# Divisibility-closed subgroup of abelian group

From Groupprops

This article describes a property that arises as the conjunction of a subgroup property: divisibility-closed subgroup with a group property imposed on theambient 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

## Contents

## Definition

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