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

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 AT-groups. Zhang’s ideas and methods also spurred some interest in the graph-theoretic community, due to a connection with CI-groups (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 AT-group 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 AT-group” by comparing those two numbers, considering both their difference and quotient. Along the way, we obtain a curious quantitative characterisation of the Fischer-Griess Monster group M.