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]
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 \le H and any homomorphism of groups \varphi:K \to G, we have \varphi(K) \le H.

Relation with other properties

Stronger properties

Weaker properties

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.
ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitive
ABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity

Suppose H \le K \le 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.
ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup condition
ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

Suppose H \le K \le 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