# Conjugacy-separable group

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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 **conjugacy-separable** if it satisfies the following equivalent conditions:

- Every element in it is a conjugacy-distinguished element.
- Given any two elements in it that are not conjugate, there exists a finite quotient group where their images are also not conjugate.
- It is a residually finite group and, under the natural embedding into its profinite completion (note that the map is an embedding because it is residually finite), the group is a conjugacy-closed subgroup of the profinite completion.

## Relation with other properties

### Stronger properties

### Weaker properties

- Residually finite group:
`For full proof, refer: Conjugacy-separable implies residually finite`