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SEMINAR: Groups and Combinatorics Seminar 16:00 Fri 06/10/2017 Weatherburn LT: Luke Morgan

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Groups and Combinatorics Seminar 16:00 Fri 06/10/2017 Weatherburn LT: Luke Morgan Other events...
Speaker: Luke Morgan (University of Western Australia) Title: Cayley digraphs and Cayley index Time and place: 16:00 Friday 06/10/2017 in Weatherburn LT

Abstract: The Cayley index of a Cayley digraph on a finite group G is the index of (the regular representation of) G in the automorphism group of that digraph. The minimum Cayley index of one is attained by digraphs called DRRs (Digraphical Regular Representations). These are classified by Babai who shows that, apart from five groups, every finite group admits a DRR. That is, apart from five exceptions, every finite group G has a Cayley digraph such that G is the full automorphism group of that digraph. In this talk, we'll consider the question of what might make the Cayley index "large" relative to the number of vertices (= the order of the group). A result of Morris shows that any Cayley digraph on a cyclic $p$-group (with $p$ an odd prime) has Cayley index super-exponential in $p$, if there exists another distinct regular subgroup. This result was later generalised to $p=2$. So these results say that if a digraph is Cayley for two distinct $p$-groups, one of which is cyclic, the Cayley index is "large". In joint work with Morris and Verret, we considered if cyclic $p$-groups are exceptional in this respect. We found that, in contrast to the previous results, every non-cyclic abelian $p$-group ($p$ odd) of order at least $p^3$ admits a Cayley digraph of Cayley index $p$ that admits two distinct regular subgroups. I'll show how this result works, and give some further questions on this topic.

Past and future seminars may be found at http://www.maths.uwa.edu.au/~glasby/S17.html
Contact Stephen Glasby <[email protected]>
Start Fri, 06 Oct 2017 16:00
End Fri, 06 Oct 2017 17:00
Submitted by Stephen Glasby <[email protected]>
Last Updated Mon, 02 Oct 2017 17:07
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