# Difference between revisions of "Isomorph-containing subgroup"

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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::isomorph-free subgroup]] || no other isomorphic subgroup (for a finite subgroup, and more generally, for a [[co-Hopfian group|co-Hopfian]] subgroup, the two properties are equivalent) || obvious || any example of a non-co-Hopfian group as a subgroup of itself -- such as the [[group of integers]] | |

− | + | |- | |

− | + | | [[Weaker than::homomorph-containing subgroup]] || contains any homomorphic image of itself in the group || obvious; isomorphs are also homomorphic images || [[cyclic maximal subgroup of dihedral group:D8]] is isomorph-containing but not homomorph-containing || {{intermediate notions short|isomorph-containing subgroup|homomorph-containing subgroup}} | |

+ | |- | ||

+ | | [[Weaker than::Subhomomorph-containing subgroup]] || contains any homomorphic image in the whole group of any subgroup of it || (via homomorph-containing) || (via homomorph-containing) || {{intermediate notions short|isomorph-containing subgroup|subhomomorph-containing subgroup}} | ||

+ | |- | ||

+ | | [[Weaker than::subisomorph-containing subgroup]] || contains any subgroup of the whole group isomorphic to any subgroup of itself || (obvious) || [[cyclic maximal subgroup of dihedral group:D8]] is isomorph-containing but not subisomorph-containing || {{intermediate notions short|isomorph-containing subgroup|subisomorph-containing subgroup}} | ||

+ | |- | ||

+ | | [[Weaker than::variety-containing subgroup]] || contains any subgroup of the whole group in the subvariety of the variety of groups generated by it || (obvious) || (via either homomorph-containing or subisomorph-containing) || {{intermediate notions short|isomorph-containing subgroup|variety-containing subgroup}} | ||

+ | |- | ||

+ | | [[Weaker than::fully invariant direct factor]] || both a [[fully invariant subgroup]] and a [[direct factor]] || [[equivalence of definitions of fully invariant direct factor]] || || {{intermediate notions short|isomorph-containing subgroup|fully invariant direct factor}} | ||

+ | |} | ||

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

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

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

− | + | |- | |

− | + | | [[Stronger than::characteristic subgroup]] || invariant under any automorphism of the whole group || [[isomorph-containing implies characteristic]] || [[characteristic not implies isomorph-containing]] || {{intermediate notions short|characteristic subgroup|isomorph-containing subgroup}} | |

− | + | |- | |

− | + | | [[Stronger than::injective endomorphism-invariant subgroup]] || invariant under any injective endomorphism of the whole group || || Can use same example as for characteristic not implies isomorph-containing, if finite || {{intermediate notions short|injective endomorphism-invariant subgroup|isomorph-containing subgroup}} | |

+ | |||

+ | | [[Stronger than::intermediately injective endomorphism-invariant subgroup]] || injective endomorphism-invariant in every intermediate subgroup || || || {{intermediate notions short|intermediately injective endomorphism-invariant subgroup|isomorph-containing subgroup}} | ||

+ | |- | ||

+ | | [[Stronger than::intermediately characteristic subgroup]] || characteristic in every intermediate subgroup || || || {{intermediate notions short|intermediately characteristic subgroup|isomorph-containing subgroup}} | ||

+ | |- | ||

+ | | [[Stronger than::normal subgroup]] || invariant under all inner automorphisms || || || {{intermediate notions short|normal subgroup|isomorph-containing subgroup}} | ||

+ | |- | ||

+ | | [[Stronger than::normal-isomorph-containing subgroup]] || contains any normal subgroup of the whole group isomorphic to it || (obvious) || any proper nontrivial subgroup of a finite simple non-abelian group || {{intermediate notions short|normal-isomorph-containing subgroup|isomorph-containing subgroup}} | ||

+ | |} | ||

==Metaproperties== | ==Metaproperties== |

## Revision as of 00:31, 18 March 2019

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]

## Definition

QUICK PHRASES: contains all isomorphic subgroups, weakly closed in any ambient group

A subgroup of a group is termed an **isomorph-containing subgroup** if it satisfies the following equivalent conditions:

- Whenever is a subgroup of isomorphic to , .
- If is a subgroup of , is weakly closed in with respect to .

### Equivalence of definitions

`Further information: Isomorph-containing iff weakly closed in any ambient group`

## 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

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

isomorph-free subgroup | no other isomorphic subgroup (for a finite subgroup, and more generally, for a co-Hopfian subgroup, the two properties are equivalent) | obvious | any example of a non-co-Hopfian group as a subgroup of itself -- such as the group of integers | |

homomorph-containing subgroup | contains any homomorphic image of itself in the group | obvious; isomorphs are also homomorphic images | cyclic maximal subgroup of dihedral group:D8 is isomorph-containing but not homomorph-containing | |FULL LIST, MORE INFO |

Subhomomorph-containing subgroup | contains any homomorphic image in the whole group of any subgroup of it | (via homomorph-containing) | (via homomorph-containing) | Homomorph-containing subgroup, Right-transitively isomorph-containing subgroup, Subisomorph-containing subgroup|FULL LIST, MORE INFO |

subisomorph-containing subgroup | contains any subgroup of the whole group isomorphic to any subgroup of itself | (obvious) | cyclic maximal subgroup of dihedral group:D8 is isomorph-containing but not subisomorph-containing | Right-transitively isomorph-containing subgroup|FULL LIST, MORE INFO |

variety-containing subgroup | contains any subgroup of the whole group in the subvariety of the variety of groups generated by it | (obvious) | (via either homomorph-containing or subisomorph-containing) | Homomorph-containing subgroup, Right-transitively isomorph-containing subgroup, Subisomorph-containing subgroup|FULL LIST, MORE INFO |

fully invariant direct factor | both a fully invariant subgroup and a direct factor | equivalence of definitions of fully invariant direct factor | Complemented homomorph-containing subgroup, Complemented isomorph-containing subgroup, Homomorph-containing subgroup|FULL LIST, MORE INFO |

### Weaker properties

## Metaproperties

### Transitivity

NO:This subgroup property isnottransitive: a subgroup with this property in a subgroup with this property, need not have the property in the whole groupABOUT 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 subgroup properties that are not transitive|View facts related to transitivity of subgroup properties | View a survey article on disproving transitivity

`For full proof, refer: Isomorph-containment is not transitive`

### 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