Group having an abelian conjugate-dense subgroup

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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
Group with two conjugacy classes				|FULL LIST, MORE INFO
Abelian group				|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Group having an abelian contranormal subgroup