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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]
Equivalent definitions in tabular format
|No.||Shorthand||A subgroup of a group is termed homomorph-containing if ...||A subgroup of a group is termed a homomorph-containing subgroup of if ...|
|1||contains every homomorphic image||it contains any homomorphic image of itself in the whole group.||for any homomorphism of groups , .|
|2||homomorphism to whole group restricts to endomorphism||every homomorphism of groups from the subgroup to the whole group restricts to an endomorphism of the suubgrop.||for any homomorphism of groups , and the restriction of to is an endomorphism of .|
|3||(definition in terms of Hom-set maps)||(too complicated to state without symbols)||the natural map (by inclusion) is a surjective map of sets.|
- 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 and normal Hall subgroups 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 information: 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
Below are some examples of a proper nontrivial subgroup that satisfy the property homomorph-containing subgroup.
Below are some examples of a proper nontrivial subgroup that does not satisfy the property homomorph-containing subgroup.
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 , both (as a subgroup of itself) and the trivial subgroup of are homomorph-containing subgroups of .|
|transitive subgroup property||No||homomorph-containment is not transitive||It is possible to have groups such that is homomorph-containing in and is homomorph-containing in but is not homomorph-containing in .|
|intermediate subgroup condition||Yes||homomorph-containment satisfies intermediate subgroup condition||If and is homomorph-containing in , then is homomorph-containing in .|
|strongly join-closed subgroup property||Yes||homomorph-containment is strongly join-closed||If are a collection of homomorph-containing subgroups of , the join of subgroups is also a homomorph-containing subgroup.|
|quotient-transitive subgroup property||Yes||homomorph-containment is quotient-transitive||If such that is homomorph-containing in and is homomorph-containing in , then is homomorph-containing in .|