# Conjugacy-closed characteristic subgroup

From Groupprops

This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: conjugacy-closed subgroup and characteristic subgroup

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## Contents

## Definition

A subgroup of a group is termed a **conjugacy-closed characteristic subgroup** if it is both a conjugacy-closed subgroup (i.e., any two elements of the subgroup that are conjugate in the whole group are conjugate in the subgroup) and a characteristic subgroup.

## Relation with other properties

### Stronger properties

### Weaker properties

- Conjugacy-closed normal subgroup
- Transitively normal subgroup
- Characteristic subgroup
- Characteristic transitively normal subgroup

## Metaproperties

### Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitiveABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity

### Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).

View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties