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]

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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	|FULL LIST, MORE INFO
Subhomomorph-containing subgroup	contains every homomorphic image of every nromal subgroup]]	subhomomorph-containing implies homomorph-containing	homomorph-containing not implies subhomomorph-containing | |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)	|FULL LIST, MORE INFO
Normal Sylow subgroup	normal and a Sylow subgroup			|FULL LIST, MORE INFO
Normal Hall subgroup	normal and a Hall subgroup			|FULL LIST, MORE INFO
Fully invariant direct factor	fully invariant and a direct factor	equivalence of definitions of fully invariant direct factor		|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	|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|FULL LIST, MORE INFO
Characteristic subgroup	invariant under all automorphisms	(via fully invariant)	(via fully invariant)	Fully invariant subgroup|FULL LIST, MORE INFO
Intermediately characteristic subgroup	characteristic in every intermediate subgroup	(via intermediately fully invariant)	(via intermediately fully invariant)	|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|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

Facts

• The omega subgroups of a group of prime power order are homomorph-containing. Further information: Omega subgroups are homomorph-containing

Metaproperties

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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 $H\leq K\leq G$ 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 $H\leq K\leq G$ 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 $H\leq K\leq G$ 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