# Powering-invariant characteristic subgroup of nilpotent group

From Groupprops

This article describes a property that arises as the conjunction of a subgroup property: powering-invariant characteristic subgroup with a group property imposed on theambient group: nilpotent 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 called a **powering-invariant characteristic subgroup of nilpotent group** if is a nilpotent group and is a powering-invariant characteristic subgroup, i.e., it is both a powering-invariant subgroup and a characteristic subgroup.

## Examples

- All the members of the lower central series of a nilpotent group are powering-invariant characteristic subgroups. In fact, lower central series members are divisibility-closed in nilpotent group.
- All the members of the upper central series of a nilpotent group are powering-invariant characteristic subgroups. In fact, upper central series members are completely divisibility-closed in nilpotent group.

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

divisibility-closed characteristic subgroup of nilpotent group | the group is nilpotent and the subgroup is a divisibility-closed characteristic subgroup: it is both divisibility-closed and characteristic | follows from divisibility-closed implies powering-invariant | |FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

characteristic subgroup of nilpotent group | the group is nilpotent and the subgroup is a characteristic subgroup | characteristic not implies powering-invariant in nilpotent group | |FULL LIST, MORE INFO | |

powering-invariant normal subgroup of nilpotent group | the group is nilpotent and the subgroup is a powering-invariant normal subgroup | follows from characteristic implies normal | any proper nonzero vector subspace of a nonzero vector space over the rationals | |FULL LIST, MORE INFO |