# C-closed normal subgroup

From Groupprops

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

View other subgroup property conjunctions | view all subgroup properties

## Contents

## Definition

### Symbol-free definition

A subgroup of a group is termed a **c-closed normal subgroup** if it is both a c-closed subgroup (it equals the centralizer of its centralizer) and a normal subgroup (it is invariant under inner automorphisms).

## Relation with other properties

### Stronger properties

## Metaproperties

### Intersection-closedness

YES:This subgroup property is intersection-closed: an arbitrary (nonempty) intersection of subgroups with this property, also has this property.ABOUT THIS PROPERTY: View variations of this property that are intersection-closed | View variations of this property that are not intersection-closedABOUT INTERSECTION-CLOSEDNESS: View all intersection-closed subgroup properties (or, strongly intersection-closed properties) | View all subgroup properties that are not intersection-closed | Read a survey article on proving intersection-closedness | Read a survey article on disproving intersection-closedness

An arbitrary intersection of c-closed normal subgroups is c-closed normal. This follows from the corresponding facts being true for c-closed subgroups and normal subgroups individually. `For full proof, refer: c-closedness in strongly intersection-closed, normality is strongly intersection-closed`

### Centralizer-closedness

This subgroup property is centralizer-closed: the centralizer of any subgroup with this property, in the whole group, again has this property

View other centralizer-closed subgroup properties

The centralizer of a c-closed normal subgroup is again c-closed normal. This again follows from the corresponding statements for both properties. `For full proof, refer: c-closedness is centralizer-closed, normality is centralizer-closed`