Prehomomorph-contained 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

Definition with symbols

Suppose $H$ is a subgroup of a group $G$. We say that $H$ is prehomomorph-contained in $G$ if for any surjective homomorphism of groups $\varphi :K\to H$ from a subgroup $K$ of $G$, we have $H\leq K$.

Relation with other properties

Weaker properties

• Intermediately strictly characteristic subgroup
• Strictly characteristic subgroup
• Isomorph-free subgroup
• Isomorph-containing subgroup
• Intermediately injective endomorphism-invariant subgroup
• Injective endomorphism-invariant subgroup
• Intermediately characteristic subgroup
• Characteristic subgroup
• Prehomomorph-dominated subgroup

Metaproperties

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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