Some infinite permutation groups and related finite linear groups

TL;DR

This paper explores the structure of infinite permutation groups with finite point stabilisers and their connections to finite linear groups, revealing classifications based on normal subgroup properties and actions on p-adic vector spaces.

Contribution

It generalizes the structure theory of infinite permutation groups under minimal normal subgroup conditions and links these groups to finite linear groups acting on p-adic spaces.

Findings

Classification of infinite permutation groups with finite stabilisers

Identification of groups with maximal finite normal subgroups or regular divisible abelian p-groups

Analysis of finite linear groups arising from these permutation groups

Abstract

This article began as a study of the structure of infinite permutation groups G in which point stabilisers are finite and all infinite normal subgroups are transitive. That led to two variations. One is the generalisation in which point stabilisers are merely assumed to satisfy min-N, the minimal condition on normal subgroups. The groups G are then of two kinds. Either they have a maximal finite normal subgroup, modulo which they have either one or two minimal non-trivial normal subgroups, or they have a regular normal subgroup M which is a divisible abelian p-group of finite rank. In the latter case the point stabilisers are finite and act irreducibly on a p-adic vector space associated with M. This leads to our second variation, which is a study of the finite linear groups that can arise.