Verbal subgroup of finite type

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Definition

Suppose $G$ is a group and $H$ is a subgroup of $G$. We say that $H$ is a verbal subgroup of finite type in $G$ if the following equivalent conditions hold:

1. There exists a single word $w$ in $n$ letters for some positive integer $n$ such that $H$ is the image of the word map corresponding to $w$.
2. There exists a finite collection $C$ of words $w$ (each with $n_{w}$ letters) such that $H$ is the union of the images of the word maps $G^{n_{w}}\to G$.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
finite verbal subgroup				|FULL LIST, MORE INFO
verbal subgroup of finite group				|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
verbal subgroup of finitely generated type	generated by a finite collection of words			|FULL LIST, MORE INFO
verbal subgroup	generated by a collection of words			|FULL LIST, MORE INFO
fully invariant subgroup	invariant under all endomorphisms	(via verbal)	(via verbal)	|FULL LIST, MORE INFO
purely definable subgroup	definable in the pure theory of the group			|FULL LIST, MORE INFO
characteristic subgroup	invariant under all automorphisms			|FULL LIST, MORE INFO
normal subgroup	invariant under all inner automorphisms			|FULL LIST, MORE INFO