Difference between revisions of "Homomorph-containing subgroup"

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==Definition==
 
==Definition==
  
A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''homomorph-containing''' if for any <math>\varphi \in \operatorname{Hom}(H,G)</math>, the image <math>\varphi(H)</math> is contained in <math>H</math>.
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===Equivalent definitions in tabular format===
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{| class="sortable" border="1"
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! No. !! Shorthand !! A [[subgroup]] of a [[group]] is termed homomorph-containing if ... !! A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed a homomorph-containing subgroup of <math>G</math> if ...
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|-
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| 1 || contains every homomorphic image || it contains any homomorphic image of itself in the whole group. || for any [[homomorphism of groups]] <math>\varphi \in \operatorname{Hom}(H,G)</math>, <math>\varphi(H) \subseteq H</math>.
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|-
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| 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]] <math>\varphi: H \to G</math>, <math>\varphi(H) \subseteq H</math> and the restriction of <math>\varphi</math> to <math>H</math> is an endomorphism of <math>H</math>.
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|-
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| 3 || (definition in terms of Hom-set maps) || (too complicated to state without symbols) || the natural map <math>\operatorname{End}(H) \to \operatorname{Hom}(H,G)</math> (by inclusion) is a surjective map of sets.
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|}
  
 
==Examples==
 
==Examples==
  
{{subgroup property see examples}}
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===Extreme examples===
  
==Relation with other properties==
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* Every group is homomorph-containing as a subgroup of itself.
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* The trivial subgroup is homomorph-containing in any group.
  
===Stronger properties===
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===Important classes of examples===
  
* [[Weaker than::Order-containing subgroup]]
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* [[Normal Sylow subgroup]]s and [[normal Hall subgroup]]s are homomorph-containing.
* [[Weaker than::Subhomomorph-containing subgroup]]
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* 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|[[Omega subgroups are homomorph-containing]]}}
* [[Weaker than::Variety-containing subgroup]]
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* The [[perfect core]] of a group is a homomorph-containing subgroup.
* [[Weaker than::Normal Sylow subgroup]]
 
* [[Weaker than::Normal Hall subgroup]]
 
* [[Weaker than::Fully invariant direct factor]]
 
* [[Weaker than::Left-transitively homomorph-containing subgroup]]
 
* [[Weaker than::Right-transitively homomorph-containing subgroup]]
 
===Weaker properties===
 
  
* [[Stronger than::Fully invariant subgroup]]: Also related:
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See also the section [[#Stronger properties]] in this page.
** [[Stronger than::Intermediately fully invariant subgroup]]
 
** [[Stronger than::Strictly characteristic subgroup]]
 
** [[Stronger than::Intermediately characteristic subgroup]]
 
** [[Stronger than::Characteristic subgroup]]
 
** [[Stronger than::Normal subgroup]]
 
* [[Stronger than::Isomorph-containing subgroup]]
 
* [[Stronger than::Homomorph-dominating subgroup]]
 
  
==Facts==
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===Examples in small finite groups===
  
* The [[omega subgroups of a group of prime power order]] are homomorph-containing. {{further|[[Omega subgroups are homomorph-containing]]}}
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{{subgroup property see examples embed|homomorph-containing subgroup}}
  
 
==Metaproperties==
 
==Metaproperties==
  
{{trim}}
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{{wikilocal-section}}
  
For any group <math>G</math>, the trivial subgroup and the whole group are both homomorph-containing.
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Here is a summary:
  
