# Characteristic subgroup of nilpotent group

From Groupprops

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

The term **characteristic subgroup of nilpotent group** is used for a situation where the whole group is a nilpotent group and the subgroup is a characteristic subgroup.

## Relation with other properties

### Stronger properties

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

characteristic subgroup of abelian group | follows from abelian implies nilpotent | follows from nilpotent not implies abelian | |FULL LIST, MORE INFO | |

characteristic direct factor of nilpotent group | Fully invariant subgroup of nilpotent group|FULL LIST, MORE INFO | |||

fully invariant subgroup of nilpotent group | |FULL LIST, MORE INFO | |||

verbal subgroup of nilpotent group | Fully invariant subgroup of nilpotent group|FULL LIST, MORE INFO | |||

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

### Weaker properties

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

normal subgroup of nilpotent group | |FULL LIST, MORE INFO | |||

nilpotent characteristic subgroup | |FULL LIST, MORE INFO |