Abelian conjugacy-closed subgroup

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed an Abelian conjugacy-closed subgroup if it satisfies the following equivalent conditions:


 * No two distinct elements of $$H$$ are conjugate in $$G$$.
 * $$H$$ is an Abelian group, and is conjugacy-closed as a subgroup: if any two elements of $$H$$ are conjugate in $$G$$, they are conjugate in $$H$$.

Weaker properties

 * Stronger than::Conjugacy-closed subgroup