Homomorph-containing subgroup: Difference between revisions

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==Definition==
==Definition==


A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''homomorph-containing''' if for any <math>\varphi \in \operatorname{Hom}(H,G)</math>, the image <math>\varphi(H)</math> is contained in <math>H</math>.
A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''homomorph-containing''' if for any <math>\varphi \in \operatorname{Hom}(H,G)</math> (i.e., any [[homomorphism of groups]] from <math>H</math> to <math>G</math>), the image <math>\varphi(H)</math> is contained in <math>H</math>.


==Examples==
==Examples==


{{subgroup property see 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, [[normal Hall subgroup]]s, as well as subgroups defined as the subgroup generated by elements of specific orders, are all homomorph-containing subgroups. See also the section [[#Stronger properties]] in this page.
 
===Examples in small finite groups===
 
{{subgroup property see examples embed}}
 
==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==
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===Stronger properties===
===Stronger properties===


{| class="wikitable" border="1"
{| class="sortable" border="1"
! property !! quick description !!proof of implication !! proof of strictness (reverse implication failure) !! intermediate notions
! 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::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::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::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 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::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::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::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::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 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"
{| class="sortable" border="1"
! property !! quick description !!proof of implication !! proof of strictness (reverse implication failure) !! intermediate notions
! 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::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::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::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::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::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::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::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}}
| [[Stronger than::homomorph-dominating subgroup]] || every homomorphic image is contained in some conjugate subgroup || || || {{intermediate notions short|homomorph-dominating subgroup|homomorph-containing subgroup}}
|}
|}


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* The [[omega subgroups of a group of prime power order]] are homomorph-containing. {{further|[[Omega subgroups are homomorph-containing]]}}
* The [[omega subgroups of a group of prime power order]] are homomorph-containing. {{further|[[Omega subgroups are homomorph-containing]]}}
==Metaproperties==
{{wikilocal-section}}
Here is a summary:
{| class="wikitable" border="1"
!Metaproperty name !! Satisfied? !! Proof !! Section in this article
|-
| [[satisfies metaproperty::trim subgroup property]] || yes || || [[#Trimness]]
|-
| [[dissatisfies metaproperty::transitive subgroup property]] || no || [[homomorph-containment is not transitive]] || [[#Transitivity]]
|-
| [[satisfies metaproperty::intermediate subgroup condition]] || yes || [[homomorph-containment satisfies intermediate subgroup condition]] || [[#Intermediate subgroup condition]]
|-
| [[satisfies metaproperty::strongly join-closed subgroup property]] || yes || [[homomorph-containment is strongly join-closed]] || [[#Join-closedness]]
|-
| [[satisfies metaproprty::quotient-transitive subgroup property]] || yes || [[homomorph-containment is quotient-transitive]] || [[#Quotient-transitivity]]
|}
{{trim}}
For any group <math>G</math>, the trivial subgroup and the whole group are both homomorph-containing.
{{intransitive}}
We can have subgroups <math>H \le K \le G</math> such that <math>H</math> is a homomorph-containing subgroup of <math>K</math> and <math>K</math> is a homomorph-containing subgroup of <math>G</math> but <math>H</math> is not a homomorph-containing subgroup of <math>G</math>. {{proofat|[[Homomorph-containment is not transitive]]}}
{{intsubcondn}}
If <math>H \le K \le G</math> and <math>H</math> is a homomorph-containing subgroup of <math>G</math>, <math>H</math> is also a homomorph-containing subgroup of <math>K</math>. {{proofat|[[Homomorph-containment satisfies intermediate subgroup condition]]}}
{{join-closed}}
If <math>H_i, i \in I</math>, are all homomorph-containing subgroups of <math>G</math>, then so is the [[join of subgroups]] <math>\langle H_i \rangle_{i \in I}</math>. {{proofat|[[Homomorph-containment is strongly join-closed]]}}
{{quot-transitive}}
If <math>H \le K \le G</math> are groups such that <math>H</math> is a homomorph-containing subgroup of <math>G</math> and <math>K/H</math> is a homomorph-containing subgroup of <math>G/H</math>, then <math>K</math> is a homomorph-containing subgroup of <math>G</math>. {{proofat|[[Homomorph-containment is quotient-transitive]]}}

Revision as of 20:28, 8 July 2011

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 H of a group G is termed homomorph-containing if for any φ∈Hom(H,G) (i.e., any homomorphism of groups from H to G), the image φ(H) is contained in H.

