# Right-transitively 2-subnormal subgroup

From Groupprops

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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

## Definition

A subgroup of a group is termed **right-transitively 2-subnormal** if whenever is a 2-subnormal subgroup of , is also 2-subnormal in .

## Formalisms

### In terms of the right transiter

This property is obtained by applying the right transiter to the property: 2-subnormal subgroup

View other properties obtained by applying the right transiter

## Relation with other properties

### Stronger properties

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

base of a wreath product | base in a wreath product of groups | base of a wreath product implies right-transitively 2-subnormal | (some of the other stronger properties, like abelian normal subgroup, provide counterexamples) | |FULL LIST, MORE INFO |

hereditarily 2-subnormal subgroup | every subgroup of it is a 2-subnormal subgroup of the whole group. | (direct) | any group that is not abelian or class two, as a subgroup of itself | |FULL LIST, MORE INFO |

central subgroup | in the center | (via hereditarily 2-subnormal) | (via hereditarily 2-subnormal) | Dedekind normal subgroup, Hereditarily 2-subnormal subgroup, Subgroup of abelian normal subgroup|FULL LIST, MORE INFO |

abelian normal subgroup | normal subgroup that is also an abelian group | (via hereditarily 2-subnormal) | (via hereditarily 2-subnormal) | Dedekind normal subgroup, Hereditarily 2-subnormal subgroup, Subgroup of abelian normal subgroup|FULL LIST, MORE INFO |

subgroup of abelian normal subgroup | subgroup inside an abelian normal subgroup | (via hereditarily 2-subnormal) | (via hereditarily 2-subnormal) | Hereditarily 2-subnormal subgroup|FULL LIST, MORE INFO |

transitively normal subgroup | any normal subgroup of it is normal in the whole group | (direct) | base of a wreath product that is nontrivial gives a counterexample | |FULL LIST, MORE INFO |

direct factor | factor in an internal direct product | (via transitively normal) | (via transitively normal) | Base of a wreath product|FULL LIST, MORE INFO |

central factor | product with centralizer is whole group | (via transitively normal) | (via transitively normal) | Base of a wreath product|FULL LIST, MORE INFO |

### Weaker properties

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

right-transitively fixed-depth subnormal subgroup | there is some natural number such that any -subnormal subgroup of the subgroup is also -subnormal in the whole group. | (set ). | any subgroup of greater subnormal depth than two inside a finite p-group | |FULL LIST, MORE INFO |

2-subnormal subgroup | normal subgroup of a normal subgroup | (direct) | follows from there exist subgroups of arbitrarily large subnormal depth | |FULL LIST, MORE INFO |

## Metaproperties

### Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitiveABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity

### Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).

View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties

### Intermediate subgroup condition

YES:This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup conditionABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition