# Group having an abelian conjugate-dense subgroup

## Contents

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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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## Definition

A group is termed a group having an abelian conjugate-dense subgroup if there exists a subgroup $H$ of $G$ such that $H$ is abelian as a group and is conjugate-dense in $G$, i.e., every element of $G$ is conjugate to an element of $H$.

Note that for a finite group, this is equivalent to being an abelian group (hence a finite abelian group) because union of all conjugates is proper and hence there is no proper conjugate-dense subgroup.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Group having a cyclic conjugate-dense subgroup