# Subhomomorph-containing subgroup

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
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 $\varphi(K) \le H$.

## 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