RECOGNITION OF FINITE QUASI-SIMPLE GROUPS BY THE DEGREES OF THEIR IRREDUCIBLE REP- RESENTATIONS HUNG NGOC NGUYEN∗ and HUNG P. TONG-VIETy ∗Department of Mathematics, The University of Akron, Akron, Ohio 44325, United States Email:
[email protected] ySchool of Mathematics, Statistics and Computer Science, University of KwaZulu- Natal, Scottsville 3209, South Africa Email:
[email protected] 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. 1 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 information in a rep- resentation and therefore, the degrees of irreducible representations are those of the key tools 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 simply as 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).