Some Exceptional Beauville Structures

We first show that every quasisimple sporadic group possesses an unmixed strongly real Beauville structure aside from the Mathieu groups M11 and M23 (and possibly 2B and M). We go on to show that no almost simple sporadic group possesses a mixed Beauville structure. We then go on to use the exceptional nature of the alternating group A6 to give a strongly real Beauville structure for this group explicitly correcting an earlier error of Fuertes and Gonzalez-Diez. In doing so we complete the classification of alternating groups that possess strongly real Beauville structures. We conclude by discussing mixed Beauville structures of the groups A6:2 and A6:2^2.Comment: v4 - case Co2 ammende

Recent Progress in the Symmetric Generation of Groups

Many groups possess highly symmetric generating sets that are naturally endowed with an underlying combinatorial structure. Such generating sets can prove to be extremely useful both theoretically in providing new existence proofs for groups and practically by providing succinct means of representing group elements. We give a survey of results obtained in the study of these symmetric generating sets. In keeping with earlier surveys on this matter, we emphasize the sporadic simple groups. ADDENDUM: This is an updated version of a survey article originally accepted for inclusion in the proceedings of the 2009 `Groups St Andrews' conference. Since the article was accepted the author has become aware of other recent work in the subject that we incorporate to provide an updated version here (the most notable addition being the contents of Section 3.4.)Comment: 14 pages, 1 figure, an updated version of a survey article accepted for the proceedings of the 2009 "Groups St Andrews" conference. v2 adds McLaughlin reference and abelian groups reference

New upper bounds on the spreads of the sporadic simple groups

We give improved upper bounds on the exact spreads of many of the larger sporadic simple groups, in some cases improving on the best known upper bound by several orders of magnitude

Recent work on Beauville surfaces, structures and groups

Beauville surfaces are a class of complex surfaces defined by letting a finite group G act on a product of Riemann surfaces. These surfaces possess many attractive geometric properties several of which are dictated by properties of the group G. In this survey we discuss the groups that may be used in this way. En route we discuss several open problems, questions and conjectures

Some examples related to Conway Groupoids

We discuss the recently introduced notion of a Conway Groupoid. In particular we consider various generalisations of the concept including infinite analogues

Symmetric Presentations of Coxeter Groups

We apply the techniques of symmetric generation to establish the standard presentations of the finite simply laced irreducible finite Coxeter groups, that is the Coxeter groups of types An, Dn and En, and show that these are naturally arrived at purely through consideration of certain natural actions of symmetric groups. We go on to use these techniques to provide explicit representations of these groups.Comment: This is the predecessor of arXiv:0901.2660v1. To appear in the Proceedings of the Edinburgh Mathematical Societ

Hypercubes as dessins d'enfant

We describe the action of the group GL2(Z) on embeddings of hypercubes on compact orientable surfaces, specifically classifying the elements of finite order that can change the genus of the underlying surface by an arbitrarily large amount. In doing so we give an explicit illustration of the kind of computations encountered in the study of dessins d'enfants in the hope that those new to the area may find such an explicit example useful