Remarks on profinite groups having few open subgroups

TL;DR

This paper explores the properties of certain profinite groups with finitely many open subgroups of each finite index, highlighting conditions under which such groups are finite and discussing their strong completeness.

Contribution

It provides examples of non-strongly complete profinite groups with limited open subgroups and characterizes finiteness conditions based on group exponent.

Findings

Profinite groups with finitely many open subgroups of each finite index can be infinite if they lack strong completeness.

A profinite group with finite exponent and finitely many open subgroups of each index must be finite.

Discussion on the relationship between strong completeness and power subgroups in profinite groups.

Abstract

Examples are given of profinite groups that are not strongly complete, and have other `bad' properties, yet have only finitely many open subgroups of each finite index. It is shown that a profinite group with the latter property must be finite if it has finite exponent. The problem of characterizing strongly complete groups in terms of their power subgroups is discussed.