Krull-Schmidt Theorem for small profinite groups

TL;DR

This paper extends the Krull-Schmidt theorem to small profinite groups, establishing their unique decomposition into indecomposables and exploring related properties like cancellation and criteria for smallness.

Contribution

It proves the Krull-Schmidt theorem for small profinite groups and investigates the cancellation property and criteria for smallness in profinite groups.

Findings

Decomposition of small profinite groups into indecomposables is unique.

Established a new criterion for smallness in profinite groups.

Analyzed cancellation properties of free pro-c groups.

Abstract

We prove that every small profinite group can be decomposed into a direct product of indecomposable profinite groups, and that such a decomposition is unique up to order and isomorphisms of the components. We also investigate the cancellation property of some free pro-$\mathcal{C}$ groups, and give a new criterion for a profinite group to be small.