Equivalence of definitions of verbal subgroup

This article gives a proof/explanation of the equivalence of multiple definitions for the term verbal subgroup
View a complete list of pages giving proofs of equivalence of definitions

Statement

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

There is a collection $C_{1}$ of words such that $H$ is precisely the span of $C_{1}$ , i.e., $H$ is the union, over all words in $C_{1}$ , of the images of the corresponding word maps.
There is a collection $C_{2}$ of words such that $H$ is precisely the subgroup generated by the span of $C_{2}$ , i.e., $H$ is the subgroup generated by the union, over all words in $C_{2}$ , of the images of the corresponding word maps.
There is a subvariety ${\mathcal {V}}$ of the variety of groups such that $H$ is the unique smallest normal subgroup of $G$ for which $G/H\in {\mathcal {V}}$ .

A subgroup $H$ satisfying these equivalent conditions is termed a verbal subgroup.

Proof

(1) implies (2)

We could take $C_{2}=C_{1}$ .

(2) implies (1)

We take $C_{1}$ as the collection of all words that can be written as words in terms of the words of $C_{2}$ . This is a form of composition of words.

(2) implies (3)

We take ${\mathcal {V}}$ as the variety defined by equations saying that each word of $C_{2}$ equals the identity element. PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

(3) implies (2)

We need to use the Birkhoff-von Neumann theorem which asserts that any variety can be defined equationally. So, we find a bunch of equations describing ${\mathcal {V}}$ . Rewrite those equations by bringing everything to one side, and define $C_{2}$ as the set of the words corresponding to these equations. PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]