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Solution to the Uniformly Fully Inert Subgroups Problem for Abelian Groups

Andrey R. Chekhlov, Peter V. Danchev

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.

Solution to the Uniformly Fully Inert Subgroups Problem for Abelian Groups

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.
Paper Structure (2 sections, 4 theorems, 3 equations)

This paper contains 2 sections, 4 theorems, 3 equations.

Table of Contents

  1. Preliminaries
  2. The Solution

Key Result

Lemma 2.1

Let $G$ be a group. Then every characteristic subgroup of $G\oplus G$ is fully invariant.

Theorems & Definitions (4)

  • Lemma 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Theorem 2.4