# Clifford-Fischer theory applied to certain groups associated with symplectic, unitary and Thompson groups.

## Date

2012

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## Abstract

The character table of a finite group is a very powerful tool to study the groups and to prove
many results. Any finite group is either simple or has a normal subgroup and hence will be of
extension type. The classification of finite simple groups, more recent work in group theory, has
been completed in 1985. Researchers turned to look at the maximal subgroups and automorphism
groups of simple groups. 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. Character
tables of finite groups can be constructed using various theoretical and computational techniques.
In this thesis we study the method developed by Bernd Fischer and known nowadays as the theory
of Clifford-Fischer matrices. This method derives its fundamentals from the Clifford theory. Let
G = N·G, where N C G and G/N = G, be a group extension. For each conjugacy class [gi]G, we
construct a non-singular square matrix Fi, called a Fischer matrix. Once we have all the Fischer
matrices together with the character tables (ordinary or projective) and fusions of the inertia factor
groups into G, the character table of G is then can be constructed easily. In this thesis we apply
the coset analysis technique (this is a method to find the conjugacy classes of group extensions)
together with theory of Clifford-Fischer matrices to calculate the ordinary character tables of seven
groups of extensions type, in which four are non-split and three are split extensions. These groups
are of the forms: 21+8
+
·A9, 37:Sp(6, 2), 26·Sp(6, 2), 25·GL(5, 2), 210:(U5(2):2), 21+6
− :((31+2:8):2)
and 22n·Sp(2n, 2) and 28·Sp(8, 2). In addition we give some general results on the non-split group 22n·Sp(2n, 2).

## Description

Thesis (Ph.D.)-University of KwaZulu-Natal, Pietermaritzburg, 2012.

## Keywords

Finite groups., Linear algebraic groups., Theses--Mathematics.