Powering-invariant subgroup of abelian group

This article describes a property that arises as the conjunction of a subgroup property: powering-invariant 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 group and $H$ is a subgroup of $G$. We say that $H$ in $G$ is a powering-invariant subgroup of abelian group if the following equivalent conditions are satisfied:

1. $G$ is an abelian group and $H$ is a powering-invariant subgroup of $G$.
2. $G$ is an abelian group and $H$ is a quotient-powering-invariant subgroup of $G$.

Examples

Non-examples

Group of integers in group of rational numbers is the standard non-example.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
characteristic subgroup of abelian group the whole group is abelian and the subgroup is characteristic. characteristic subgroup of abelian group is powering-invariant any non-characteristic subgroup of a finite abelian group. |FULL LIST, MORE INFO
direct factor of abelian group
subgroup of finite abelian group

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
powering-invariant normal subgroup of nilpotent group follows from abelian implies nilpotent and abelian implies every subgroup is normal follows from nilpotent not implies abelian |FULL LIST, MORE INFO
powering-invariant subgroup