# Normal iff potential endomorphism kernel

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 in a group :

- is a normal subgroup of .
- There exists a group containing such that is an endomorphism kernel in .

## Facts used

- Endomorphism kernel implies normal
- Normality satisfies intermediate subgroup condition

## Proof

### (2) implies (1) (the easy direction)

If (2) holds, is a normal subgroup of by Fact (1). By Fact (2), then, is also a normal subgroup of .

### (1) implies (2) (the hard direction)

Let be the external direct product (we could also take the restricted external direct product) of and a countably infinite number of copies of , i.e.,:

Identify with the first direct factor (i.e., treating it as an internal direct product locally) and with the subgroup in the first direct factor.

By construction is the kernel of the endomorphism of that sends each direct factor to the next, with the first map being the quotient map by , and the remaining maps being identity maps. Explicitly, if is the quotient map, the mapping we are talking about is:

(where denotes the identity element).