# Difference between revisions of "Homomorph-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]

## 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

property quick description proof of implication proof of strictness (reverse implication failure) intermediate notions
Order-containing subgroup contains every subgroup whose order divides its order order-containing implies homomorph-containing homomorph-containing not implies order-containing Right-transitively homomorph-containing subgroup, Subhomomorph-containing subgroup|FULL LIST, MORE INFO
Subhomomorph-containing subgroup contains every homomorphic image of every subgroup subhomomorph-containing implies homomorph-containing homomorph-containing not implies subhomomorph-containing Right-transitively homomorph-containing subgroup|FULL LIST, MORE INFO
Variety-containing subgroup contains every subgroup of the whole group in the variety it generates (via subhomomorph-containing) (via subhomomorph-containing) Right-transitively homomorph-containing subgroup, Subhomomorph-containing subgroup|FULL LIST, MORE INFO
Normal Sylow subgroup normal and a Sylow subgroup Complemented homomorph-containing subgroup, Normal Hall subgroup, Normal subgroup having no nontrivial homomorphism to its quotient group, Order-containing subgroup, Variety-containing subgroup|FULL LIST, MORE INFO
Normal Hall subgroup normal and a Hall subgroup Complemented homomorph-containing subgroup, Normal subgroup having no nontrivial homomorphism to its quotient group, Order-containing subgroup, Variety-containing subgroup|FULL LIST, MORE INFO
Fully invariant direct factor fully invariant and a direct factor equivalence of definitions of fully invariant direct factor Complemented homomorph-containing subgroup, Left-transitively homomorph-containing subgroup, Normal subgroup having no nontrivial homomorphism to its quotient group|FULL LIST, MORE INFO
Left-transitively homomorph-containing subgroup if whole group is homomorph-containing in some group, so is the subgroup homomorph-containment is not transitive |FULL LIST, MORE INFO
Right-transitively homomorph-containing subgroup any homomorph-containing subgroup of it is homomorph-containing in the whole group |FULL LIST, MORE INFO
Normal subgroup having no nontrivial homomorphism to its quotient group no nontrivial homomorphism to the quotient group |FULL LIST, MORE INFO

### Weaker properties

property quick description proof of implication proof of strictness (reverse implication failure) intermediate notions
Fully invariant subgroup invariant under all endomorphisms homomorph-containing implies fully invariant fully invariant not implies homomorph-containing Intermediately fully invariant subgroup, Sub-homomorph-containing subgroup|FULL LIST, MORE INFO
Intermediately fully invariant subgroup fully invariant in every intermediate subgroup |FULL LIST, MORE INFO
Strictly characteristic subgroup invariant under all surjective endomorphisms (via fully invariant) (via fully invariant) Fully invariant subgroup, Intermediately strictly characteristic subgroup, Normal-homomorph-containing subgroup, Sub-homomorph-containing subgroup|FULL LIST, MORE INFO
Characteristic subgroup invariant under all automorphisms (via fully invariant) (via fully invariant) Fully invariant subgroup, Intermediately fully invariant subgroup, Intermediately injective endomorphism-invariant subgroup, Intermediately strictly characteristic subgroup, Isomorph-containing subgroup, Normal-homomorph-containing subgroup, Sub-homomorph-containing subgroup|FULL LIST, MORE INFO
Intermediately characteristic subgroup characteristic in every intermediate subgroup (via intermediately fully invariant) (via intermediately fully invariant) Intermediately fully invariant subgroup, Intermediately injective endomorphism-invariant subgroup, Intermediately strictly characteristic subgroup, Isomorph-containing subgroup|FULL LIST, MORE INFO
Normal subgroup invariant under all inner automorphisms, kernel of homomorphism (via fully invariant) (via fully invariant) Characteristic subgroup, Fully invariant subgroup, Isomorph-containing subgroup, Normal-homomorph-containing subgroup, Sub-homomorph-containing subgroup|FULL LIST, MORE INFO
Isomorph-containing subgroup contains all isomorphic subgroups homomorph-containing implies isomorph-containing isomorph-containing not implies homomorph-containing |FULL LIST, MORE INFO
Homomorph-dominating subgroup every homomorphic image is contained in some conjugate subgroup |FULL LIST, MORE INFO

## 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:

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