Homomorph-containing subgroup

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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]

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Definition

A subgroup $H$ of a group $G$ is termed homomorph-containing if for any $\varphi \in \operatorname {Hom} (H,G)$, the image $\varphi (H)$ is contained in $H$.

Relation with other properties

Weaker properties

• Intermediately fully characteristic subgroup
• Fully characteristic subgroup
• Strictly characteristic subgroup
• Intermediately characteristic subgroup
• Characteristic subgroup
• Isomorph-free subgroup in case the subgroup is co-Hopfian as a group: it is not isomorphic to any proper subgroup of itself.

Facts

• The omega subgroups of a group of prime power order are homomorph-containing. Further information: Omega subgroups are homomorph-containing

Metaproperties

Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).
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Intermediate subgroup condition

YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
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Join-closedness

YES: This subgroup property is join-closed: an arbitrary (nonempty) join of subgroups with this property, also has this property.
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ABOUT JOIN-CLOSEDNESS: View all join-closed subgroup properties (or, strongly join-closed properties) | View all subgroup properties that are not join-closed | Read a survey article on proving join-closedness | Read a survey article on disproving join-closedness

Quotient-transitivity

This subgroup property is quotient-transitive: the corresponding quotient property is transitive.
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