University of Sydney Algebra Seminar

Neil Saunders (University of Sydney)

Friday 14 October, 12:05-12:55pm, Carslaw 175

Minimal faithful permutation representations of finite groups

The minimal degree of a finite group \(G\) is the smallest non-negative integer \(n\) such that \(G\) embeds in \(\mbox{Sym}(n)\). This defines an invariant of the group \(\mu(G)\). In this talk, I will present some interesting examples of calculating \(\mu(G)\) and examine how this invariant behaves under taking direct products and homomorphic images.

In particular, I will focus on the problem of determining the smallest degree for which we obtain a strict inequality \(\mu(G \times H) < \mu(G) + \mu(H)\), for two groups \(G\) and \(H\). The answer to this question also leads us to consider the problem of exceptional permutation groups. These are groups \(G\) that possess a normal subgroup \(N\) such that \(\mu(G/N)>\mu(G)\). They are somewhat mysterious in the sense that a particular homomorphic image becomes 'harder' to faithfully represent than the group itself. I will present some recent examples of exceptional groups and detail recent developments in the 'abelian quotients conjecture' which states that \(\mu(G/N) < \mu(G)\), whenever \(G/N\) is abelian.