# Subgroup of nilpotent group

From Groupprops

This article describes a property that arises as the conjunction of a subgroup property: 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 **subgroup of nilpotent group** is used for a situation of a subgroup of a group where the whole group is a nilpotent group.

## Relation with other properties

### Stronger properties

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

subgroup of abelian group | Normal subgroup of nilpotent group|FULL LIST, MORE INFO | |||

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

### Weaker properties

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

hypernormalized subgroup | |FULL LIST, MORE INFO | |||

subnormal subgroup | nilpotent implies every subgroup is subnormal | Asymptotically fixed-depth join-transitively subnormal subgroup|FULL LIST, MORE INFO | ||

subgroup of solvable group | nilpotent implies solvable | |FULL LIST, MORE INFO |