# 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]

## 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

### Examples in small finite groups

Below are some examples of a proper nontrivial subgroup that satisfy the property [[{{{1}}}]].

Group partSubgroup partQuotient part
2-Sylow subgroup of general linear group:GL(2,3)General linear group:GL(2,3)Semidihedral group:SD16
A3 in A4Alternating group:A4Cyclic group:Z3
A3 in A5Alternating group:A5Cyclic group:Z3
A3 in S3Symmetric group:S3Cyclic group:Z3Cyclic group:Z2
A3 in S4Symmetric group:S4Cyclic group:Z3
A4 in A5Alternating group:A5Alternating group:A4
A4 in S4Symmetric group:S4Alternating group:A4Cyclic group:Z2
Center of M16M16Cyclic group:Z4Klein four-group
Center of central product of D8 and Z4Central product of D8 and Z4Cyclic group:Z4Klein four-group
Center of dihedral group:D16Dihedral group:D16Cyclic group:Z2Dihedral group:D8
Center of dihedral group:D8Dihedral group:D8Cyclic group:Z2Klein four-group
Center of direct product of D8 and Z2Direct product of D8 and Z2Klein four-groupKlein four-group
Center of nontrivial semidirect product of Z4 and Z4Nontrivial semidirect product of Z4 and Z4Klein four-groupKlein four-group
Center of quaternion groupQuaternion groupCyclic group:Z2Klein four-group
Center of semidihedral group:SD16Semidihedral group:SD16Cyclic group:Z2Dihedral group:D8
Center of special linear group:SL(2,3)Special linear group:SL(2,3)Cyclic group:Z2Alternating group:A4
Center of special linear group:SL(2,5)Special linear group:SL(2,5)Cyclic group:Z2Alternating group:A5
Center of unitriangular matrix group:UT(3,p)Unitriangular matrix group:UT(3,p)Group of prime orderElementary abelian group of prime-square order
Central subgroup generated by a non-square in nontrivial semidirect product of Z4 and Z4Nontrivial semidirect product of Z4 and Z4Cyclic group:Z2Quaternion group
Cyclic four-subgroups of symmetric group:S4Symmetric group:S4Cyclic group:Z4
Cyclic maximal subgroup of dihedral group:D16Dihedral group:D16Cyclic group:Z8Cyclic group:Z2
Cyclic maximal subgroup of dihedral group:D8Dihedral group:D8Cyclic group:Z4Cyclic group:Z2
Cyclic maximal subgroup of semidihedral group:SD16Semidihedral group:SD16Cyclic group:Z8Cyclic group:Z2
Cyclic maximal subgroups of quaternion groupQuaternion groupCyclic group:Z4Cyclic group:Z2
D8 in A6Alternating group:A6Dihedral group:D8
D8 in D16Dihedral group:D16Dihedral group:D8Cyclic group:Z2
D8 in S4Symmetric group:S4Dihedral group:D8
D8 in SD16Semidihedral group:SD16Dihedral group:D8Cyclic group:Z2
Derived subgroup of M16M16Cyclic group:Z2Direct product of Z4 and Z2
Derived subgroup of dihedral group:D16Dihedral group:D16Cyclic group:Z4Klein four-group
Derived subgroup of nontrivial semidirect product of Z4 and Z4Nontrivial semidirect product of Z4 and Z4Cyclic group:Z2Direct product of Z4 and Z2
Diagonally embedded Z4 in direct product of Z8 and Z2Direct product of Z8 and Z2Cyclic group:Z4Cyclic group:Z4
Direct product of Z4 and Z2 in M16M16Direct product of Z4 and Z2Cyclic group:Z2
First agemo subgroup of direct product of Z4 and Z2Direct product of Z4 and Z2Cyclic group:Z2Klein four-group
First omega subgroup of direct product of Z4 and Z2Direct product of Z4 and Z2Klein four-groupCyclic group:Z2
Group of integers in group of rational numbersGroup of rational numbersGroup of integersGroup of rational numbers modulo integers
Klein four-subgroup of M16M16Klein four-groupCyclic group:Z4
Klein four-subgroup of alternating group:A4Alternating group:A4Klein four-groupCyclic group:Z3
Klein four-subgroup of alternating group:A5Alternating group:A5Klein four-group
Klein four-subgroups of dihedral group:D8Dihedral group:D8Klein four-groupCyclic group:Z2
Non-central Z4 in M16M16Cyclic group:Z4Cyclic group:Z4
Non-characteristic order two subgroups of direct product of Z4 and Z2Direct product of Z4 and Z2Cyclic group:Z2Cyclic group:Z4
Non-normal Klein four-subgroups of symmetric group:S4Symmetric group:S4Klein four-group
Non-normal subgroups of M16M16Cyclic group:Z2
Non-normal subgroups of dihedral group:D8Dihedral group:D8Cyclic group:Z2
Normal Klein four-subgroup of symmetric group:S4Symmetric group:S4Klein four-groupSymmetric group:S3
Q8 in SD16Semidihedral group:SD16Quaternion groupCyclic group:Z2
Q8 in central product of D8 and Z4Central product of D8 and Z4Quaternion groupCyclic group:Z2
S2 in S3Symmetric group:S3Cyclic group:Z2
S2 in S4Symmetric group:S4Cyclic group:Z2
subgroups satisfying this property
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• Property "Stronger than" (as page type) with input value "{{{1}}}" contains invalid characters or is incomplete and therefore can cause unexpected results during a query or annotation process.

Below are some examples of a proper nontrivial subgroup that does not satisfy the property [[{{{1}}}]].

