Homomorph-containing subgroup

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Definition

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

Examples

Extreme examples

• Every group is homomorph-containing as a subgroup of itself.
• The trivial subgroup is homomorph-containing in any group.

Important classes of examples

Normal Sylow subgroups, normal Hall subgroups, as well as subgroups defined as the subgroup generated by elements of specific orders, are all homomorph-containing subgroups. The omega subgroups of a group of prime power order are homomorph-containing. Further information: Omega subgroups are homomorph-containing

See also the section #Stronger properties in this page.

Examples in small finite groups

Below are some examples of a proper nontrivial subgroup that satisfy the property homomorph-containing subgroup.

Below are some examples of a proper nontrivial subgroup that does not satisfy the property homomorph-containing subgroup.

	Group part	Subgroup part	Quotient part
Center of dihedral group:D8	Dihedral group:D8	Cyclic group:Z2	Klein four-group

Metaproperties

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Here is a summary:

Metaproperty name	Satisfied?	Proof	Statement with symbols
trim subgroup property	Yes		For any group $G$, both $G$ (as a subgroup of itself) and the trivial subgroup of $G$ are homomorph-containing subgroups of $G$.
transitive subgroup property	No	homomorph-containment is not transitive	It is possible to have groups $H\leq K\leq G$ such that $H$ is homomorph-containing in $K$ and $K$ is homomorph-containing in $G$ but $H$ is not homomorph-containing in $G$.
intermediate subgroup condition	Yes	homomorph-containment satisfies intermediate subgroup condition	If $H\leq K\leq G$ and $H$ is homomorph-containing in $G$, then $H$ is homomorph-containing in $K$.
strongly join-closed subgroup property	Yes	homomorph-containment is strongly join-closed	If $H_{i},i\in I$ are a collection of homomorph-containing subgroups of $G$, the join of subgroups $\langle H_{i}\rangle _{i\in I}$ is also a homomorph-containing subgroup.
quotient-transitive subgroup property	Yes	homomorph-containment is quotient-transitive	If $H\leq K\leq G$ such that $H$ is homomorph-containing in $G$ and $K/H$ is homomorph-containing in $G/H$, then $K$ is homomorph-containing in $G$.

Relation with other properties

Stronger properties

Property	Meaning	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 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	Meaning	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