{{intransitive}}
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{| class="sortable" border="1"
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!Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
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|-
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| [[satisfies metaproperty::trim subgroup property]] || Yes || || For any group <math>G</math>, both <math>G</math> (as a subgroup of itself) and the trivial subgroup of <math>G</math> are homomorph-containing subgroups of <math>G</math>.
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|-
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| [[dissatisfies metaproperty::transitive subgroup property]] || No || [[homomorph-containment is not transitive]] || It is possible to have groups <math>H \le K \le G</math> such that <math>H</math> is homomorph-containing in <math>K</math> and <math>K</math> is homomorph-containing in <math>G</math> but <math>H</matH> is not homomorph-containing in <math>G</math>.
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|-
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| [[satisfies metaproperty::intermediate subgroup condition]] || Yes || [[homomorph-containment satisfies intermediate subgroup condition]] || If <math>H \le K \le G</math> and <math>H</math> is homomorph-containing in <math>G</math>, then <math>H</math> is homomorph-containing in <math>K</math>.
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|-
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| [[satisfies metaproperty::strongly join-closed subgroup property]] || Yes || [[homomorph-containment is strongly join-closed]] || If <math>H_i, i \in I</math> are a collection of homomorph-containing subgroups of <math>G</math>, the [[join of subgroups]] <math>\langle H_i \rangle_{i \in I}</math> is also a homomorph-containing subgroup.
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|-
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| [[satisfies metaproperty::quotient-transitive subgroup property]] || Yes || [[homomorph-containment is quotient-transitive]] || If <math>H \le K \le G</math> such that <math>H</math> is homomorph-containing in <math>G</math> and <math>K/H</math> is homomorph-containing in <matH>G/H</math>, then <math>K</math> is homomorph-containing in <math>G</math>.
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|}
  
We can have subgroups <math>H \le K \le G</math> such that <math>H</math> is a homomorph-containing subgroup of <math>K</math> and <math>K</math> is a homomorph-containing subgroup of <math>G</math> but <math>H</math> is not a homomorph-containing subgroup of <math>G</math>. {{proofat|[[Homomorph-containment is not transitive]]}}
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==Relation with other properties==
  
{{intsubcondn}}
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===Stronger properties===
  
If <math>H \le K \le G</math> and <math>H</math> is a homomorph-containing subgroup of <math>G</math>, <math>H</math> is also a homomorph-containing subgroup of <math>K</math>. {{proofat|[[Homomorph-containment satisfies intermediate subgroup condition]]}}
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{| class="sortable" border="1"
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
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|-
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| [[Weaker than::order-containing subgroup]] || contains every subgroup whose order divides its order || [[order-containing implies homomorph-containing]] || [[homomorph-containing not implies order-containing]] || {{intermediate notions short|homomorph-containing subgroup|order-containing subgroup}}
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|-
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| [[Weaker than::subhomomorph-containing subgroup]] || contains every homomorphic image of every subgroup || [[subhomomorph-containing implies homomorph-containing]] || [[homomorph-containing not implies subhomomorph-containing]] || {{intermediate notions short|homomorph-containing subgroup|subhomomorph-containing subgroup}}
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|-
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| [[Weaker than::variety-containing subgroup]] || contains every subgroup of the whole group in the variety it generates || (via subhomomorph-containing) || (via subhomomorph-containing) || {{intermediate notions short|homomorph-containing subgroup|variety-containing subgroup}}
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|-
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| [[Weaker than::normal Sylow subgroup]] || normal and a [[Sylow subgroup]] || || || {{intermediate notions short|homomorph-containing subgroup|normal Sylow subgroup}}
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|-
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| [[Weaker than::normal Hall subgroup]] || normal and a [[Hall subgroup]] || || || {{intermediate notions short|homomorph-containing subgroup|normal Hall subgroup}}
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|-
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| [[Weaker than::fully invariant direct factor]] || [[fully invariant subgroup|fully invariant]] and a [[direct factor]] || [[equivalence of definitions of fully invariant direct factor]] || || {{intermediate notions short|homomorph-containing subgroup|fully invariant direct factor}}
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|-
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| [[Weaker than::left-transitively homomorph-containing subgroup]] || if whole group is homomorph-containing in some group, so is the subgroup || || [[homomorph-containment is not transitive]] || {{intermediate notions short|homomorph-containing subgroup|left-transitively homomorph-containing subgroup}}
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|-
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| [[Weaker than::right-transitively homomorph-containing subgroup]] || any homomorph-containing subgroup of it is homomorph-containing in the whole group || || || {{intermediate notions short|homomorph-containing subgroup|right-transitively homomorph-containing subgroup}}
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|-
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| [[Weaker than::normal subgroup having no nontrivial homomorphism to its quotient group]] || no nontrivial [[homomorphism]] to the [[quotient group]] || || || {{intermediate notions short|homomorph-containing subgroup|normal subgroup having no nontrivial homomorphism to its quotient group}}
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|}
  
{{join-closed}}
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===Weaker properties===
 
