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 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.

	Group part	Subgroup part	Quotient part
A3 in S3	Symmetric group:S3	Cyclic group:Z3	Cyclic group:Z2
A4 in S4	Symmetric group:S4	Alternating group:A4	Cyclic group:Z2
Center of nontrivial semidirect product of Z4 and Z4	Nontrivial semidirect product of Z4 and Z4	Klein four-group	Klein four-group
Center of quaternion group	Quaternion group	Cyclic group:Z2	Klein four-group
Center of special linear group:SL(2,3)	Special linear group:SL(2,3)	Cyclic group:Z2	Alternating group:A4
Center of special linear group:SL(2,5)	Special linear group:SL(2,5)	Cyclic group:Z2	Alternating group:A5
D8 in SD16	Semidihedral group:SD16	Dihedral group:D8	Cyclic group:Z2
Direct product of Z4 and Z2 in M16	M16	Direct product of Z4 and Z2	Cyclic group:Z2
First omega subgroup of direct product of Z4 and Z2	Direct product of Z4 and Z2	Klein four-group	Cyclic group:Z2
Klein four-subgroup of alternating group:A4	Alternating group:A4	Klein four-group	Cyclic group:Z3
SL(2,3) in GL(2,3)	General linear group:GL(2,3)	Special linear group:SL(2,3)	Cyclic group:Z2

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 central product of D8 and Z4	Central product of D8 and Z4	Cyclic group:Z4	Klein four-group
Center of dihedral group:D16	Dihedral group:D16	Cyclic group:Z2	Dihedral group:D8
Center of dihedral group:D8	Dihedral group:D8	Cyclic group:Z2	Klein four-group
Center of direct product of D8 and Z2	Direct product of D8 and Z2	Klein four-group	Klein four-group
Center of semidihedral group:SD16	Semidihedral group:SD16	Cyclic group:Z2	Dihedral group:D8
Center of unitriangular matrix group:UT(3,p)	Unitriangular matrix group:UT(3,p)	Group of prime order	Elementary abelian group of prime-square order
Central subgroup generated by a non-square in nontrivial semidirect product of Z4 and Z4	Nontrivial semidirect product of Z4 and Z4	Cyclic group:Z2	Quaternion group
Cyclic maximal subgroup of dihedral group:D16	Dihedral group:D16	Cyclic group:Z8	Cyclic group:Z2
Cyclic maximal subgroup of dihedral group:D8	Dihedral group:D8	Cyclic group:Z4	Cyclic group:Z2
Cyclic maximal subgroup of semidihedral group:SD16	Semidihedral group:SD16	Cyclic group:Z8	Cyclic group:Z2
Cyclic maximal subgroups of quaternion group	Quaternion group	Cyclic group:Z4	Cyclic group:Z2
D8 in D16	Dihedral group:D16	Dihedral group:D8	Cyclic group:Z2
Derived subgroup of M16	M16	Cyclic group:Z2	Direct product of Z4 and Z2
Derived subgroup of dihedral group:D16	Dihedral group:D16	Cyclic group:Z4	Klein four-group
Derived subgroup of nontrivial semidirect product of Z4 and Z4	Nontrivial semidirect product of Z4 and Z4	Cyclic group:Z2	Direct product of Z4 and Z2
Diagonally embedded Z4 in direct product of Z8 and Z2	Direct product of Z8 and Z2	Cyclic group:Z4	Cyclic group:Z4
First agemo subgroup of direct product of Z4 and Z2	Direct product of Z4 and Z2	Cyclic group:Z2	Klein four-group
Group of integers in group of rational numbers	Group of rational numbers	Group of integers	Group of rational numbers modulo integers
Klein four-subgroup of M16	M16	Klein four-group	Cyclic group:Z4
Klein four-subgroups of dihedral group:D8	Dihedral group:D8	Klein four-group	Cyclic group:Z2
Non-central Z4 in M16	M16	Cyclic group:Z4	Cyclic group:Z4
Non-characteristic order two subgroups of direct product of Z4 and Z2	Direct product of Z4 and Z2	Cyclic group:Z2	Cyclic group:Z4
Normal Klein four-subgroup of symmetric group:S4	Symmetric group:S4	Klein four-group	Symmetric group:S3
Q8 in SD16	Semidihedral group:SD16	Quaternion group	Cyclic group:Z2
Q8 in central product of D8 and Z4	Central product of D8 and Z4	Quaternion group	Cyclic group:Z2
S2 in S3	Symmetric group:S3	Cyclic group:Z2
Subgroup generated by a non-commutator square in nontrivial semidirect product of Z4 and Z4	Nontrivial semidirect product of Z4 and Z4	Cyclic group:Z2	Dihedral group:D8
Z2 in V4	Klein four-group	Cyclic group:Z2	Cyclic group:Z2
Z4 in direct product of Z4 and Z2	Direct product of Z4 and Z2	Cyclic group:Z4	Cyclic group:Z2

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