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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==

===~~Stronger properties~~Important classes of examples===

* [[~~Weaker than::Order-containing ~~Normal Sylow subgroup]]~~* ~~s and [[~~Weaker than::Subhomomorph-containing ~~normal Hall subgroup]]~~* [[Weaker than::Variety~~s 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 subgroup~~omega 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===~~.

==~~Facts~~=Examples in small finite groups===

==Metaproperties==

Here is a summary:

{| class="~~wikitable~~sortable" border="1"!Metaproperty name !! Satisfied? !! Proof !! ~~Section in this article~~Statement 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-closedness~~join of subgroups]]<math>\langle H_i \rangle_{i \in I}</math> is also a homomorph-containing subgroup.

|-

| [[satisfies ~~metaproprty~~metaproperty::quotient-transitive subgroup property]] || ~~yes ~~Yes || [[homomorph-containment is quotient-transitive]] || ~~[[#Quotient~~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-~~transitivity]]~~containing in <math>G</math>.

|}

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