# Difference between revisions of "Homomorph-containing subgroup"

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(New page: {{wikilocal}} {{subgroup property}} ==Definition== A subgroup <math>H</math> of a group <math>G</math> is termed '''homomorph-containing''' if for any <math>\varphi \in \operator...) |
(→Equivalent definitions in tabular format) |
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==Definition== | ==Definition== | ||

− | A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed | + | ===Equivalent definitions in tabular format=== |

+ | |||

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

+ | ! No. !! Shorthand !! A [[subgroup]] of a [[group]] is termed homomorph-containing if ... !! A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed a homomorph-containing subgroup of <math>G</math> if ... | ||

+ | |- | ||

+ | | 1 || contains every homomorphic image || it contains any homomorphic image of itself in the whole group. || for any [[homomorphism of groups]] <math>\varphi \in \operatorname{Hom}(H,G)</math>, <math>\varphi(H) \subseteq H</math>. | ||

+ | |- | ||

+ | | 2 || homomorphism to whole group restricts to endomorphism || every homomorphism of groups from the subgroup to the whole group restricts to an endomorphism of the suubgrop. || for any [[homomorphism of groups]] <math>\varphi: H \to G</math>, <math>\varphi(H) \subseteq H</math> and the restriction of <math>\varphi</math> to <math>H</math> is an endomorphism of <math>H</math>. | ||

+ | |- | ||

+ | | 3 || (definition in terms of Hom-set maps) || (too complicated to state without symbols) || the natural map <math>\operatorname{End}(H) \to \operatorname{Hom}(H,G)</math> (by inclusion) is a surjective map of sets. | ||

+ | |} | ||

+ | |||

+ | ==Examples== | ||

+ | |||

+ | ===Extreme examples=== | ||

+ | |||

+ | * Every group is homomorph-containing as a subgroup of itself. | ||

+ | * The trivial subgroup is homomorph-containing in any group. | ||

+ | |||

+ | ===Important classes of examples=== | ||

+ | |||

+ | * [[Normal Sylow subgroup]]s and [[normal Hall subgroup]]s are homomorph-containing. | ||

+ | * Subgroups defined as the subgroup generated by elements of specific orders, are all homomorph-containing subgroups. The [[omega subgroups of a group of prime power order]] are such examples. {{further|[[Omega subgroups are homomorph-containing]]}} | ||

+ | * The [[perfect core]] of a group is a homomorph-containing subgroup. | ||

+ | |||

+ | See also the section [[#Stronger properties]] in this page. | ||

+ | |||

+ | ===Examples in small finite groups=== | ||

+ | |||

+ | {{subgroup property see examples embed|homomorph-containing subgroup}} | ||

+ | |||

+ | ==Metaproperties== | ||

+ | |||

+ | {{wikilocal-section}} | ||

+ | |||

+ | Here is a summary: | ||

+ | |||

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

+ | !Metaproperty name !! Satisfied? !! Proof !! Statement with symbols | ||

+ | |- | ||

+ | | [[satisfies metaproperty::trim subgroup property]] || Yes || || For any group <math>G</math>, both <math>G</math> (as a subgroup of itself) and the trivial subgroup of <math>G</math> are homomorph-containing subgroups of <math>G</math>. | ||

+ | |- | ||

+ | | [[dissatisfies metaproperty::transitive subgroup property]] || No || [[homomorph-containment is not transitive]] || It is possible to have groups <math>H \le K \le G</math> such that <math>H</math> is homomorph-containing in <math>K</math> and <math>K</math> is homomorph-containing in <math>G</math> but <math>H</matH> is not homomorph-containing in <math>G</math>. | ||

+ | |- | ||

+ | | [[satisfies metaproperty::intermediate subgroup condition]] || Yes || [[homomorph-containment satisfies intermediate subgroup condition]] || If <math>H \le K \le G</math> and <math>H</math> is homomorph-containing in <math>G</math>, then <math>H</math> is homomorph-containing in <math>K</math>. | ||

+ | |- | ||

+ | | [[satisfies metaproperty::strongly join-closed subgroup property]] || Yes || [[homomorph-containment is strongly join-closed]] || If <math>H_i, i \in I</math> are a collection of homomorph-containing subgroups of <math>G</math>, the [[join of subgroups]] <math>\langle H_i \rangle_{i \in I}</math> is also a homomorph-containing subgroup. | ||

+ | |- | ||

+ | | [[satisfies metaproperty::quotient-transitive subgroup property]] || Yes || [[homomorph-containment is quotient-transitive]] || If <math>H \le K \le G</math> such that <math>H</math> is homomorph-containing in <math>G</math> and <math>K/H</math> is homomorph-containing in <matH>G/H</math>, then <math>K</math> is homomorph-containing in <math>G</math>. | ||

+ | |} | ||

==Relation with other properties== | ==Relation with other properties== | ||

− | === | + | ===Stronger properties=== |

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

− | + | ! Property !! Meaning !! 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 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=== |

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

+ | ! Property !! Meaning !! 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}} | ||

+ | |} |

## Latest revision as of 14:52, 9 March 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]

## Definition

### Equivalent definitions in tabular format

No. | Shorthand | A subgroup of a group is termed homomorph-containing if ... | A subgroup of a group is termed a homomorph-containing subgroup of if ... |
---|---|---|---|

1 | contains every homomorphic image | it contains any homomorphic image of itself in the whole group. | for any homomorphism of groups , . |

2 | homomorphism to whole group restricts to endomorphism | every homomorphism of groups from the subgroup to the whole group restricts to an endomorphism of the suubgrop. | for any homomorphism of groups , and the restriction of to is an endomorphism of . |

3 | (definition in terms of Hom-set maps) | (too complicated to state without symbols) | the natural map (by inclusion) is a surjective map of sets. |

## Examples

### Extreme examples

- Every group is homomorph-containing as a subgroup of itself.
- The trivial subgroup is homomorph-containing in any group.

### Important classes of examples

- Normal Sylow subgroups and normal Hall subgroups are homomorph-containing.
- Subgroups defined as the subgroup generated by elements of specific orders, are all homomorph-containing subgroups. The omega subgroups of a group of prime power order are such examples.
`Further information: Omega subgroups are homomorph-containing` - The perfect core of a group is a homomorph-containing subgroup.

See also the section #Stronger properties in this page.

### Examples in small finite groups

Below are some examples of a proper nontrivial subgroup that satisfy the property homomorph-containing subgroup.

Below are some examples of a proper nontrivial subgroup that *does not* satisfy the property homomorph-containing subgroup.

## 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 | Statement with symbols |
---|---|---|---|

trim subgroup property | Yes | For any group , both (as a subgroup of itself) and the trivial subgroup of are homomorph-containing subgroups of . | |

transitive subgroup property | No | homomorph-containment is not transitive | It is possible to have groups such that is homomorph-containing in and is homomorph-containing in but is not homomorph-containing in . |

intermediate subgroup condition | Yes | homomorph-containment satisfies intermediate subgroup condition | If and is homomorph-containing in , then is homomorph-containing in . |

strongly join-closed subgroup property | Yes | homomorph-containment is strongly join-closed | If are a collection of homomorph-containing subgroups of , the join of subgroups is also a homomorph-containing subgroup. |

quotient-transitive subgroup property | Yes | homomorph-containment is quotient-transitive | If such that is homomorph-containing in and is homomorph-containing in , then is homomorph-containing in . |