# Normal subgroup of prime order

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): group of prime order

View a complete list of such conjunctions

## Contents

## Definition

A subgroup of a group is termed a **normal subgroup of prime order** if is a normal subgroup of and is also a group of prime order, i.e., the order of is a prime number.

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

characteristic subgroup of prime order | characteristic subgroup that is also a group of prime order | |FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

simple normal subgroup | normal subgroup that is also a simple group | |FULL LIST, MORE INFO | ||

minimal normal subgroup | nontrivial normal subgroup that does not contain any other nontrivial normal subgroup | |FULL LIST, MORE INFO | ||

cyclic normal subgroup | normal subgroup that is also a cyclic group | |FULL LIST, MORE INFO | ||

abelian normal subgroup | normal subgroup that is also an abelian group | |FULL LIST, MORE INFO |