Group partSubgroup partQuotient part
2-Sylow subgroup of general linear group:GL(2,3)General linear group:GL(2,3)Semidihedral group:SD16
A3 in A4Alternating group:A4Cyclic group:Z3
A3 in A5Alternating group:A5Cyclic group:Z3
A3 in S3Symmetric group:S3Cyclic group:Z3Cyclic group:Z2
A3 in S4Symmetric group:S4Cyclic group:Z3
A4 in A5Alternating group:A5Alternating group:A4
A4 in S4Symmetric group:S4Alternating group:A4Cyclic group:Z2
Center of M16M16Cyclic group:Z4Klein four-group
Center of central product of D8 and Z4Central product of D8 and Z4Cyclic group:Z4Klein four-group
Center of dihedral group:D16Dihedral group:D16Cyclic group:Z2Dihedral group:D8
Center of dihedral group:D8Dihedral group:D8Cyclic group:Z2Klein four-group
Center of direct product of D8 and Z2Direct product of D8 and Z2Klein four-groupKlein four-group
Center of semidihedral group:SD16Semidihedral group:SD16Cyclic group:Z2Dihedral group:D8
Center of unitriangular matrix group:UT(3,p)Unitriangular matrix group:UT(3,p)Group of prime orderElementary abelian group of prime-square order
Central subgroup generated by a non-square in nontrivial semidirect product of Z4 and Z4Nontrivial semidirect product of Z4 and Z4Cyclic group:Z2Quaternion group
Cyclic four-subgroups of symmetric group:S4Symmetric group:S4Cyclic group:Z4
Cyclic maximal subgroup of dihedral group:D16Dihedral group:D16Cyclic group:Z8Cyclic group:Z2
Cyclic maximal subgroup of dihedral group:D8Dihedral group:D8Cyclic group:Z4Cyclic group:Z2
Cyclic maximal subgroup of semidihedral group:SD16Semidihedral group:SD16Cyclic group:Z8Cyclic group:Z2
Cyclic maximal subgroups of quaternion groupQuaternion groupCyclic group:Z4Cyclic group:Z2
D8 in A6Alternating group:A6Dihedral group:D8
D8 in D16Dihedral group:D16Dihedral group:D8Cyclic group:Z2
D8 in S4Symmetric group:S4Dihedral group:D8
D8 in SD16Semidihedral group:SD16Dihedral group:D8Cyclic group:Z2
Derived subgroup of M16M16Cyclic group:Z2Direct product of Z4 and Z2
Derived subgroup of dihedral group:D16Dihedral group:D16Cyclic group:Z4Klein four-group
Derived subgroup of nontrivial semidirect product of Z4 and Z4Nontrivial semidirect product of Z4 and Z4Cyclic group:Z2Direct product of Z4 and Z2
Diagonally embedded Z4 in direct product of Z8 and Z2Direct product of Z8 and Z2Cyclic group:Z4Cyclic group:Z4
Direct product of Z4 and Z2 in M16M16Direct product of Z4 and Z2Cyclic group:Z2
First agemo subgroup of direct product of Z4 and Z2Direct product of Z4 and Z2Cyclic group:Z2Klein four-group
First omega subgroup of direct product of Z4 and Z2Direct product of Z4 and Z2Klein four-groupCyclic group:Z2
Group of integers in group of rational numbersGroup of rational numbersGroup of integersGroup of rational numbers modulo integers
Klein four-subgroup of M16M16Klein four-groupCyclic group:Z4
Klein four-subgroup of alternating group:A4Alternating group:A4Klein four-groupCyclic group:Z3
Klein four-subgroup of alternating group:A5Alternating group:A5Klein four-group
Klein four-subgroups of dihedral group:D8Dihedral group:D8Klein four-groupCyclic group:Z2
Non-central Z4 in M16M16Cyclic group:Z4Cyclic group:Z4
Non-characteristic order two subgroups of direct product of Z4 and Z2Direct product of Z4 and Z2Cyclic group:Z2Cyclic group:Z4
Non-normal Klein four-subgroups of symmetric group:S4Symmetric group:S4Klein four-group
Non-normal subgroups of M16M16Cyclic group:Z2
Non-normal subgroups of dihedral group:D8Dihedral group:D8Cyclic group:Z2
Normal Klein four-subgroup of symmetric group:S4Symmetric group:S4Klein four-groupSymmetric group:S3
Q8 in SD16Semidihedral group:SD16Quaternion groupCyclic group:Z2
Q8 in central product of D8 and Z4Central product of D8 and Z4Quaternion groupCyclic group:Z2
S2 in S3Symmetric group:S3Cyclic group:Z2
S2 in S4Symmetric group:S4Cyclic group:Z2
S3 in S4Symmetric group:S4Symmetric group:S3
SL(2,3) in GL(2,3)General linear group:GL(2,3)Special linear group:SL(2,3)Cyclic group:Z2
Subgroup generated by a non-commutator square in nontrivial semidirect product of Z4 and Z4Nontrivial semidirect product of Z4 and Z4Cyclic group:Z2Dihedral group:D8
Subgroup generated by double transposition in symmetric group:S4Symmetric group:S4Cyclic group:Z2
subgroups dissatisfying property {{{1}}}
• Property "Dissatisfies property" (as page type) with input value "{{{1}}}" contains invalid characters or is incomplete and therefore can cause unexpected results during a query or annotation process.
• Property "Weaker than" (as page type) with input value "{{{1}}}" contains invalid characters or is incomplete and therefore can cause unexpected results during a query or annotation process.

## 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 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 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 characteristic 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, Intermediately characteristic 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