# 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) Fully invariant subgroup of abelian group|FULL LIST, MORE INFO
verbal subgroup Abelian verbal subgroup, Verbal subgroup of nilpotent group|FULL LIST, MORE INFO
subgroup of abelian group Abelian-potentially verbal subgroup, Characteristic subgroup of abelian group, Fully invariant 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 Characteristic subgroup of abelian group, Divisibility-closed subgroup of abelian group|FULL LIST, MORE INFO