Solution to the Uniformly Fully Inert Subgroups Problem for Abelian Groups

TL;DR

This paper proves that in any Abelian group, every uniformly fully inert subgroup is essentially equivalent to a fully invariant subgroup, confirming a long-standing conjecture.

Contribution

It provides a complete proof of the Dardano-Dikranjan-Rinauro-Salce conjecture for all Abelian groups, establishing a key structural property.

Findings

Confirmed the conjecture for all Abelian groups

Showed uniformly fully inert subgroups are commensurable with fully invariant subgroups

Resolved a significant open problem in group theory

Abstract

A famous conjecture attributed to Dardano-Dikranjan-Rinauro-Salce states that any uniformly fully inert subgroup of a given group is commensurable with a fully invariant subgroup (see, respectively, [5] and [6]). In this short note, we completely settle this problem in the affirmative for an arbitrary Abelian group.