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 $G$:

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

Facts used

1. Endomorphism kernel implies normal
2. Normality satisfies intermediate subgroup condition

Proof

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

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

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

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 $\pi :G\to G/H$ is the quotient map, the mapping we are talking about is:

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

(where $e$ denotes the identity element).