On pro-p Cappitt groups with finite exponent

TL;DR

This paper investigates the structure of pro-p Cappitt groups, proving they have finite exponent under certain conditions, their commutator subgroup is procyclic and central, and classifies pro-2 Cappitt groups of exponent 4 as Dedekind groups.

Contribution

It establishes new structural properties of pro-p Cappitt groups, including finite exponent and classification results, extending Cappitt's generalized Dedekind groups to the pro-p setting.

Findings

Non-abelian pro-p Cappitt groups with closed torsion subgroup have finite exponent.

The commutator subgroup of a pro-p Cappitt group is procyclic and central.

Pro-2 Cappitt groups of exponent 4 are pro-2 Dedekind groups.

Abstract

A pro-p Cappitt group is a pro-p group G such that the subgroup topologically generated by all non-normal closed subgroups is a proper subgroup of G. In this paper we prove that non-abelian pro-p Cappitt groups whose torsion subgroup is closed has finite exponent. We also prove that in a pro-p Cappitt group its subgroup commutator is a procyclic central subgroup. Finally we show that pro-2 Cappitt groups of exponent 4 are pro-2 Dedekind groups. These results are pro-p versions of the generalized Dedekind groups studied by Cappitt.