# Difference between revisions of "Intermediately characteristic subgroup"

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| [[Weaker than::isomorph-free subgroup]] || no other isomorphic subgroup || (obvious) || any non-co-Hopfian group, such as the [[group of integers]], as a subgroup of itself|| {{intermediate notions short|intermediately characteristic subgroup|isomorph-free subgroup}} | | [[Weaker than::isomorph-free subgroup]] || no other isomorphic subgroup || (obvious) || any non-co-Hopfian group, such as the [[group of integers]], as a subgroup of itself|| {{intermediate notions short|intermediately characteristic subgroup|isomorph-free subgroup}} | ||

|- | |- | ||

− | | [[Weaker than::isomorph-containing subgroup]] || contains | + | | [[Weaker than::isomorph-containing subgroup]] || contains every isomorphic subgroup || || || {{intermediate notions short|intermediately characteristic subgroup|isomorph-containing subgroup}} |

+ | |- | ||

+ | | [[Weaker than::homomorph-containing subgroup]] || contains every homomorphic image || || || {{intermediate notions short|intermediately characteristic subgroup|homomorph-containing subgroup}} | ||

+ | |- | ||

+ | | [[Weaker than::subisomorph-containing subgroup]] || contains every subgroup isommorphic to a subgroup of it || || || {{intermediate notions short|intermediately characteristic subgroup|subisomorph-containing subgroup}} | ||

+ | |- | ||

+ | | [[Weaker than::intermediately fully invariant subgroup]] || fully invariant in every intermediate subgroup || || || {{intermediate notions short|intermediately characteristic subgroup|intermediately fully invariant subgroup}} | ||

|- | |- | ||

| [[Weaker than::intermediately injective endomorphism-invariant subgroup]] || [[injective endomorphism-invariant subgroup|injective endomorphism-invariant]] in every intermediate subgroup || (obvious) || || {{intermediate notions short|intermediately characteristic subgroup|intermediately injective endomorphism-invariant subgroup}} | | [[Weaker than::intermediately injective endomorphism-invariant subgroup]] || [[injective endomorphism-invariant subgroup|injective endomorphism-invariant]] in every intermediate subgroup || (obvious) || || {{intermediate notions short|intermediately characteristic subgroup|intermediately injective endomorphism-invariant subgroup}} |

## Revision as of 13:47, 1 June 2020

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

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

## Definition

### Symbol-free definition

A subgroup of a group is said to be **intermediately characteristic** if it is characteristic not only in the whole group but also in every intermediate subgroup.

### Definition with symbols

A subgroup of a group is said to be **intermediately characteristic** if for any intermediate subgroup (such that ), is characteristic in .

## Formalisms

### In terms of the intermediately operator

This property is obtained by applying the intermediately operator to the property: characteristic subgroup

View other properties obtained by applying the intermediately operator

The subgroup property of being **intermediately characteristic** can be obtained by applying the intermediately operator to the subgroup property of being characteristic.

## Examples

VIEW: subgroups of groups satisfying this property | subgroups of groups dissatisfying this propertyVIEW: Related subgroup property satisfactions | Related subgroup property dissatisfactions

## Relation with other properties

### Stronger properties

### Weaker properties

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

intersection of finitely many intermediately characteristic subgroups | intersection of finitely many intermediately characteristic subgroups | (obvious) | follows from intermediate characteristicity is not finite-intersection-closed | |FULL LIST, MORE INFO |

sub-intermediately characteristic subgroup | there is a chain of subgroups from the subgroup to the whole group, with each member intermediately characteristic in its successor | (obvious)) | follows from intermediate characteristicity is not transitive | |FULL LIST, MORE INFO |

characteristic subgroup | invariant under all automorphisms | (obvious) | characteristicity does not satisfy intermediate subgroup condition | Intersection of finitely many intermediately characteristic subgroups|FULL LIST, MORE INFO |

normal subgroup | invariant under all inner automorphisms (conjugations) | (via characteristic) | (via characteristic) | Characteristic subgroup|FULL LIST, MORE INFO |

### Related properties

## 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)

Here is a summary:

Metaproperty name | Satisfied? | Proof | Difficulty level (0-5) | Statement with symbols |
---|---|---|---|---|

transitive subgroup property | No | intermediate characteristicity is not transitive | It is possible to have groups such that is intermediately characteristic in and is intermediately characteristic in but is not intermediately characteristic in . | |

quotient-transitive subgroup property | Yes | intermediate characteristicity is quotient-transitive | Suppose we have groups such that is intermediately characteristic in and is intermediately characteristic in . Then, is intermediately characteristic in . | |

intermediate subgroup condition | Yes | (obvious) | 0 | Suppose are groups such that is intermediately characteristic in . Then, is intermediately characteristic in . |

finite-intersection-closed subgroup property | No | intersection of two isomorph-free subgroups need not be intermediately characteristic | It is possible to have a group and intermediately characteristic subgroups of such that is not intermediately characteristic in (i.e., there exists containing as a non-characteristic subgroup). In fact, we can choose an example where both and are isomorph-free in . | |

strongly join-closed subgroup property | Yes | intermediate characteristicity is strongly join-closed | Suppose is a family of intermediately characteristic subgroups of a group . Then, thee join of subgroups is also intermediately characteristic in . |

## Effect of property operators

### Right transiter

It turns out that any intermediately characteristic subgroup of a transfer-closed characteristic subgroup is again intermediately characteristic. This follows from some simple reasoning and the fact that characteristicity is itself transitive. `Further information: Intermediately characteristic of transfer-closed characteristic implies intermediately characteristic`

Hence, the right transiter of the property of being intermediately characteristic is weaker than the property of being transfer-closed characteristic.