SEMINAR: Groups and Combinatorics Seminar: Alex Bors 4pm Sep 27 in Weatherburn LT


Groups and Combinatorics Seminar: Alex Bors 4pm Sep 27 in Weatherburn LT 
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Speaker: Alexander Bors (University of Western Australia)
Title: Automorphism orbits and element orders in finite groups
Time and place: 4pm Friday 27 Sep 2019, Weatherburn LT
Abstract: Joint with Michael Giudici and Cheryl E. Praeger.
In contrast to other kinds of structures (such as graphs), for groups G, the assumption that the automorphism group Aut(G) acts transitively on G is not interesting to study, as only the trivial group satisfies it. Various weakenings of this condition have been proposed and studied, though. For example, in a paper from 1992, Zhang extensively studied finite groups G with the property that for every element order o in G, the action of Aut(G) on order o elements in G is transitive. He called such finite groups ATgroups. Zhang’s ideas and methods also spurred some interest in the graphtheoretic community, due to a connection with CIgroups (groups G such that any two isomorphic Cayley graphs over G are “naturally isomorphic” via an automorphism of G).
In this talk, we present results on finite groups G that are “close to being AT groups”, essentially showing that such groups are “almost soluble” (i.e., they have a soluble normal subgroup of bounded index). A finite group G is an ATgroup if and only if the numbers of Aut(G)orbits on G and of distinct element orders in G respectively are equal. Hence we measure the “closeness of G to being an ATgroup” by comparing those two numbers, considering both their difference and quotient. Along the way, we obtain a curious quantitative characterisation of the FischerGriess Monster group M.
Contact 
Stephen Glasby
<[email protected]>

Start 
Fri, 27 Sep 2019 16:00

End 
Fri, 27 Sep 2019 17:00

Submitted by 
Stephen Glasby <[email protected]>

Last Updated 
Wed, 25 Sep 2019 13:53

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