Representations and characters have been proved to be very useful tools to study finite groups. I will discuss some results on the relationship between fields of values of characters of a finite group and the structure of the group, from a classical result of Burnside on real-valued characters and Navarro-Tiep's result on rational-valued characters of odd degree to very recent results of my collaborators and myself on (almost) p-rational characters.
A subset X of a group G invariably generates G if we are free to replace each element of X by an arbitrary conjugate, and we must always obtain a generating set of G. This concept was introduced by Dixon in the early nineties with motivations from computational Galois theory. We will review these motivations and their intimate connections with permutation groups. We will then present some new results concerning the probability of generating invariably a finite simple group. For instance, we will see that two random elements of a finite simple group of Lie type of bounded rank invariably generate with probability bounded away from zero.
The concept of a synchronizing permutation group was introduced nearly 15 years ago as a possible way of approaching The Černý Conjecture. Such groups must be primitive. In an attempt to understand synchronizing groups, a whole hierarchy of properties for a permutation group has been developed, namely, 2-transitive groups, ℚI-groups, spreading, separating, synchronizing, almost synchronizing and primitive. Many surprising connections with other areas of mathematics such as finite geometry, graph theory, and design theory have arisen in the study of these properties. In this survey talk I will give an overview of the hierarchy and discuss what is known about which groups lie where.
A graph is vertex-transitive if its group of automorphism acts transitively on its vertices. A very important concept in the study of these graphs is that of local action, that is, the permutation group induced by a vertex-stabilizer on the corresponding neighbourhood. I will explain some of its importance and discuss some attempts to generalize it to the case of directed graphs.
A complete mapping of a group G is a bijection f : G → G such that the map x f(x) is also a bijection. Hall and Paige conjectured in 1955 that every finite group satisfying a certain necessary condition has a complete mapping; this was proved in 2009 by Wilcox, Evans, and Bray using the classification of finite simple groups. I will discuss recent joint work with Freddie Manners and Rudi Mrazovic in which we asymptotically count complete mappings using something like the circle method.
We show that the reduction mod p of an orthogonal linear representation is orthogonal.
Given a finite group G and a set A of generators, the diameter diam(Γ(G,A)) of the Cayley graph Γ(G,A) is the smallest 𝓁 such that every element of G can be expressed as a word of length at most 𝓁 in A ⋃ A-1. We are concerned with bounding diam(G):= maxA diam(Γ(G,A)). It has long been conjectured that the diameter of the symmetric group of degree n is polynomially bounded in n. In 2011, Helfgott and Seress gave a quasipolynomial bound exp((log n)4+ε). We will discuss a recent, much simplified version of the proof.
This video is of the London Mathematical Society and European Mathematical Society‘s Joint Mathematical Weekend in 2015.
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