# Normal equals strongly image-potentially characteristic

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

## Contents

## Statement

The following are equivalent for a subgroup of a group :

- is a normal subgroup of .
- is a
**strongly image-potentially characteristic subgroup**of in the following sense: there exists a group and a surjective homomorphism such that both the kernel of and are characteristic subgroups of .

## Related facts

- NPC theorem
- NRPC theorem
- Finite NPC theorem
- Finite NIPC theorem
- Fact about amalgam-characteristic subgroups: finite normal implies amalgam-characteristic, periodic normal implies amalgam-characteristic, central implies amalgam-characteristic

## Facts used

## Proof

**PLACEHOLDER FOR INFORMATION TO BE FILLED IN**: [SHOW MORE] , slight modification of NRPC theorem.