Fischer-Clifford theory for split and non-split group extensions.

Abstract

The character table of a finite group provides considerable amount of information about the group, and hence is of great importance in Mathematics as well as in Physical Sciences. Most of the maximal subgroups of the finite simple groups and their automorphisms are of extensions of elementary abelian groups, so methods have been developed for calculating the character tables of extensions of elementary abelian groups. Character tables of finite groups can be constructed using various techniques. However Bernd Fischer presented a powerful and interesting technique for calculating the character tables of group extensions. This technique, which is known as the technique of the Fischer-Clifford matrices, derives its fundamentals from the Clifford theory. If G=N.G is an appropriate extension of N by G, the method involves the construction of a nonsingular matrix for each conjugacy class of G/N~G. The character table of G can then be determined from these Fischer-Clifford matrices and the character table of certain subgroups of G, called inertia factor groups. In this dissertation, we described the Fischer-Clifford theory and apply it to both split and non-split group extensions. First we apply the technique to the split extensions 2,7:Sp6(2) and 2,8:SP6(2) which are maximal subgroups of Sp8(2) and 2,8:08+(2) respectively. This technique has also been discussed and used by many other researchers, but applied only to split extensions or to the case when every irreducible character of N can be extended to an irreducible character of its inertia group in G. However the same method can not be used to construct character tables of certain non-split group extensions. In particular, it can not be applied to the non-split extensions of the forms 3,7.07(3) and 3,7.(0,7(3):2) which are maximal subgroups of Fischer's largest sporadic simple group Fi~24 and its automorphism group Fi24 respectively. In an attempt to generalize these methods to such type of non-split group extensions, we need to consider the projective representations and characters. We have shown that how the technique of Fischer-Clifford matrices can be applied to any such type of non-split extensions. However in order to apply this technique, the projective characters of the inertia factors must be known and these can be difficult to determine for some groups. We successfully applied the technique of Fischer-Clifford matrices and determined the Fischer-Clifford matrices and hence the character tables of the non-split extensions 3,7.0,7(3) and 3,7.(0,7(3):2). The character tables computed in this thesis have been accepted for incorporation into GAP and will be available in the latest versions.