12th meeting: Permutation Groups

University of Bristol, May 17th 2017

School of Mathematics, University Walk, lectures will take place in room SM2.

After the last talk we will go for drinks and dinner.

Owing to exceptional budget constraints we politely request that any postgraduate who will be requesting that we cover their travel expenses seek-out the cheapest means of getting here possible (typically a pre-booked train is fine).

14:00 – 15:00 Joanna Fawcett (University of Cambridge) Finite k-connected-homogeneous graphs

A graph is homogeneous if any isomorphism between finite induced subgraphs extends to an automorphism of the entire graph. The finite homogeneous graphs have been completely classified, and only a few families of examples arise. In this talk, we will discuss several methods of relaxing the hypothesis of homogeneity. In particular, we will discuss some recent progress on classifying the finite k-connected-homogeneous graphs where k>3. This is joint work with A. Devillers, C.H. Li, C.E. Praeger and J.-X. Zhou.

15:00 – 15:30 Break

15:30 – 16:30 Melissa Lee (Imperial College) Bases of quasisimple linear groups and Pyber’s conjecture

A base of a group G acting faithfully on a set  Ω is a subset B⊆Ω such that the pointwise stabiliser of B in G is trivial. The minimal base size of G is denoted by b(G).

A well-known conjecture made by Pyber in 1993 states that there is an absolute constant c such that if G acts primitively on Ω, then b(G) < clog|G|/log n, where |Ω|=n.

Following the contributions of several authors, the conjecture was finally established in 2016 by Duyan, Halasi and Maróti.

A result that played a major role in the proof of Pyber’s conjecture for primitive linear groups was given by Liebeck and Shalev, who proved that there is a constant C such that if G is a quasisimple group acting irreducibly on a finite vector space V, then either b(G)C, or G is an alternating or classical group acting on its natural module. In this talk, I will cover the history of Pyber’s conjecture, especially in the context of primitive linear groups, and present some results on the determination of the constant C for bases of quasisimple groups. I will also discuss an application by Liebeck of the latter result, which improves the known upper bounds for b(G) when G is an irreducible primitive linear group.

16:30 – 17:30 Simon Smith (University of Lincoln) The box product of two permutation groups

There are a number of ways in which one may take the product of two groups. Products which possess some kind of “universal” property (like the free and wreath products), or those which preserve some of the important properties of the input groups, are rare and precious.

Arguably, the most important product in permutation group theory is the wreath product, acting in its product action. The reason for this is that, unlike other products, it preserves a fundamental property called primitivity. Primitive permutation groups are indecomposable in some sense, and for finite groups they are the basic building blocks from which all permutation groups are comprised.

I am going to talk about a new product, called the box product. It is fundamentally different to the wreath product in product action. Nevertheless, it preserves primitivity under astonishingly similar conditions. Moreover, the box product has a “universal” property, and under natural conditions on groups M and N, the product of M and N is simple. The product can be used to easily solve a well-known open problem from topological group theory, and has an important role to play in the classification of infinite permutation groups.