# Nilpotent subnormal subgroup

From Groupprops

This article describes a property that arises as the conjunction of a subgroup property: subnormal subgroup with a group property (itself viewed as a subgroup property): nilpotent group

View a complete list of such conjunctions

## Contents

## Definition

A subgroup of a group is termed a **nilpotent subnormal subgroup** if it is nilpotent as a group and subnormal as a subgroup.

## Examples

Here are some examples of subgroups in basic/important groups satisfying the property:

Group part | Subgroup part | Quotient part | |
---|---|---|---|

A3 in S3 | Symmetric group:S3 | Cyclic group:Z3 | Cyclic group:Z2 |

Z2 in V4 | Klein four-group | Cyclic group:Z2 | Cyclic group:Z2 |

Here are some examples of subgroups in relatively less basic/important groups satisfying the property:

Here are some examples of subgroups in even more complicated/less basic groups satisfying the property:

## Relation with other properties

### Stronger properties

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

abelian subnormal subgroup | the subgroup is an abelian group and is a subnormal subgroup of the whole group | |FULL LIST, MORE INFO | ||

abelian normal subgroup | the subgroup is an abelian group and is a normal subgroup of the whole group | Abelian subnormal subgroup|FULL LIST, MORE INFO | ||

abelian characteristic subgroup | the subgroup is an abelian group and is a characteristic subgroup of the whole group | Abelian subnormal subgroup|FULL LIST, MORE INFO | ||

nilpotent normal subgroup | nilpotent and a normal subgroup | |FULL LIST, MORE INFO | ||

hereditarily normal subgroup | every subgroup of it is normal | |FULL LIST, MORE INFO | ||

nilpotent characteristic subgroup | nilpotent and a characteristic subgroup | |FULL LIST, MORE INFO |

### Weaker properties

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

subnormal subgroup | Hereditarily subnormal subgroup, Right-transitively fixed-depth subnormal subgroup|FULL LIST, MORE INFO | |||

hereditarily subnormal subgroup | |FULL LIST, MORE INFO | |||

right-transitively fixed-depth subnormal subgroup | |FULL LIST, MORE INFO | |||

solvable subnormal subgroup | |FULL LIST, MORE INFO |