A survey on Clifford-Fischer theory

Summary

Abstract

Bernd Fischer presented a powerful and interesting technique, known as Clifford- Fischer theory, for calculating the character tables of group extensions. This technique derives its fundamentals from the Clifford theory. The present article surveys the developments of Clifford-Fischer theory applied to group extensions (split and non-split) and in particular we focus on the contributions of the second author and his research groups including students.

Introduction

The character table of a finite group is a very powerful tool to study the group structure and to prove many results. Any finite group is either simple or has a non-trivial normal subgroup and hence will be of extension type (non-trivial). The classification of finite simple groups, more recent work in group theory, has been completed in 1985 and since then the researchers concentrated on the generation, subgroup structures of the finite simple groups and their automorphism groups. Few also studied the interplay between finite simple groups and combinatorial structures. A knowledge of the character table of a finite group G provides considerable information about G and hence it is of importance in the Physical Sciences as well as in Pure Mathematics. Character tables of finite groups can be constructed using various theoretical and computational techniques. The character tables of all the maximal subgroups of the sporadic simple groups are known, except for some maximal subgroups of the Monster M and the Baby Monster B. There are several well-developed methods for calculating the character tables of group extensions and in particular when the kernel of the extension is an elementary abelian group. For example, the Schreier-Sims algorithm, the Todd-Coxeter coset enumeration method, the Burnside-Dixon algorithm and various other techniques. Bernd Fischer [18, 19, 20] presented a powerful and interesting technique for calculating the character tables of group extensions. This technique, which is known as Clifford-Fischer matrices, derives its fundamentals from the Clifford theory [17].