# Potentially normal-subhomomorph-containing equals normal

From Groupprops

This article gives a proof/explanation of the equivalence of multiple definitions for the term normal subgroup

View a complete list of pages giving proofs of equivalence of definitions

## Statement

The following are equal for a subgroup of a group :

- is a normal subgroup of .
- There exists a group containing such that is a Normal-subhomomorph-containing subgroup (?) of .
- There exists a group containing such that is a Normal-homomorph-containing subgroup (?) of .
- There exists a group containing such that is a Strictly characteristic subgroup (?) of .

## Proof

The same construction as used for the NPC theorem works.