# Powering-invariant subgroup of abelian group

From Groupprops

This article describes a property that arises as the conjunction of a subgroup property: powering-invariant 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

Suppose is an group and is a subgroup of . We say that in is a **powering-invariant subgroup of abelian group** if the following equivalent conditions are satisfied:

- is an abelian group and is a powering-invariant subgroup of .
- is an abelian group and is a quotient-powering-invariant subgroup of .

## 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 |