Subhomomorph-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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WARNING: POTENTIAL TERMINOLOGICAL CONFUSION: Please don't confuse this with sub-homomorph-containing subgroup

Definition

A subgroup $H$ of a group $G$ is termed a subhomomorph-containing subgroup if for any subgroup $K\leq H$ and any homomorphism of groups $\varphi :K\to G$, we have $\varphi (K)\leq H$.

Relation with other properties

Stronger properties

• Order-containing subgroup
• Variety-containing subgroup

Weaker properties

• Homomorph-containing subgroup
• Subisomorph-containing subgroup
• Subhomomorph-dominating subgroup
• Transfer-closed fully invariant subgroup
• Intermediately fully invariant subgroup
• Fully invariant subgroup
• Intermediately strictly characteristic subgroup
• Strictly characteristic subgroup
• Transfer-closed characteristic subgroup
• Intermediately characteristic subgroup
• Characteristic subgroup

Metaproperties

Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.
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Suppose $H\leq K\leq G$ are groups such that $H$ is a subhomomorph-containing subgroup of $K$ and $K$ is a subhomomorph-containing subgroup of $G$. Then, $H$ is a subhomomorph-containing subgroup of $G$. For full proof, refer: Subhomomorph-containment is transitive

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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Suppose $H\leq K\leq G$ are groups such that $H$ is a subhomomorph-containing subgroup of $G$. Then, $H$ is a subhomomorph-containing subgroup of $K$. For full proof, refer: Subhomomorph-containment satisfies intermediate subgroup condition