Homomorph-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]

Definition

A subgroup H of a group G is termed homomorph-containing if for any \varphi \in \operatorname{Hom}(H,G), the image \varphi(H) is contained in H.

Examples

VIEW: subgroups of groups satisfying this property | subgroups of groups dissatisfying this property
VIEW: Related subgroup property satisfactions | Related subgroup property dissatisfactions

Relation with other properties

Stronger properties

Weaker properties

Facts

Metaproperties

BEWARE! This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)

Here is a summary:

Metaproperty name Satisfied? Proof Section in this article
trim subgroup property yes #Trimness
transitive subgroup property no homomorph-containment is not transitive #Transitivity
intermediate subgroup condition yes homomorph-containment satisfies intermediate subgroup condition #Intermediate subgroup condition
strongly join-closed subgroup property yes homomorph-containment is strongly join-closed #Join-closedness
quotient-transitive subgroup property yes homomorph-containment is quotient-transitive #Quotient-transitivity

Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).
View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties

For any group G, the trivial subgroup and the whole group are both homomorph-containing.

Transitivity

NO: This subgroup property is not transitive: a subgroup with this property in a subgroup with this property, need not have the 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 subgroup properties that are not transitive|View facts related to transitivity of subgroup properties | View a survey article on disproving transitivity

We can have subgroups H \le K \le G such that H is a homomorph-containing subgroup of K and K is a homomorph-containing subgroup of G but H is not a homomorph-containing subgroup of G. For full proof, refer: Homomorph-containment is not 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

If H \le K \le G and H is a homomorph-containing subgroup of G, H is also a homomorph-containing subgroup of K. For full proof, refer: Homomorph-containment satisfies intermediate subgroup condition

Join-closedness

YES: This subgroup property is join-closed: an arbitrary (nonempty) join of subgroups with this property, also has this property.
ABOUT THIS PROPERTY: View variations of this property that are join-closed | View variations of this property that are not join-closed
ABOUT JOIN-CLOSEDNESS: View all join-closed subgroup properties (or, strongly join-closed properties) | View all subgroup properties that are not join-closed | Read a survey article on proving join-closedness | Read a survey article on disproving join-closedness

If H_i, i \in I, are all homomorph-containing subgroups of G, then so is the join of subgroups \langle H_i \rangle_{i \in I}. For full proof, refer: Homomorph-containment is strongly join-closed

Quotient-transitivity

This subgroup property is quotient-transitive: the corresponding quotient property is transitive.
View a complete list of quotient-transitive subgroup properties

If H \le K \le G are groups such that H is a homomorph-containing subgroup of G and K/H is a homomorph-containing subgroup of G/H, then K is a homomorph-containing subgroup of G. For full proof, refer: Homomorph-containment is quotient-transitive