Ben Fairbairn

A note on Beauville p-groups

We examine which p-groups of order ⩽p 6 are Beauville. We completely classify them for groups of order ⩽p 4. We also show that the proportion of 2-generated groups of order p 5 that are Beauville tends to 1 as p tends to infinity; this is not true, however, for groups of order p 6. For each prime p we determine the smallest nonabelian Beauville p-group.

Some exceptional Beauville structures

Abstract. We show that every quasisimple sporadic group apart from the Mathieu groups and is a strongly real Beauville group. We further show that none of the almost simple sporadic groups or any of the groups of the form or are mixed Beauville groups.

More on Strongly Real Beauville 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. A particularly interesting subclass are the ‘strongly real’ Beauville surfaces that have an analogue of complex conjugation defined on them. In this survey we discuss these objects and in particular the groups that may be used to define them. En route we discuss several open problems, questions and conjectures and in places make some progress made on addressing these.

Strongly real Beauville groups

A strongly real Beauville group is a Beauville group that defines a real Beauville surface. Here we discuss efforts to find examples of these groups, emphasising on the one extreme finite simple groups and on the other abelian and nilpotent groups. We will also discuss the case of characteristically simple groups and almost simple groups. En route we shall discuss several questions, open problems and conjectures as well as giving several new examples of infinite families of strongly real Beauville groups.

New examples of Mixed Beauville Groups

Abstract We generalise a construction of mixed Beauville groups first given by Bauer, Catanese and Grunewald. We go on to give several examples of infinite families of characteristically simple groups that satisfy the hypotheses of our theorem and thus provide a wealth of new examples of mixed Beauville groups.

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.

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.

A Note on Monomial Representations of Linear Groups

A matrix is said to be monomial if every row and column has only one nonzero entry. Let G be a group. A representation ρ: G → GL n (ℂ) is said to be a monomial representation of G if there exists a basis with respect to which ρ(g) is a monomial matrix for every g ∈ G. We use elementary methods to classify the irreducible monomial representations of the groups L 2(q), L 3(q) and their natural decorations.

A new infinite family of non-abelian strongly real Beauville p-groups for every odd prime p

We explicitly construct infinitely many a non-abelian strongly real Beauville

Louis Joel Mordell's time in London

p