# Transitively normal subgroup

From Groupprops

BEWARE!This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

## Definition

### Equivalent definitions in tabular format

No. | Shorthand | A subgroup of a group is termed transitively normal if ... | A subgroup of a group is termed transitively normal if ... |
---|---|---|---|

1 | transitively normal | every normal subgroup of the subgroup is normal in the whole group. | whenever is a normal subgroup of , is also a normal subgroup of . |

2 | normal automorphisms restriction | every normal automorphism of the whole group restricts to a normal automorphism of the subgroup. | for any normal automorphism of , the restriction of to is a normal automorphism of . |

3 | normal CEP-subgroup | it is a normal subgroup as well as a CEP-subgroup: every normal subgroup of it is an intersection with it of a normal subgroup of the whole group | is a normal subgroup of and for any normal subgroup of there exists a normal subgroup of such that . |

### Equivalence of definitions

`Further information: Equivalence of definitions of transitively normal subgroup`

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]

This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: normal subgroup and CEP-subgroup

View other subgroup property conjunctions | view all subgroup properties

This is a variation of normality|Find other variations of normality | Read a survey article on varying normality

CAUTIONARY NOTE: There is a paper where the termtransitively normalis used for what we call intermediately subnormal-to-normal subgroup

## Examples

### Extreme examples

- Every group is transitively normal as a subgroup of itself.
- The trivial subgroup is transitively normal in any group.

### Examples in small finite groups

Below are some examples of a proper nontrivial subgroup that satisfy the property transitively normal subgroup.

Below are some examples of a proper nontrivial subgroup that *does not* satisfy the property transitively normal subgroup.

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

Klein four-subgroups of dihedral group:D8 | Dihedral group:D8 | Klein four-group | Cyclic group:Z2 |

## Metaproperties

BEWARE!This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)

Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|

transitive subgroup property | Yes | transitive normality is transitive | If are groups such that is transitively normal in and is transitively normal in , then is transitively normal in . |

intermediate subgroup condition | Yes | transitive normality satisfies intermediate subgroup condition | If and is transitively normal in , then is transitively normal in . |

finite-intersection-closed subgroup property | No | transitive normality is not finite-intersection-closed | It is possible to have a group and transitively normal subgroups of such that is not transitively normal in . |

finite-join-closed subgroup property | No | transitive normality is not finite-join-closed | It is possible to have a group and transitively normal subgroups of such that the join (which is the same as the product ) is not transitively normal. |

centralizer-closed subgroup property | No | transitive normality is not centralizer-closed | It is possible to have a group and a transitively normal subgroup of such that the centralizer is not transitively normal. |

image condition | Yes | transitive normality satisfies image condition | If is a transitively normal subgroup of and is a surjective homomorphism, then is a transitively normal subgroup of . |

quotient-transitive subgroup property | No | transitive normality is not quotient-transitive | It is possible to have groups such that is transitively normal in and is transitively normal in , but is not transitively normal in . |

## Relation with other properties

### Conjunction with other properties

Here are some conjunctions with other subgroup properties:

Conjunction | Other component of conjunction | Intermediate notions | Additional comments |
---|---|---|---|

characteristic transitively normal subgroup | characteristic subgroup | |FULL LIST, MORE INFO | |

c-closed transitively normal subgroup | c-closed subgroup (equals its double centralizer) | |FULL LIST, MORE INFO |

Conjunctions with group properties:

Conjunction | Property of the group | Intermediate notions | Additional comments |
---|---|---|---|

cyclic normal subgroup | Cyclic group | Abelian hereditarily normal subgroup, Hereditarily normal subgroup|FULL LIST, MORE INFO | for a cyclic subgroup, being normal is equivalent to being transitively normal. |

abelian hereditarily normal subgroup | Abelian group | Hereditarily normal subgroup|FULL LIST, MORE INFO | for an abelian subgroup, being transitively normal is equivalent to being hereditarily normal. |

hereditarily normal subgroup | Dedekind group (every subgroup is normal) | SCAB-subgroup|FULL LIST, MORE INFO | |

nilpotent transitively normal subgroup | Nilpotent group | |FULL LIST, MORE INFO |

### Stronger properties

### Weaker properties

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

normal subgroup | (by definition) | normality is not transitive | |FULL LIST, MORE INFO | |

CEP-subgroup | every normal subgroup of it arises as its intersection with a normal subgroup of whole group | |FULL LIST, MORE INFO | ||

subgroup whose commutator with any subset is normal | its commutator with any subset of whole group is normal in whole group | commutator of a transitively normal subgroup and a subset implies normal | commutator with any subset is normal not implies transitively normal | |FULL LIST, MORE INFO |

## Formalisms

BEWARE!This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)

### Function restriction expression

This subgroup property is a function restriction-expressible subgroup property: it can be expressed by means of the function restriction formalism, viz there is a function restriction expression for it.

Find other function restriction-expressible subgroup properties | View the function restriction formalism chart for a graphic placement of this property

Function restriction expression | is a transitively normal subgroup of if ... | This means that transitive normality is ... | Additional comments |
---|---|---|---|

normal automorphism normal automorphism | every normal automorphism of restricts to a normal automorphism of | the balanced subgroup property for normal automorphisms | Hence, it is a t.i. subgroup property, both transitive and identity-true |

inner automorphism normal automorphism | every inner automorphism of restricts to a normal automorphism of | ||

class-preserving automorphism normal automorphism | every class-preserving automorphism of restricts to a normal automorphism of | ||

subgroup-conjugating automorphism normal automorphism | every subgroup-conjugating automorphism of restricts to a normal automorphism of | ||

strong monomial automorphism normal automorphism | every strong monomial automorphism of restricts to a normal automorphism of |