On small profinite groups

TL;DR

This paper investigates properties of small and strongly complete profinite groups, proving that Frattini covers preserve smallness and extending isomorphism results for elementarily equivalent groups under strong completeness.

Contribution

It demonstrates that Frattini covers of small profinite groups are small and extends isomorphism criteria to all strongly complete profinite groups, not just finitely generated ones.

Findings

Frattini covers of small profinite groups are small.

Elementarily equivalent strongly complete profinite groups are isomorphic.

Extension of previous results to broader classes of profinite groups.

Abstract

A profinite group is called small if it has only finitely many open subgroups of index n for each positive integer n. We show that every Frattini cover of a small profinite group is small. A profinite group is called strongly complete if every subgroup of finite index is open. We show that two profinite groups that are elementarily equivalent, in the first-order language of groups, are isomorphic if one of them is strongly complete, extending a result of Moshe Jarden and Alexander Lubotzky which treats the case of finitely generated profinite groups.