Subhomomorph-containing subgroup

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

For its use outside the wiki, please define the term when using it. If you are aware of an equivalent standard term, please leave a comment on the talk page
VIEW: Definitions built on this | Facts about this: (facts closely related to Subhomomorph-containing subgroup, all facts related to Subhomomorph-containing subgroup) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
Learn more about terminology local to the wiki | view a complete list of such terminology

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]

|Get subgroup property lookup help |Get exploration suggestions
VIEW RELATED: Subgroup property implications | Subgroup property non-implications | Subgroup metaproperty satisfactions | Subgroup metaproperty dissatisfactions | Subgroup property satisfactions |Subgroup property dissatisfactions
VIEW FACTS THAT:use, or start with, a subgroup satisfying the property|prove, or show that, a subgroup satisfies the property

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 $\phi :K\to G$, we have $\phi (K)\le H$.

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.
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