# 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

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Definition

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

## 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 |