# Abelian conjugacy-closed subgroup

This article describes a property that arises as the conjunction of a subgroup property: conjugacy-closed subgroup with a group property (itself viewed as a subgroup property): Abelian group
View a complete list of such conjunctions

## Definition

### 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 $H$.