Cheryl E Praeger - Coprime actions of finite groups - Pure Mathematics Colloquium
|Dates:||16 April 2021|
|Times:||11:00 - 12:00|
|What is it:||Seminar|
|Organiser:||Department of Mathematics|
|Who is it for:||University staff, External researchers|
|Speaker:||Cheryl E Praeger|
Speaker: Cheryl E Praeger, University of Western Australia
Series: Pure Mathematics Colloquium
Title: Coprime actions of finite groups
Abstract: Gabriel Navarro asked the following question: Suppose that, for a finite linear group H acting completely reducibly on a finite vector space V, 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. Is there an 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 in this situation H cannot be irreducible on V.
Viewed differently this tells us that a point-stabiliser in a finite affine primitive permutation group 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.
Cheryl E Praeger
Organisation: University of Western Australia
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