# Finite 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): finite group

View a complete list of such conjunctions

## Contents

## Definition

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

## Relation with other properties

### Stronger properties

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

Finite characteristic subgroup | the subgroup is both a finite group and a characteristic subgroup | (characteristic implies subnormal, via normal) | use normal not implies characteristic (finite examples) | Finite normal subgroup|FULL LIST, MORE INFO |

Finite normal subgroup | the subgroup is both a finite group and a normal subgroup | (normal implies subnormal) | normality is not transitive (finite examples) | |FULL LIST, MORE INFO |

Subnormal subgroup of finite group | subnormal subgroup where the whole group is a finite group | |FULL LIST, MORE INFO |

### Weaker properties

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

Finitely generated subnormal subgroup | |FULL LIST, MORE INFO | |||

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