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==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>(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===

==~~Facts~~=Weaker properties===

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