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

Statement

The following are equivalent for a subgroup $H$ in a group $G$ :

$H$ is a normal subgroup of $G$ .
There exists a group $K$ containing $G$ such that $H$ is an endomorphism kernel in $K$ .

Proof

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

$K = G \times G / H \times G / H \times \dots$

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

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

$(a, g_{1}, g_{2}, \dots, g_{n}, \dots) \mapsto (e, π (a), g_{1}, g_{2}, \dots, g_{n - 1}, \dots)$

(where $e$ denotes the identity element).