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Homomorph-containing subgroup

4,163 bytes added, 20:39, 8 July 2011
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>.
{{subgroup property see ===Extreme examples}}===
==Relation with other properties==* Every group is homomorph-containing as a subgroup of itself.* The trivial subgroup is homomorph-containing in any group.
===Stronger propertiesImportant classes of examples===
* [[Weaker than::Order-containing Normal Sylow subgroup]]* s and [[Weaker than::Subhomomorph-containing normal Hall subgroup]]* [[Weaker than::Varietys are homomorph-containing subgroup]].* [[Weaker than::Normal Sylow Subgroups defined as the subgroup]]* generated by elements of specific orders, are all homomorph-containing subgroups. The [[Weaker than::Normal Hall subgroupomega subgroups of a group of prime power order]]* are such examples. {{further|[[Weaker than::Fully invariant direct factor]]* [[Weaker than::Left-transitively Omega subgroups are homomorph-containing subgroup]]}}* The [[Weaker than::Right-transitively perfect core]] of a group is a homomorph-containing subgroup]]===Weaker properties===.
* See also the section [[#Stronger than::Fully invariant subgroup]]: Also related:** [[Stronger than::Intermediately fully invariant subgroup]]** [[Stronger than::Strictly characteristic subgroup]]** [[Stronger than::Intermediately characteristic subgroup]]** [[Stronger than::Characteristic subgroup]]** [[Stronger than::Normal subgroup]]* [[Stronger than::Isomorph-containing subgroup]]* [[Stronger than::Homomorph-dominating subgroupproperties]]in this page.
==Facts=Examples in small finite groups===
* The [[omega subgroups of a group of prime power order]] are homomorph-containing. {{furthersubgroup property see examples embed|[[Omega subgroups are homomorph-containing]]subgroup}}
Here is a summary:
{| class="wikitablesortable" border="1"!Metaproperty name !! Satisfied? !! Proof !! Section in this articleStatement with symbols
| [[satisfies metaproperty::trim subgroup property]] || yes Yes || || [[#Trimness]]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 No || [[homomorph-containment is not transitive]] || [[#Transitivity]]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 Yes || [[homomorph-containment satisfies intermediate subgroup condition]] || [[#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 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-closednessjoin of subgroups]]<math>\langle H_i \rangle_{i \in I}</math> is also a homomorph-containing subgroup.
| [[satisfies metaproprtymetaproperty::quotient-transitive subgroup property]] || yes Yes || [[homomorph-containment is quotient-transitive]] || [[#QuotientIf <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-transitivity]]containing in <math>G</math>.
For any group <math>G</math>, the trivial subgroup and the whole group are both homomorph-containing.==Relation with other properties==
{{intransitive}}===Stronger properties===
We can have subgroups <math>H \le K \le G</math> such that <math>H</math> is a {| 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 <math>K</math> 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 <math>K</math> is a [[Sylow subgroup]] || || || {{intermediate notions short|homomorph-containing subgroup of <math>G</math> but <math>H</math> is not |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 <math>G</math>. definitions of fully invariant direct factor]] || || {{proofatintermediate notions short|homomorph-containing subgroup|fully invariant direct factor}}|-|[[HomomorphWeaker 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}}|}
{{intsubcondn}}===Weaker properties===
If <math>H \le K \le G</math> and <math>H</math> is a {| 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 of <math>G</math>, <math>H</math> is also a |homomorph-containing subgroup of <math>K</math>. {{proofat}}|-|[[Homomorph-containment satisfies intermediate Stronger than::intermediately fully invariant subgroup condition]]|| 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) || {{joinintermediate notions short|strictly characteristic subgroup|homomorph-closedcontaining subgroup}}|-If <math>H_i, i \in I</math>, are | [[Stronger than::characteristic subgroup]] || invariant under all [[automorphism]]s || (via fully invariant) || (via fully invariant) || {{intermediate notions short|characteristic subgroup|homomorph-containing subgroups of <math>G</math>, then so is the subgroup}}|-| [[join of subgroupsStronger than::intermediately characteristic subgroup]] <math>\langle H_i \rangle_{i \|| characteristic in I}</math>. every intermediate subgroup || (via intermediately fully invariant) || (via intermediately fully invariant) || {{proofatintermediate notions short|[[Homomorphintermediately characteristic subgroup|homomorph-containment is strongly join-closed]]containing subgroup}}|-| [[Stronger than::normal subgroup]] || invariant under all [[inner automorphism]]s, kernel of homomorphism || (via fully invariant) || (via fully invariant) || {{quotintermediate notions short|normal subgroup|homomorph-transitivecontaining subgroup}}|-If <math>H \le K \le G</math> are groups such that <math>H</math> is a | [[Stronger than::isomorph-containing subgroup]] || contains all isomorphic subgroups || [[homomorph-containing subgroup of <math>G</math> and <math>K/H</math> is a implies isomorph-containing]] || [[isomorph-containing not implies homomorph-containing]] || {{intermediate notions short|isomorph-containing subgroup of <math>G/H</math>, then <math>K</math> is a |homomorph-containing subgroup of <math>G</math>. {{proofat}}|-|[[HomomorphStronger than::homomorph-containment dominating subgroup]] || every homomorphic image is quotientcontained in some conjugate subgroup || || || {{intermediate notions short|homomorph-dominating subgroup|homomorph-transitive]]containing subgroup}}|}
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