# 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

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 conjugate-dense subgroup** if there exists a subgroup of such that is abelian as a group and is conjugate-dense in , i.e., every element of is conjugate to an element of .

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 |