# Divisibility-closed subgroup of abelian group

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This article describes a property that arises as the conjunction of a subgroup property: divisibility-closed 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

A subgroup $H$ of a group $G$ is termed a divisibility-closed subgroup of abelian group if $G$ is an abelian group and $H$ is a divisibility-closed subgroup of $G$, i.e., for any prime number $p$ such that $G$ is a $p$-divisible group, we have that $H$ is also a $p$-divisible group.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
completely divisibility-closed subgroup of abelian group
verbal subgroup of abelian group verbal subgroup of abelian group implies divisibility-closed

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
powering-invariant subgroup of abelian group the subgroup is a powering-invariant subgroup and the group is abelian divisibility-closed implies powering-invariant |FULL LIST, MORE INFO
divisibility-closed subgroup of nilpotent group divisibility-closed and the whole group is a nilpotent group abelian implies nilpotent nilpotent not implies abelian, use the whole group as a subgroup of itself |FULL LIST, MORE INFO
powering-invariant subgroup of nilpotent group Divisibility-closed subgroup of nilpotent group|FULL LIST, MORE INFO
divisibility-closed subgroup |FULL LIST, MORE INFO
powering-invariant subgroup |FULL LIST, MORE INFO