# Difference between revisions of "Homomorph-containing subgroup"

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===Stronger properties=== | ===Stronger properties=== | ||

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

− | + | ! property !! quick description !!proof of implication !! proof of strictness (reverse implication failure) !! intermediate notions | |

− | + | |- | |

− | + | | [[Weaker than::Order-containing subgroup]] || contains every subgroup whose order divides its order || [[order-containing implies homomorph-containing]] || [[homomorph-containing not implies order-containing]] || {{intermediate notions short|homomorph-containing subgroup|order-containing subgroup}} | |

− | + | |- | |

− | + | | [[Weaker than::Subhomomorph-containing subgroup]] || contains every homomorphic image of every nromal subgroup]] || [[subhomomorph-containing implies homomorph-containing]] || [[homomorph-containing not implies subhomomorph-containing]] | {{intermediate notions short|homomorph-containing subgroup|subhomomorph-containing subgroup}} | |

− | + | |- | |

− | + | | [[Weaker than::Variety-containing subgroup]] || contains every subgroup of the whole group in the variety it generates || (via subhomomorph-containing) || (via subhomomorph-containing) || {{intermediate notions short|homomorph-containing subgroup|variety-containing subgroup}} | |

+ | |- | ||

+ | | [[Weaker than::Normal Sylow subgroup]] || normal and a [[Sylow subgroup]] || || || {{intermediate notions short|homomorph-containing subgroup|normal Sylow subgroup}} | ||

+ | |- | ||

+ | | [[Weaker than::Normal Hall subgroup]] || normal and a [[Hall subgroup]] || || || {{intermediate notions short|homomorph-containing subgroup|normal Hall subgroup}} | ||

+ | |- | ||

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

+ | |- | ||

+ | | [[Weaker than::Left-transitively homomorph-containing subgroup]] || if whole group is homomorph-containing in some group, so is the subgroup || || [[homomorph-containment is not transitive]] || {{intermediate notions short|homomorph-containing subgroup|left-transitively homomorph-containing subgroup}} | ||

+ | |- | ||

+ | | [[Weaker than::Right-transitively homomorph-containing subgroup]] || any homomorph-containing subgroup of it is homomorph-containing in the whole group || || || {{intermediate notions short|homomorph-containing subgroup|right-transitively homomorph-containing subgroup}} | ||

+ | |- | ||

+ | | [[Weaker than::Normal subgroup having no nontrivial homomorphism to its quotient group]] || no nontrivial [[homomorphism]] to the [[quotient group]] || || || {{intermediate notions short|homomorph-containing subgroup|normal subgroup having no nontrivial homomorphism to its quotient group}} | ||

+ | |} | ||

+ | |||

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

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

− | + | ! property !! quick description !!proof of implication !! proof of strictness (reverse implication failure) !! intermediate notions | |

− | + | |- | |

− | + | | [[Stronger than::Fully invariant subgroup]] || invariant under all [[endomorphism]]s || [[homomorph-containing implies fully invariant]] || [[fully invariant not implies homomorph-containing]] || {{intermediate notions short|fully invariant subgroup|homomorph-containing subgroup}} | |

− | + | |- | |

− | + | |[[Stronger than::Intermediately fully invariant subgroup]] || fully invariant in every intermediate subgroup || || || {{intermediate notions short|intermediately fully invariant subgroup|homomorph-containing subgroup}} | |

− | + | |- | |

− | + | | [[Stronger than::Strictly characteristic subgroup]] || invariant under all [[surjective endomorphism]]s || (via fully invariant) || (via fully invariant) || {{intermediate notions short|strictly characteristic subgroup|homomorph-containing subgroup}} | |

+ | |- | ||

+ | | [[Stronger than::Characteristic subgroup]] || invariant under all [[automorphism]]s || (via fully invariant) || (via fully invariant) || {{intermediate notions short|characteristic subgroup|homomorph-containing subgroup}} | ||

+ | |- | ||

+ | | [[Stronger than::Intermediately characteristic subgroup]] || characteristic in every intermediate subgroup || (via intermediately fully invariant) || (via intermediately fully invariant) || {{intermediate notions short|intermediately characteristic subgroup|homomorph-containing subgroup}} | ||

+ | |- | ||

+ | | [[Stronger than::Normal subgroup]] || invariant under all [[inner automorphism]]s, kernel of homomorphism || (via fully invariant) || (via fully invariant) || {{intermediate notions short|normal subgroup|homomorph-containing subgroup}} | ||

+ | |- | ||

+ | | [[Stronger than::Isomorph-containing subgroup]] || contains all isomorphic subgroups || [[homomorph-containing implies isomorph-containing]] || [[isomorph-containing not implies homomorph-containing]] || {{intermediate notions short|isomorph-containing subgroup|homomorph-containing subgroup}} | ||

+ | |- | ||

+ | | [[Stronger than::Homomorph-dominating subgroup]] || every homomorphic image is contained in some conjugate subgroup || || || {{intermediate notions short|homomorph-dominating subgroup|homomorph-containing subgroup}} | ||

+ | |} | ||

==Facts== | ==Facts== |

## Revision as of 22:10, 12 November 2009

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

A subgroup of a group is termed **homomorph-containing** if for any , the image is contained in .

## 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

## Facts

- The omega subgroups of a group of prime power order are homomorph-containing.
`Further information: Omega subgroups are homomorph-containing`

## 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:

### 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

For any group , the trivial subgroup and the whole group are both homomorph-containing.

### 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

We can have subgroups such that is a homomorph-containing subgroup of and is a homomorph-containing subgroup of but is not a homomorph-containing subgroup of . `For full proof, refer: Homomorph-containment is not transitive`

### 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

If and is a homomorph-containing subgroup of , is also a homomorph-containing subgroup of . `For full proof, refer: Homomorph-containment satisfies intermediate subgroup condition`

### Join-closedness

YES:This subgroup property is join-closed: an arbitrary (nonempty) join of subgroups with this property, also has this property.ABOUT THIS PROPERTY: View variations of this property that are join-closed | View variations of this property that are not join-closedABOUT JOIN-CLOSEDNESS: View all join-closed subgroup properties (or, strongly join-closed properties) | View all subgroup properties that are not join-closed | Read a survey article on proving join-closedness | Read a survey article on disproving join-closedness

If , are all homomorph-containing subgroups of , then so is the join of subgroups . `For full proof, refer: Homomorph-containment is strongly join-closed`

### Quotient-transitivity

This subgroup property is quotient-transitive: the corresponding quotient property is transitive.

View a complete list of quotient-transitive subgroup properties

If are groups such that is a homomorph-containing subgroup of and is a homomorph-containing subgroup of , then is a homomorph-containing subgroup of . `For full proof, refer: Homomorph-containment is quotient-transitive`