Group having an abelian contranormal 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 G is termed a group having an abelian contranormal subgroup if there exists a subgroup H of G that is abelian as a group and is a contranormal subgroup of G, i.e., the normal closure of H in G equals G.

Relation with other properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Abelian group
Simple group
Group that is the normal closure of a singleton subset
Group with two conjugacy classes
Group having a cyclic conjugate-dense subgroup
Group having an abelian conjugate-dense subgroup