# Proper normal subgroup

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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 of a group is termed a **proper normal subgroup** if:

- i.e. is not the whole of
- i.e. is a normal subgroup of