Conjugacy classes of some projective linear groups

Abstract

Given a finite set X of distinct symbols the symmetric group Sx and the alternating group Ax are obtained without further constructions. More interesting groups are contrived, however, by imposing a certain structure on the set X and observing the subgroups formed by those elements of Sx that preserve this structure. In this thesis we concern ourselves with one such imposition viz. that defining the notion of a finite projective plane. We look at the different subgroups of Sx arising in this manner, with particular emphasis on the projective linear groups and their action on the projective plane. We conclude this work with a detailed study of the structure of the projective linear groups of orders 168 and 5616, respectively. Of particular interest to us are the distinct conjugacy classes of these groups, and the manner in which they relate to one another, within each particular group.