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