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.

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A practical method is described for deciding whether or not a finite-dimensional module for a group over a finite field is reducible or not. In the reducible case, an explicit submodule is found. The method is a generalisation of the Parker-Norton 'Meataxe' algorithm, but it does not depend for its efficiency on the field being small. The principal tools involved are the calculation of the nullspace and the characteristic polynomial of a matrix over a finite field, and the factorisation of the latter. Related algorithms to determine absolute irreducibility and module isomorphism for irreducibles are also described. Details of an implementation in the GAP system, together with some performance analyses are included.

We prove that any Artin group of large type is shortlex automatic with respect to its standard generating set, and that the set of all geodesic words over the same generating set satisfies the Falsification by Fellow‐Traveller Property (FFTP) and hence is regular.

The class of co-context-free groups is studied. A co-context-free group is defined as one whose coword problem (the complement of its word problem) is context-free. This class is larger than the subclass of context-free groups, being closed under the taking of finite direct products, restricted standard wreath products with context-free top groups, and passing to finitely generated subgroups and finite index overgroups. No other examples of co-context-free groups are known. It is proved that the only examples amongst polycyclic groups or the Baumslag-Solitar groups are virtually abelian. This is done by proving that languages with certain purely arithmetical properties cannot be context-free; this result may be of independent interest.

A graph having 2 7 vertices is described, whose automorphism group is transitive on vertices and undirected edges, but not on directed edges.The object of this Note is to provide an example of a finite graphs whose automorphism group is transitive on vertices and undirected edges, but not on directed edges. In other words, the graph is edge transitive but not arc transitive. The question of the existence of such a graph was raised by Tutte on p. 60 of [ 2 ] , and he showed that it must have even valency if it exists. An infinite family of such graphs was constructed by Bouwer in [ 11, one for each valency 2n(n 2 2 ) . His smallest example has 54 vertices and valency 4. The example constructed here has 27 vertices and valency 4, and so it appears to be the smallest known example at present. It has diameter 4 and girth 5 , and it is not bipartite. I have been informed by the referee that a further example of such a graph has been constructed by P. Kornya.We adopt the following procedure. Start with a transitive permutation group on a finite set SZ, denoted by Gn. Let A be an orbit of G on SZ X SZ, other than the diagonal orbit, and let A' be the paired orbit { ( p , a) I (a, p ) E A}. Then G acts transitively on the arcs of the digraph having point set SZ and arc set A. Now let D be the digraph with point set SZ and arc set A U A', and let r be the graph obtained from D by viewing each symmetric pair of arcs {(a,p),(p,a)} as an undirected edge {a$}. If A f A', then G acts transitively on the edges of r, but not on the arcs of D , which are the directed edges of r. The problem is that this does not automatically yield an example of the type that we are seeking, because the full automorphism group of r may be larger than G, and may act transitively on the arcs of D . Indeed, this seems to happen most of the time. The smallest example that I have been able to find in which this does not occur is one in which Gn is a group of order 54 and degree 27, having a regular nonabelian subgroup. The relevant A has

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