Relatively projective pro-$p$ groups
Dan Haran, Pavel A. Zalesskii

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- groups of big cardinality. However, a subgroup of a free pro- product is projective relative to the Kurosh family of subgroups. In this paper we prove the converse of this fact.
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Taxonomy
TopicsAdvanced Topology and Set Theory · Homotopy and Cohomology in Algebraic Topology · Rings, Modules, and Algebras
