Homomorph-containing subgroup: Difference between revisions

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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 $\varphi \in \operatorname {Hom} (H,G)$ , $\varphi (H)\subseteq H$ .
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

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 $H\leq K\leq G$ 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

@@ Line 4: / Line 4: @@
 ==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>.
+===Equivalent definitions in tabular format===
+{| class="sortable" border="1"
+! 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 ...
+|-
+| 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>.
+|-
+| 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>.
+|-
+| 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.
+|}
 ==Examples==
-{{subgroup property see 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 subgroup]]s and [[normal Hall subgroup]]s 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|[[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===
-==Relation with other properties==
+{{subgroup property see examples embed|homomorph-containing subgroup}}
+==Metaproperties==
+{{wikilocal-section}}
-===Stronger properties===
+Here is a summary:
-{| class="wikitable" border="1"
+{| class="sortable" border="1"
-! property !! quick description !!proof of implication !! proof of strictness (reverse implication failure) !! intermediate notions
+!Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
 |-
-| [[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}}
+| [[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>.
 |-
-| [[Weaker than::Subhomomorph-containing subgroup]] || contains every homomorphic image of every nromal subgroup]] || [[subhomomorph-containing implies homomorph-containing]] || [[homomorph-containing not implies subhomomorph-containing]] | {{intermediate notions short|homomorph-containing subgroup|subhomomorph-containing subgroup}}
+| [[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>.
 |-
-| [[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}}
+| [[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>.
 |-
-| [[Weaker than::Normal Sylow subgroup]] || normal and a [[Sylow subgroup]] || || || {{intermediate notions short|homomorph-containing subgroup|normal Sylow subgroup}}
+| [[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.
 |-
-| [[Weaker than::Normal Hall subgroup]] || normal and a [[Hall subgroup]] || || || {{intermediate notions short|homomorph-containing subgroup|normal Hall subgroup}}
+| [[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>.
-|-
-| [[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}}
-|-
-| [[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}}
-|-
-| [[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}}
-|-
-| [[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}}
 |}
-===Weaker properties===
+==Relation with other properties==
+===Stronger properties===
-{| class="wikitable" border="1"
+{| class="sortable" border="1"
-! property !! quick description !!proof of implication !! proof of strictness (reverse implication failure) !! intermediate notions
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[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}}
 |-
-| [[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}}
+| [[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}}
 |-
-|[[Stronger than::Intermediately fully invariant subgroup]] || fully invariant in every intermediate subgroup || || || {{intermediate notions short|intermediately fully invariant subgroup|homomorph-containing subgroup}}
+| [[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}}
 |-
-| [[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}}
+| [[Weaker than::normal Sylow subgroup]] || normal and a [[Sylow subgroup]] || || || {{intermediate notions short|homomorph-containing subgroup|normal Sylow subgroup}}
 |-
-| [[Stronger than::Characteristic subgroup]] || invariant under all [[automorphism]]s || (via fully invariant) || (via fully invariant) || {{intermediate notions short|characteristic subgroup|homomorph-containing subgroup}}
+| [[Weaker than::normal Hall subgroup]] || normal and a [[Hall subgroup]] || || || {{intermediate notions short|homomorph-containing subgroup|normal Hall subgroup}}
 |-
-| [[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}}
+| [[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}}
 |-
-| [[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}}
+| [[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}}
 |-
-| [[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}}
+| [[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}}
 |-
-| [[Stronger than::Homomorph-dominating subgroup]] || every homomorphic image is contained in some conjugate subgroup || || || {{intermediate notions short|homomorph-dominating subgroup|homomorph-containing subgroup}}
+| [[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}}
 |}
-==Facts==
+===Weaker properties===
-* The [[omega subgroups of a group of prime power order]] are homomorph-containing. {{further|[[Omega subgroups are homomorph-containing]]}}
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-==Metaproperties==
+|-
+| [[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}}
-{{wikilocal-section}}
+|-
+|[[Stronger than::intermediately fully invariant subgroup]] || fully invariant in every intermediate subgroup || || || {{intermediate notions short|intermediately fully invariant subgroup|homomorph-containing subgroup}}
-Here is a summary:
+|-
+| [[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}}
-{| class="wikitable" border="1"
-!Metaproperty name !! Satisfied? !! Proof !! Section in this article
 |-
-| [[satisfies metaproperty::trim subgroup property]] || yes || || [[#Trimness]]
+| [[Stronger than::characteristic subgroup]] || invariant under all [[automorphism]]s || (via fully invariant) || (via fully invariant) || {{intermediate notions short|characteristic subgroup|homomorph-containing subgroup}}
 |-
-| [[dissatisfies metaproperty::transitive subgroup property]] || no || [[homomorph-containment is not transitive]] || [[#Transitivity]]
+| [[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}}
 |-
-| [[satisfies metaproperty::intermediate subgroup condition]] || yes || [[homomorph-containment satisfies intermediate subgroup condition]] || [[#Intermediate subgroup condition]]
+| [[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}}
 |-
-| [[satisfies metaproperty::strongly join-closed subgroup property]] || yes || [[homomorph-containment is strongly join-closed]] || [[#Join-closedness]]
+| [[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}}
 |-
-| [[satisfies metaproprty::quotient-transitive subgroup property]] || yes || [[homomorph-containment is quotient-transitive]] || [[#Quotient-transitivity]]
+| [[Stronger than::homomorph-dominating subgroup]] || every homomorphic image is contained in some conjugate subgroup || || || {{intermediate notions short|homomorph-dominating subgroup|homomorph-containing subgroup}}
 |}
-{{trim}}
-For any group <math>G</math>, the trivial subgroup and the whole group are both homomorph-containing.
-{{intransitive}}
-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]]}}
-{{intsubcondn}}
-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]]}}
-{{join-closed}}
-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]]}}