Classical Groups, Derangements and Primes: (Series Number 25 Australian Mathematical Society Lecture Series)

A classical theorem of Jordan states that every finite transitive permutation group contains a derangement. This existence result has interesting and unexpected applications in many areas of mathematics, including graph theory, number theory and topology. Various generalisations have been studied in more recent years, with a particular focus on the existence of derangements with special properties. Written for academic researchers and postgraduate students working in related areas of algebra, this introduction to the finite classical groups features a comprehensive account of the conjugacy and geometry of elements of prime order. The development is tailored towards the study of derangements in finite primitive classical groups; the basic problem is to determine when such a group G contains a derangement of prime order r, for each prime divisor r of the degree of G. This involves a detailed analysis of the conjugacy classes and subgroup structure of the finite classical groups.

Table of Contents:
Preface; Notational conventions; 1. Introduction; 2. Finite classical groups; 3. Conjugacy classes; 4. Subspace actions; 5. Non-subspace actions; 6. Low-dimensional classical groups; Appendix A. Number-theoretic miscellanea; Appendix B. Tables; References; Index.

About the Author :
Timothy C. Burness is a Senior Lecturer in Pure Mathematics at the University of Bristol. His main area of research is in group theory; he is interested in simple groups, both finite and algebraic, with a particular focus on subgroup structure and representation theory. He has published numerous research articles in group theory, including two monographs in the Memoirs of the American Mathematical Society. He was formerly a Junior Research Fellow at St John's College, Oxford and a Lady Davis Fellow at the Hebrew University of Jerusalem. Michael Giudici is an Associate Professor at the University of Western Australia and is currently Deputy Director of the Centre for the Mathematics of Symmetry and Computation. His main research lies in permutation groups and the objects upon which they act. He has published numerous journal articles in group theory, graph theory and finite geometry. He previously held an Australian Research Fellowship and an Australian Postdoctoral Fellowship. In 2005 he received the Kirkman Medal from the Institute of Combinatorics and its Applications.

Review :
'This book should be an indispensable reference for anybody doing research, or wanting to do research, in this area. Even for nonspecialists, however, chapters 2 and 3 should be a useful source of information on classical groups.' Mark Hunacek, The Mathematical Gazette