Character Tables Of Some Groups of Extension Type

Abstract

The main aim of this mini-thesis is to give a description of some of the basic methods and techniques that have been developed to calculate the character tables of groups of extension type. We restrict our attention to split extensions G of the normal subgroup N of G by the subgroup G with the property that every irreducible character of N can be extended to an irreducible character of its inertia group in G. This is particularly true when N is abelian. We are therefore interested in this special case for which Bernd Fischer developed the theory of Fischer matrices based on the Clifford Theory, to calculate the character tables for both split and non-split extensions. Before the character table can be determined, the conjugacy classes of our group extensions are calculated using the method of coset analysis. As mentioned earlier we concentrate on examples of split extensions G in which N is always abelian, that is, either cyclic or elementary abelian. A brief outline of the classical theory of characters pertinent to this study, is followed by a detailed discussion of the Clifford theory which provides the basis for the theory of Fischer matrices. Some of the properties of these Fischer matrices which make their calculation much easier, are also given. In our final chapter, we give four examples illustrating the use of both the classical theory as well as the Fischer matrices to calculate the character tables of our examples which are all maximal subgroups of their respective groups.