Normal iff potential endomorphism kernel

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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).