Verbal subgroup of abelian group

This article describes a property that arises as the conjunction of a subgroup property: verbal subgroup with a group property imposed on the ambient group: abelian group
View a complete list of such conjunctions | View a complete list of conjunctions where the group property is imposed on the subgroup

Definition

Suppose $G$ is an abelian group and $H$ is a subgroup of $G$. We say that $H$ is a verbal subgroup of abelian group if it satisfies the following equivalent conditions:

1. $H$ is a verbal subgroup of $G$.
2. There exists an integer $m$ such that $H$ is precisely the set of $m^{th}$ powers in $G$.

Equivalence of definitions

Further information: verbal subgroup equals power subgroup in abelian group

Relation with other properties

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
fully invariant subgroup of abelian group			see fully invariant not implies verbal in finite abelian group	|FULL LIST, MORE INFO
characteristic subgroup of abelian group		(via fully invariant)	(via fully invariant)	|FULL LIST, MORE INFO
verbal subgroup				|FULL LIST, MORE INFO
subgroup of abelian group				|FULL LIST, MORE INFO
divisibility-closed subgroup		verbal subgroup of abelian group implies divisibility-closed		|FULL LIST, MORE INFO
powering-invariant subgroup of abelian group				|FULL LIST, MORE INFO