# Fully invariant subgroup of nilpotent group

From Groupprops

This article describes a property that arises as the conjunction of a subgroup property: fully invariant 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 termed a **fully invariant subgroup of nilpotent group** if the whole group is a nilpotent group and the subgroup is a fully invariant subgroup.

## Relation with other properties

### Stronger properties

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

fully invariant subgroup of abelian group | abelian implies nilpotent | follows from nilpotent not implies abelian (use the group as a subgroup of itself) | |FULL LIST, MORE INFO | |

verbal subgroup of nilpotent group | follows from verbal implies fully invariant | follows from fully invariant not implies verbal in finite abelian group | |FULL LIST, MORE INFO | |

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

verbal subgroup of abelian group | follows by combining abelian implies nilpotent and verbal implies fully invariant | follows from nilpotent not implies abelian (use the group as a subgorup of itself) | Verbal subgroup of nilpotent group|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 subgroup is a characteristic subgroup and the whole group is nilpotent | follows from fully invariant implies characteristic | follows from characteristic not implies fully invariant in finite abelian group | |FULL LIST, MORE INFO |

normal subgroup of nilpotent group | the subgroup is a normal subgroup and the whole group is nilpotent | (via characteristic) | (via characteristic) | Characteristic subgroup of nilpotent group|FULL LIST, MORE INFO |

subgroup of nilpotent group | the whole group is nilpotent | |FULL LIST, MORE INFO | ||

fully invariant subgroup | invariant under all endomorphisms | any non-nilpotent group as a subgroup of itself | |FULL LIST, MORE INFO |