# Normal subgroup of prime index

From Groupprops

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

View other subgroup property conjunctions | view all subgroup properties

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

View other subgroup property conjunctions | view all subgroup properties

## Contents

## Definition

A **normal maximal subgroup** or **normal subgroup of prime index** is defined in the following equivalent ways:

- It is a normal subgroup that is also maximal among proper subgroups: it is not contained in any strictly bigger proper subgroup.
- It is a normal subgroup and its index is a prime number.

## Relation with other properties

### Stronger properties

- Subgroup of index two:
`For full proof, refer: Subgroup of index two is normal` - Subgroup of least prime index:
`For full proof, refer: Subgroup of least prime index is normal`

### Weaker properties

- Maximal normal subgroup: Note that the properties are equivalent for a solvable group, and more generally, for an imperfect group.
- Subgroup of prime index: Note that the properties are equivalent for a nilpotent group.
`For full proof, refer: Nilpotent implies every maximal subgroup is normal` - Maximal subgroup: Note that the properties are equivalent for a nilpotent group.
`For full proof, refer: Nilpotent implies every maximal subgroup is normal`