Divisibility-closed subgroup of abelian group

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				|FULL LIST, MORE INFO
divisibility-closed subgroup				|FULL LIST, MORE INFO
powering-invariant subgroup				|FULL LIST, MORE INFO