Relatively projective pro-$p$ groups

TL;DR

This paper investigates the projectivity of subgroups within free pro-$p$ products, establishing a converse to a known property related to the Kurosh Subgroup Theorem for large pro-$p$ groups.

Contribution

It proves that subgroups of free pro-$p$ products are projective relative to the Kurosh family, providing a converse to a previously known result.

Findings

Subgroups of free pro-$p$ products are projective relative to the Kurosh family.

The converse of the known property is established.

Enhances understanding of subgroup structure in pro-$p$ groups.

Abstract

It is known that the Kurosh Subgroup Theorem does not hold for pro-$p$ groups of big cardinality. However, a subgroup $G$ of a free pro-$p$ product is projective relative to the Kurosh family of subgroups. In this paper we prove the converse of this fact.