123 research outputs found
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
Some examples related to Conway Groupoids and their generalisations
We discuss the recently introduced notion of a Conway Groupoid. In particular
we consider various generalisations of the concept including infinite analogues
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