Proper normal subgroup

This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: proper subgroup and normal subgroup
View other subgroup property conjunctions | view all subgroup properties

Definition

Symbol-free definition

A subgroup of a group is termed a proper normal subgroup if it satisfies both these conditions:

• It is a proper subgroup i.e. it is not the whole group
• It is a normal subgroup

Definition with symbols

A subgroup $H$ of a group $G$ is termed a proper normal subgroup if:

• $H \ne G$ i.e. $H$ is not the whole of $G$
• $H \triangleleft G$ i.e. $H$ is a normal subgroup of $G$

Relation with other properties

Stronger properties

• Maximal normal subgroup

Incomparable properties

• Nontrivial normal subgroup