Type: Seminar

Distribution: World

Expiry: 14 Oct 2011

Auth: athomas(.pmstaff;2039.2002)@p615.pc.maths.usyd.edu.au

There will be an Algebra Seminar on Friday 14 October, given by Neil Saunders. --------------------------------------------------------------------------- Speaker: Neil Saunders (University of Sydney) Date: Friday 14 October Time: 12:05-12:55pm Venue: Carslaw 175 Title: Minimal faithful permutation representations of finite groups Abstract: The minimal degree of a finite group G is the smallest non-negative integer n such that G embeds in 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 x 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. ---------------------------------------------------------------------------