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, normal Hall subgroups, as well as subgroups defined as the subgroup generated by elements of specific orders, are all homomorph-containing subgroups. 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 [[{{{1}}}]].


Below are some examples of a proper nontrivial subgroup that does not satisfy the property [[{{{1}}}]].


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 G, both G (as a subgroup of itself) and the trivial subgroup of G are homomorph-containing subgroups of G.
transitive subgroup property No homomorph-containment is not transitive It is possible to have groups H≤K≤G such that H is homomorph-containing in K and K is homomorph-containing in G but H is not homomorph-containing in G.
intermediate subgroup condition Yes homomorph-containment satisfies intermediate subgroup condition If H≤K≤G and H is homomorph-containing in G, then H is homomorph-containing in K.
strongly join-closed subgroup property Yes homomorph-containment is strongly join-closed If Hi,i∈I are a collection of homomorph-containing subgroups of G, the join of subgroups ⟨Hi⟩i∈I is also a homomorph-containing subgroup.
quotient-transitive subgroup property Yes homomorph-containment is quotient-transitive If H≤K≤G such that H is homomorph-containing in G and K/H is homomorph-containing in G/H, then K is homomorph-containing in G.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
order-containing subgroup contains every subgroup whose order divides its order order-containing implies homomorph-containing homomorph-containing not implies order-containing |FULL LIST, MORE INFO
subhomomorph-containing subgroup contains every homomorphic image of every subgroup subhomomorph-containing implies homomorph-containing homomorph-containing not implies subhomomorph-containing |FULL LIST, MORE INFO
variety-containing subgroup contains every subgroup of the whole group in the variety it generates (via subhomomorph-containing) (via subhomomorph-containing) |FULL LIST, MORE INFO
normal Sylow subgroup normal and a Sylow subgroup |FULL LIST, MORE INFO
normal Hall subgroup normal and a Hall subgroup |FULL LIST, MORE INFO
fully invariant direct factor fully invariant and a direct factor equivalence of definitions of fully invariant direct factor |FULL LIST, MORE INFO
left-transitively homomorph-containing subgroup if whole group is homomorph-containing in some group, so is the subgroup homomorph-containment is not transitive |FULL LIST, MORE INFO
right-transitively homomorph-containing subgroup any homomorph-containing subgroup of it is homomorph-containing in the whole group |FULL LIST, MORE INFO
normal subgroup having no nontrivial homomorphism to its quotient group no nontrivial homomorphism to the quotient group |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
fully invariant subgroup invariant under all endomorphisms homomorph-containing implies fully invariant fully invariant not implies homomorph-containing |FULL LIST, MORE INFO
intermediately fully invariant subgroup fully invariant in every intermediate subgroup |FULL LIST, MORE INFO
strictly characteristic subgroup invariant under all surjective endomorphisms (via fully invariant) (via fully invariant) Fully invariant subgroup|FULL LIST, MORE INFO
characteristic subgroup invariant under all automorphisms (via fully invariant) (via fully invariant) Fully invariant subgroup|FULL LIST, MORE INFO
intermediately characteristic subgroup characteristic in every intermediate subgroup (via intermediately fully invariant) (via intermediately fully invariant) |FULL LIST, MORE INFO
normal subgroup invariant under all inner automorphisms, kernel of homomorphism (via fully invariant) (via fully invariant) Characteristic subgroup, Fully invariant subgroup|FULL LIST, MORE INFO
isomorph-containing subgroup contains all isomorphic subgroups homomorph-containing implies isomorph-containing isomorph-containing not implies homomorph-containing |FULL LIST, MORE INFO
homomorph-dominating subgroup every homomorphic image is contained in some conjugate subgroup |FULL LIST, MORE INFO

Facts

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