-complement, and therefore the nilpotency of a group is determined by its degree multiset. Later, T. Hawkes [14] provided a counterexample showing that the same assertion does not hold for super-solvability. It is still unknown whether the solvability of a finite group is determined by its degree multiset, see [30, Problem 11.8].