On the structure of just infinite profinite groups

TL;DR

This paper characterizes just infinite profinite groups by a finiteness condition on open normal subgroups relative to open subgroups, extending previous results from pro-p groups to all profinite groups.

Contribution

It generalizes a known characterization of just infinite pro-p groups to all profinite groups, providing new insights into their structural properties.

Findings

Characterization of just infinite profinite groups via open subgroup conditions

Extension of previous pro-p group results to all profinite groups

Structural features of profinite groups related to the just infinite property

Abstract

A profinite group $G$ is just infinite if every closed normal subgroup of $G$ is of finite index. We prove that an infinite profinite group is just infinite if and only if, for every open subgroup $H$ of $G$, there are only finitely many open normal subgroups of $G$ not contained in $H$. This extends a result recently established by Barnea, Gavioli, Jaikin-Zapirain, Monti and Scoppola, who proved the same characterisation in the case of pro-$p$ groups. We also use this result to establish a number of features of the general structure of profinite groups with regard to the just infinite property.