Group having an abelian conjugate-dense subgroup

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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
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