Fischer Clifford matrices and character tables of certain groups associated with simple groups O+10(2) [the simple orthogonal group of dimension 10 over GF (2)], HS and Ly.

Abstract

The character table of any finite group provides a considerable amount of information about a group and the use of character tables is of great importance in Mathematics and Physical Sciences. Most of the maximal subgroups of finite simple groups and their automorphisms are extensions of elementary abelian groups. Various techniques have been used to compute character tables, however Bernd Fischer came up with the most powerful and informative technique of calculating character tables of group extensions. This method is known as the Fischer-Clifford Theory and uses Fischer-Clifford matrices, as one of the tools, to compute character tables. This is derived from the Clifford theory. Here G is an extension of a group N by a finite group G, that is G = N.G. We then construct a non-singular matrix for each conjugacy class of G/N =G. These matrices, together with partial character tables of certain subgroups of G, known as the inertia groups, are used to compute the full character table of G. In this dissertation, we discuss Fischer-Clifford theory and apply it to both split and non-split extensions. We first, under the guidance of Dr Mpono, studied the group 27:S8 as a maximal subgroup of 27:SP(6,2), to familiarize ourselves to Fischer-Clifford theory. We then looked at 26:A8 and 28:O+8 (2) as maximal subgroups of 28:O+8 (2) and O+10(2) respectively and these were both split extensions. Split extensions have also been discussed quite extensively, for various groups, by different researchers in the past. We then turned our attention to non-split extensions. We started with 24.S6 and 25.S6 which were maximal subgroups of HS and HS:2 respectively. Except for some negative signs in the first column of the Fischer-Clifford matrices we used the Fisher-Clifford theory as it is. The Fischer-Clifford theory, is also applied to 53.L(3, 5), which is a maximal subgroup of the Lyon's group Ly. To be able to use the Fisher-Clifford theory we had to consider projective representations and characters of inertia factor groups. This is not a simple method and quite some smart computations were needed but we were able to determine the character table of 53.L(3,5). All character tables computed in this dissertation will be sent to GAP for incorporation.