- CARMA SEMINAR
- Speaker: Dr Neil Saunders, The University of Sydney
- Title: Minimal Faithful Permutation Representations of Finite Groups
- Location: Room V129, Mathematics Building (Callaghan Campus) The University of Newcastle
- Time and Date: 4:00 pm, Thu, 15th Dec 2011
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 $\mu(G \times H) < \mu(G) + \mu(H)$, for two groups $G$ and $H$. The answer to this questions 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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