Pseudofinite primitive permutation groups of finite SU-rank

Abstract: A (definably) primitive permutation group (G,X) is a group G together with a transitive faithful and definable action on X such that there are no proper nontrivial (definable) G-invariant equivalence relations on X. Definably primitive permutation groups of finite Morley rank are well-studied: in particular, it is shown by Macpherson and Pillay that such a group with infinite point stabilisers is actually primitive and by Borovik and Cherlin that, given such a group (G,X), the Morley rank of G can be bounded in terms of the Morley rank of X. We show similar results for a pseudofinite definably primitive permutation group (G,X) of finite SU-rank: we first show that (G,X) is primitive if and only if the point stabilisers are infinite. This then allows us to apply a classification result by Liebeck, Macpherson and Tent on (G,X) so that we may bound the SU-rank of G in terms of the SU-rank of X. This is joint work in with Nick Ramsey.

group theorylogic

Audience: researchers in the topic

Online logic seminar

Series comments: Description: Seminar on all areas of logic