Algebra Seminar: Saunders -- Minimal faithful permutation representations of finite groups

There will be an Algebra Seminar on Friday 14 October, given by Neil Saunders.  

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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.  

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