# Abelian conjugacy-closed subgroup

From Groupprops

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

## Contents

## Definition

### Definition with symbols

A subgroup of a group is termed an **Abelian conjugacy-closed subgroup** if it satisfies the following equivalent conditions:

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