8 Apr 2021
12:00 pm - 1:00 pm
Thursday, 8 April @ 1200 (AEST) (Melbourne)
Thursday, 8 April @ 1000 (AWST) (Perth)
Thursday, 8 April @ 1000 (CST) (Beijing, China)
Thursday, 8 April @ 0730 (IST) (New Delhi, India)
Thursday, 8 April @ 0300 (BST) (London)
Wednesday, 7 April @ 2200 (EDT) (New York)
Wednesday, 7 April @ 1900 (PDT) (Seattle)
Presenter: Emeritus Prof. Cheryl E. Praeger, University of Western Australia
Title: Coprime Actions of Finite Groups
Abstract: Suppose that, H is a finite linear group acting completely reducibly on a finite vector space V, and there are vectors a and b not fixed by H such that the H-orbits containing a and b have coprime lengths m and n. Gabriel Navarro asked whether there must be another H-orbit on vectors of length mn? We answered, by showing that the H-orbit containing a + b has length mn, and by showing, moreover, that such a group H must leave invariant some proper subspace of V – H cannot be irreducible.
Viewed differently our result speaks about finite primitive permutation groups: it tells us that for affine groups a point-stabiliser cannot have a pair of orbits of coprime lengths. On the other hand, stabilisers in different kinds of finite primitive permutation groups can have coprime orbits. Considering such groups led us to resolving a question of Peter Neumann from 1973, arising from a theorem of Marie Weiss n 1935. This is joint work with Silvio Dolfi, Bob Guralnick and Pablo Spiga.
Structure: 45 minutes seminar with 15 minutes question time
Seminar Recording & Slides: