Centralizers and conjugacy classes in finite classical groups

Abstract

Let G be a classical group defined over a finite field. The aim of this thesis is to obtain a precise and explicit solution to each of the following closely related problems:• List a representative for each conjugacy class of G.• Given x ∈ G, describe the centralizer CG(x) of x in G, by giving its group structure and a generating set.• Given x, y ∈ G, establish whether x and y are conjugate in G and, if they are, find explicit z ∈ G such that z−1xz = y. We present comprehensive theoretical solutions to all three problems. Their solution is often a necessary and vital component of algorithms for computational group theory. Hence we seek explicit solutions which can be employed widely. To achieve this outcome, we use our theoretical solutions to formulate practical algorithms to solve the problems. In parallel to our theoretical work, we have developed in Magma complete implementations of these algorithms.