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Homomorph-containing subgroup

6,999 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>. ==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==
===Weaker Stronger properties===
* {| class="sortable" border="1"! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions|-| [[Stronger Weaker than::Fully characteristic order-containing subgroup]] || contains every subgroupwhose order divides its order || [[order-containing implies homomorph-containing]] || [[homomorph-containing not implies order-containing]]|| {{intermediate notions short|homomorph-containing subgroup|order-containing subgroup}}|-* | [[Stronger Weaker than::Strictly characteristic 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}}|-* | [[Stronger Weaker than::Characteristic 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}}* |-| [[IsomorphWeaker than::left-free transitively homomorph-containing subgroup]] || if whole group is homomorph-containing in case some group, so is the subgroup || || [[homomorph-containment is not transitive]] || {{intermediate notions short|homomorph-containing subgroup|left-transitively homomorph-containing subgroup}}|-| [[coWeaker than::right-transitively homomorph-containing subgroup]] || any homomorph-containing subgroup of it is homomorph-Hopfian containing in the whole group|co| || || {{intermediate notions short|homomorph-containing subgroup|right-transitively homomorph-containing subgroup}}|-Hopfian as a | [[Weaker than::normal subgroup having no nontrivial homomorphism to its quotient group]]: it is not isomorphic || no nontrivial [[homomorphism]] to any proper the [[quotient group]] || || || {{intermediate notions short|homomorph-containing subgroup|normal subgroup of itself.having no nontrivial homomorphism to its quotient group}}|}
==Facts=Weaker properties===
* The {| 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 [[omega subgroups of a group of prime power orderautomorphism]]s || (via fully invariant) || (via fully invariant) || {{intermediate notions short|characteristic subgroup|homomorph-containing subgroup}}|-| [[Stronger than::intermediately characteristic subgroup]] are || 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) || {{furtherintermediate notions short|normal subgroup|homomorph-containing subgroup}}|-|[[Omega Stronger than::isomorph-containing subgroup]] || contains all isomorphic subgroups are || [[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}}|}
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