This book classifies the maximal subgroups of the almost simple finite classical groups in dimension up to 12; it also describes the maximal subgroups of the almost simple finite exceptional groups with socle one of Sz(q), G2(q), 2G2(q) or 3D4(q). Theoretical and computational tools are used throughout, with downloadable Magma code provided. The exposition contains a wealth of information on the structure and action of the geometric subgroups of classical groups, but the reader will also encounter methods for analysing the structure and maximality of almost simple subgroups of almost simple groups. Additionally, this book contains detailed information on using Magma to calculate with representations over number fields and finite fields. Featured within are previously unseen results and over 80 tables describing the maximal subgroups, making this volume an essential reference for researchers. It also functions as a graduate-level textbook on finite simple groups, computational group theory and representation theory.

In this paper we describe a new method for finding generators of a centralizer of an involution. This technique works well in practice and can cope with cases when the existing method fails. As an application, we show how we can use our method to obtain a presentation of 2 2Á U 6 2.

Most of the maximal subgroups of the Monster are now known, but in many cases they are hard to calculate in. We produce explicit 'small' representations of all the maximal subgroups which are not 2-local. The representations we construct are available on the World Wide Web at http://brauer.maths. qmul.ac.uk/Atlas/.

The Hall-Paige conjecture asserts that a finite group has a complete mapping if and only if its Sylow subgroups are not cyclic. The conjecture is now proved, and one aim of this paper is to document the final step in the proof (for the sporadic simple group J 4 ).We apply this result to prove that primitive permutation groups of simple diagonal type with three or more simple factors in the socle are non-synchronizing. We also give the simpler proof that, for groups of affine type, or simple diagonal type with two socle factors, synchronization and separation are equivalent.Synchronization and separation are conditions on permutation groups which are stronger than primitivity but weaker than 2-homogeneity, the second of these being stronger than the first. Empirically it has been found that groups which are synchronizing but not separating are rather rare. It follows from our results that such groups must be primitive of almost simple type.

Define a sequence (s n ) of two-variable words in variables x, y as follows:It is shown that a finite group G is soluble if and only if s n is a law of G for all but finitely many values of n.

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