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

Equivalent definitions in tabular format

No.	Shorthand	A subgroup of a group is termed homomorph-containing if ...	A subgroup $H$ of a group $G$ is termed a homomorph-containing subgroup of $G$ if ...
1	contains every homomorphic image	it contains any homomorphic image of itself in the whole group.	for any homomorphism of groups $\phi \in \mathrm{H}\mathrm{o}\mathrm{m}(H,G)$, $\phi (H)\subseteq H$.
2	homomorphism to whole group restricts to endomorphism	every homomorphism of groups from the subgroup to the whole group restricts to an endomorphism of the suubgrop.	for any homomorphism of groups $\phi :H\to G$, $\phi (H)\subseteq H$ and the restriction of $\phi$ to $H$ is an endomorphism of $H$.
3	(definition in terms of Hom-set maps)	(too complicated to state without symbols)	the natural map $\mathrm{E}\mathrm{n}\mathrm{d}(H)\to \mathrm{H}\mathrm{o}\mathrm{m}(H,G)$ (by inclusion) is a surjective map of sets.

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 and normal Hall subgroups are homomorph-containing.
• 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 such examples. Further information: Omega subgroups are homomorph-containing
• The perfect core of a group is a homomorph-containing subgroup.

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

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	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 $H\le K\le G$ 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 $H\le K\le G$ 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 $⟨H_i{⟩}_{i\in I}$ is also a homomorph-containing subgroup.
quotient-transitive subgroup property	Yes	homomorph-containment is quotient-transitive	If $H\le K\le G$ 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