# 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.