SYMMETRIC PRESENTATIONS OF FINITE GROUPS AND RELATED TOPICS

A progenitor is an infinite semi-direct product of the form m∗n : N, where N ≤ Sn and m∗n : N is a free product of n copies of a cyclic group of order m. A progenitor of this type, in particular 2∗n : N, gives finite non-abelian simple groups and groups involving these, including alternating groups, classical groups, and the sporadic group. We have conducted a systematic search of finite homomorphic images of numerous progenitors. In this thesis we have presented original symmetric presentations of the sporadic simple groups, M12, J1 as homomorphic images of the progenitor 2∗12 : (2×A5), M22 and M22 : 2 as homomorphic images of 2∗14 : (23 : 7) and J2 as a homomorphic image of 2∗160 : PSL(2, 11). We have also given original symmetric presentations of a number of alternating and classical groups and symmetric groups such as PSL(2, 7), PSL(2, 19), PSL(2, 41), PSL(2, 8), A8, S7, and S8. We have also searched for finite homomorphic images of the monomial progenitor: 23 : 3 :m 23 : A6 and found the original symmetric presentations of the image 26 : Sym(5). We construct the following images by using our technique of double coset enumeration: 34 : (22 : S3) over (33 : (3 : 2)),J1 over (2 × A5),(24) : (S5 : 2) over S5, 73 : S3 : 2 over (21 × (S6),27 : PSL(2, 7) over PSL(2, 7), and 27 : (7 : 3) over (23 : (7 : 3). In addition, we give isomorphism class of each image that we have discovered

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

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

On a homotopy relation between the 2-local geometry and the Bouc complex for the sporadic group McL

We study the homotopy relation between the standard 2-local geometry and the Bouc complex for the sporadic finite simple group McL.Comment: 8 pages, 2 tables, final version to appear in Archiv der Mathemati

Monomial Modular Representations and Construction of the Held Group

AbstractMonomial representations of familiar finite groups over finite fields are used to construct (infinite) semi-direct products of free products of cyclic groups by groups of monomial automorphisms. Finite homomorphic images of theseprogenitorsin which the actions on the group of automorphisms and on the cyclic components are faithful are sought. The smallest non-trivial images of this type are often sporadic simple groups. The technique is demonstrated by three examples over the fieldsZ3,Z5, andZ7, which produce the Mathieu groupM11, the unitary groupU3(5):2, and the Held group, respectively