Subgroups of finite index and the just infinite property

TL;DR

This paper investigates conditions under which subgroups of just infinite groups inherit the just infinite property, providing characterizations for non-virtually abelian and certain virtually abelian groups.

Contribution

It offers a new description of the just infinite property for subgroups in non-virtually abelian groups using maximal subgroups, and extends results to specific virtually abelian groups.

Findings

Hereditary just infiniteness characterized by maximal subgroups

Normal subgroups' just infiniteness linked to maximal subgroup properties

Results applicable to virtually abelian pro-p groups and discrete analogues

Abstract

A residually finite (profinite) group $G$ is just infinite if every non-trivial (closed) normal subgroup of $G$ is of finite index. This paper considers the problem of determining whether a (closed) subgroup $H$ of a just infinite group is itself just infinite. If $G$ is not virtually abelian, we give a description of the just infinite property for normal subgroups in terms of maximal subgroups. In particular, we show that such a group $G$ is hereditarily just infinite if and only if all maximal subgroups of finite index are just infinite. We also obtain results for certain families of virtually abelian groups, including all virtually abelian pro-$p$ groups and their discrete analogues.