Conjugacy-separable group

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This article is about a definition in group theory that is standard among the group theory community (or sub-community that dabbles in such things) but is not very basic or common for people outside.
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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:

  1. Every element in it is a conjugacy-distinguished element.
  2. Given any two elements in it that are not conjugate, there exists a finite quotient group where their images are also not conjugate.
  3. 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.

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Facts