# Normal subgroup of prime order

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

## Definition

A subgroup $H$ of a group $G$ is termed a normal subgroup of prime order if $H$ is a normal subgroup of $G$ and $H$ is also a group of prime order, i.e., the order of $H$ 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