 
If <math>H_i, i \in I</math>, are all homomorph-containing subgroups of <math>G</math>, then so is the [[join of subgroups]] <math>\langle H_i \rangle_{i \in I}</math>. {{proofat|[[Homomorph-containment is strongly join-closed]]}}
 
 
 
{{quot-transitive}}
 
  
If <math>H \le K \le G</math> are groups such that <math>H</math> is a homomorph-containing subgroup of <math>G</math> and <math>K/H</math> is a homomorph-containing subgroup of <math>G/H</math>, then <math>K</math> is a homomorph-containing subgroup of <math>G</math>. {{proofat|[[Homomorph-containment is quotient-transitive]]}}
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{| class="sortable" border="1"
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
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|-
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| [[Stronger than::fully invariant subgroup]] || invariant under all [[endomorphism]]s || [[homomorph-containing implies fully invariant]] || [[fully invariant not implies homomorph-containing]] || {{intermediate notions short|fully invariant subgroup|homomorph-containing subgroup}}
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|-
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|[[Stronger than::intermediately fully invariant subgroup]] || fully invariant in every intermediate subgroup || || || {{intermediate notions short|intermediately fully invariant subgroup|homomorph-containing subgroup}}
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|-
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| [[Stronger than::strictly characteristic subgroup]] || invariant under all [[surjective endomorphism]]s || (via fully invariant) || (via fully invariant) || {{intermediate notions short|strictly characteristic subgroup|homomorph-containing subgroup}}
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|-
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| [[Stronger than::characteristic subgroup]] || invariant under all [[automorphism]]s || (via fully invariant) || (via fully invariant) || {{intermediate notions short|characteristic subgroup|homomorph-containing subgroup}}
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|-
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| [[Stronger than::intermediately characteristic subgroup]] || characteristic in every intermediate subgroup || (via intermediately fully invariant) || (via intermediately fully invariant) || {{intermediate notions short|intermediately characteristic subgroup|homomorph-containing subgroup}}
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|-
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| [[Stronger than::normal subgroup]] || invariant under all [[inner automorphism]]s, kernel of homomorphism || (via fully invariant) || (via fully invariant) || {{intermediate notions short|normal subgroup|homomorph-containing subgroup}}
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|-
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| [[Stronger than::isomorph-containing subgroup]] || contains all isomorphic subgroups || [[homomorph-containing implies isomorph-containing]] || [[isomorph-containing not implies homomorph-containing]] || {{intermediate notions short|isomorph-containing subgroup|homomorph-containing subgroup}}
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|-
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| [[Stronger than::homomorph-dominating subgroup]] || every homomorphic image is contained in some conjugate subgroup || || || {{intermediate notions short|homomorph-dominating subgroup|homomorph-containing subgroup}}
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|}

Latest revision as of 14:52, 9 March 2020

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
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

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 \varphi \in \operatorname{Hom}(H,G), \varphi(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 \varphi: H \to G, \varphi(H) \subseteq H and the restriction of \varphi to H is an endomorphism of H.
3 (definition in terms of Hom-set maps) (too complicated to state without symbols) the natural map \operatorname{End}(H) \to \operatorname{Hom}(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

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 partSubgroup partQuotient part
A3 in S3Symmetric group:S3Cyclic group:Z3Cyclic group:Z2
A4 in S4Symmetric group:S4Alternating group:A4Cyclic group:Z2
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 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
D8 in SD16Semidihedral group:SD16Dihedral group:D8Cyclic group:Z2
Direct product of Z4 and Z2 in M16M16Direct product of Z4 and Z2Cyclic group:Z2
First omega subgroup of direct product of Z4 and Z2Direct product of Z4 and Z2Klein four-groupCyclic group:Z2
Klein four-subgroup of alternating group:A4Alternating group:A4Klein four-groupCyclic 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 partSubgroup partQuotient part
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 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 D16Dihedral group:D16Dihedral 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
First agemo subgroup of direct product of Z4 and Z2Direct product of Z4 and Z2Cyclic group:Z2Klein four-group
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-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
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
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
Z2 in V4Klein four-groupCyclic group:Z2Cyclic group:Z2
Z4 in direct product of Z4 and Z2Direct product of Z4 and Z2Cyclic group:Z4Cyclic 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 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 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