# Difference between revisions of "Finite subnormal subgroup"

From Groupprops

(New page: {{group-subgroup property conjunction|subnormal subgroup|finite group}} ==Definition== A subgroup of a group is termed a '''finite subnormal subgroup''' if it is [[finite group|f...) |
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===Stronger properties=== | ===Stronger properties=== | ||

− | + | {| class="sortable" border="1" | |

− | + | ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |

+ | |- | ||

+ | | [[Weaker than::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) || {{intermediate notions short|finite subnormal subgroup|finite characteristic subgroup}} | ||

+ | |- | ||

+ | | [[Weaker than::Finite normal subgroup]] || the subgroup is both a [[finite group]] and a [[normal subgroup]] || (normal implies subnormal) || [[normality is not transitive]] (finite examples) || {{intermediate notions short|finite subnormal subgroup|finite normal subgroup}} | ||

+ | |- | ||

+ | | [[Weaker than::Subnormal subgroup of finite group]] || [[subnormal subgroup]] where the whole group is a [[finite group]] || || || {{intermediate notions short|finite subnormal subgroup|subnormal subgroup of finite group}} | ||

+ | |} | ||

===Weaker properties=== | ===Weaker properties=== | ||

− | + | {| class="sortable" border="1" | |

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

+ | |- | ||

+ | | [[Stronger than::Finitely generated subnormal subgroup]] || || || || {{intermediate notions short|finitely generated subnormal subgroup|finite subnormal subgroup}} | ||

+ | |- | ||

+ | | [[Stronger than::Subnormal subgroup]] || || || || {{intermediate notions short|subnormal subgroup|finite subnormal subgroup}} | ||

+ | |} | ||

===Related properties=== | ===Related properties=== | ||

− | * [[ | + | * [[Subnormal subgroup of finite index]] |

+ | |||

+ | ==Facts== | ||

+ | |||

+ | * [[Join of two finite subnormal subgroups is subnormal]] |

## Latest revision as of 22:33, 12 May 2010

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 |