Conjugacy-distinguished element

This article defines a property of elements in groups

Definition

An element $g$ in a group $G$ is termed conjugacy-distinguished if whenever $h$ is an element of $G$ not conjugate to $g$, there exists a normal subgroup of finite index $N\leq G$, such that the images of $g$ and $h$ under the quotient map $G\to G/N$ are not conjugate in $G/N$.

A group where every element is conjugacy-distinguished is termed a conjugacy-separable group.