# Perfect normal subgroup

From Groupprops

This article describes a property that arises as the conjunction of a subgroup property: normal subgroup with a group property (itself viewed as a subgroup property): perfect group

View a complete list of such conjunctions

This article describes a property that arises as the conjunction of a subgroup property: permutable subgroup with a group property (itself viewed as a subgroup property): perfect group

View a complete list of such conjunctions

## Contents

## Definition

### Symbol-free definition

A subgroup of a group is termed a **perfect normal subgroup** if it satisfies the following equivalent conditions:

- It is perfect as a group and normal as a subgroup.
- It is perfect as a group and permutable as a subgroup.

### Definition with symbols

A subgroup of a group is termed a **perfect normal subgroup** if .

## Relation with other properties

### Stronger properties

### Weaker properties

- Subgroup whose commutator subgroup equals its intersection with whole commutator subgroup
- Subgroup whose focal subgroup equals its commutator subgroup
- Subgroup whose focal subgroup equals its intersection with the commutator subgroup
- Perfect 2-subnormal subgroup
- Perfect subnormal subgroup
- Normal subgroup contained in the perfect core