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

Stronger properties

• Order-containing subgroup
• Subhomomorph-containing subgroup
• Variety-containing subgroup
• Normal Sylow subgroup
• Normal Hall subgroup

Weaker properties

• Fully characteristic subgroup: Also related:
  • Intermediately fully characteristic subgroup
  • Strictly characteristic subgroup
  • Intermediately characteristic subgroup
  • Characteristic subgroup
• Isomorph-containing subgroup

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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ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

Join-closedness

YES: This subgroup property is join-closed: an arbitrary (nonempty) join of subgroups with this property, also has this property.
ABOUT THIS PROPERTY: View variations of this property that are join-closed | View variations of this property that are not join-closed
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.
View a complete list of quotient-transitive subgroup properties