Metacyclic groups of odd order

Abstract

The work in this thesis was largely motivated by the aim of producing computer libraries of finite soluble primitive permutation groups with metacyclic point stabilizers. A classical result of Galois reduces the problem to that of determining all metacyclic irreducible linear groups over finite prime fields. The central topic of this thesis is a description of a theoretical approach to the problem for groups of odd order. The first part of the thesis is devoted to the determination of the abstract isomorphism types of metacyclic groups of odd order. We propose (four-generator) presentations for such groups and obtain a practical solution of the isomorphism problem for these presentations. We then proceed to investigate faithful irreducible representations of metacyclic groups of odd order. We discuss a natural correspondence between faithful irreducible representations of such a group and irreducible representations of the centre of the Fitting subgroup with core-free kernel. This produces, in principle, a solution of the linear isomorphism problem for metacyclic irreducible linear groups of odd order. We also attempt by a direct approach to determine, up to linear isomorphism, metacyclic primitive linear groups of arbitrary order over finite fields. It is expected that the results we obtained will provide a theoretical basis for a practical algorithm to list representatives of the linear isomorphism types of odd order metacyclic irreducible linear groups over finite fields.