Isomorph-containing subgroup: Difference between revisions

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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

QUICK PHRASES: contains all isomorphic subgroups, weakly closed in any ambient group

A subgroup $H$ of a group $G$ is termed an isomorph-containing subgroup if it satisfies the following equivalent conditions:

Whenever $K \leq G$ is a subgroup of $G$ isomorphic to $H$ , $K \leq H$ .
If $G$ is a subgroup of $L$ , $H$ is weakly closed in $G$ with respect to $L$ .

Equivalence of definitions

Further information: Isomorph-containing iff weakly closed in any ambient group

Examples

VIEW: subgroups of groups satisfying this property | subgroups of groups dissatisfying this property
VIEW: Related subgroup property satisfactions | Related subgroup property dissatisfactions

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
isomorph-free subgroup	no other isomorphic subgroup (for a finite subgroup, and more generally, for a co-Hopfian subgroup, the two properties are equivalent)	obvious	any example of a non-co-Hopfian group as a subgroup of itself -- such as the group of integers
homomorph-containing subgroup	contains any homomorphic image of itself in the group	obvious; isomorphs are also homomorphic images	cyclic maximal subgroup of dihedral group:D8 is isomorph-containing but not homomorph-containing	\|FULL LIST, MORE INFO
Subhomomorph-containing subgroup	contains any homomorphic image in the whole group of any subgroup of it	(via homomorph-containing)	(via homomorph-containing)	\|FULL LIST, MORE INFO
subisomorph-containing subgroup	contains any subgroup of the whole group isomorphic to any subgroup of itself	(obvious)	cyclic maximal subgroup of dihedral group:D8 is isomorph-containing but not subisomorph-containing	\|FULL LIST, MORE INFO
variety-containing subgroup	contains any subgroup of the whole group in the subvariety of the variety of groups generated by it	(obvious)	(via either homomorph-containing or subisomorph-containing)	\|FULL LIST, MORE INFO
fully invariant direct factor	both a fully invariant subgroup and a direct factor	equivalence of definitions of fully invariant direct factor		\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
characteristic subgroup	invariant under any automorphism of the whole group	isomorph-containing implies characteristic	characteristic not implies isomorph-containing	\|FULL LIST, MORE INFO
injective endomorphism-invariant subgroup	invariant under any injective endomorphism of the whole group		Can use same example as for characteristic not implies isomorph-containing, if finite	\|FULL LIST, MORE INFO	intermediately injective endomorphism-invariant subgroup	injective endomorphism-invariant in every intermediate subgroup	\|FULL LIST, MORE INFO
intermediately characteristic subgroup	characteristic in every intermediate subgroup			\|FULL LIST, MORE INFO
normal subgroup	invariant under all inner automorphisms			Characteristic subgroup\|FULL LIST, MORE INFO
normal-isomorph-containing subgroup	contains any normal subgroup of the whole group isomorphic to it	(obvious)	any proper nontrivial subgroup of a finite simple non-abelian group	\|FULL LIST, MORE INFO

Metaproperties

Transitivity

NO: This subgroup property is not transitive: a subgroup with this property in a subgroup with this property, need not have the property in the whole group
ABOUT THIS PROPERTY: View variations of this property that are transitive|View variations of this property that are not transitive
ABOUT TRANSITIVITY: View a complete list of subgroup properties that are not transitive|View facts related to transitivity of subgroup properties | View a survey article on disproving transitivity

For full proof, refer: Isomorph-containment is not transitive

Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).
View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties

Intermediate subgroup condition

YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup condition
ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

@@ Line 21: / Line 21: @@
 ===Stronger properties===
-* [[Weaker than::Isomorph-free subgroup]]: For a finite subgroup, and more generally, for a [[co-Hopfian group|co-Hopfian]] subgroup, the two properties are equivalent.
+{| class="sortable" border="1"
-* [[Weaker than::Homomorph-containing subgroup]]: Also related:
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-** [[Weaker than::Subhomomorph-containing subgroup]]
+|-
-** [[Weaker than::Subisomorph-containing subgroup]]
+|  [[Weaker than::isomorph-free subgroup]] || no other isomorphic subgroup (for a finite subgroup, and more generally, for a [[co-Hopfian group|co-Hopfian]] subgroup, the two properties are equivalent) || obvious || any example of a non-co-Hopfian group as a subgroup of itself -- such as the [[group of integers]]
-** [[Weaker than::Variety-containing subgroup]]
+|-
-* [[Weaker than::Fully invariant direct factor]]: {{proofat|[[Equivalence of definitions of fully invariant direct factor]]}}
+| [[Weaker than::homomorph-containing subgroup]] || contains any homomorphic image of itself in the group || obvious; isomorphs are also homomorphic images || [[cyclic maximal subgroup of dihedral group:D8]] is isomorph-containing but not homomorph-containing || {{intermediate notions short|isomorph-containing subgroup|homomorph-containing subgroup}}
+|-
+| [[Weaker than::Subhomomorph-containing subgroup]] || contains any homomorphic image in the whole group of any subgroup of it || (via homomorph-containing) || (via homomorph-containing) || {{intermediate notions short|isomorph-containing subgroup|subhomomorph-containing subgroup}}
+|-
+| [[Weaker than::subisomorph-containing subgroup]] || contains any subgroup of the whole group isomorphic to any subgroup of itself || (obvious) || [[cyclic maximal subgroup of dihedral group:D8]] is isomorph-containing but not subisomorph-containing || {{intermediate notions short|isomorph-containing subgroup|subisomorph-containing subgroup}}
+|-
+| [[Weaker than::variety-containing subgroup]] || contains any subgroup of the whole group in the subvariety of the variety of groups generated by it || (obvious) || (via either homomorph-containing or subisomorph-containing) || {{intermediate notions short|isomorph-containing subgroup|variety-containing subgroup}}
+|-
+| [[Weaker than::fully invariant direct factor]] || both a [[fully invariant subgroup]] and a [[direct factor]] || [[equivalence of definitions of fully invariant direct factor]] ||  || {{intermediate notions short|isomorph-containing subgroup|fully invariant direct factor}}
+|}
 ===Weaker properties===
-* [[Stronger than::Characteristic subgroup]]: {{proofofstrictimplicationat|[[Isomorph-containing implies characteristic]]|[[Characteristic not implies isomorph-containing]]}} Also related:
+{| class="sortable" border="1"
-** [[Stronger than::Intermediately injective endomorphism-invariant subgroup]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-** [[Stronger than::Injective endomorphism-invariant subgroup]]
+|-
-** [[Stronger than::Intermediately characteristic subgroup]]
+| [[Stronger than::characteristic subgroup]] || invariant under any automorphism of the whole group || [[isomorph-containing implies characteristic]] || [[characteristic not implies isomorph-containing]] || {{intermediate notions short|characteristic subgroup|isomorph-containing subgroup}}
-** [[Stronger than::Normal subgroup]]
+|-
-* [[Stronger than::Normal-isomorph-containing subgroup]]
+| [[Stronger than::injective endomorphism-invariant subgroup]] || invariant under any injective endomorphism of the whole group || || Can use same example as for characteristic not implies isomorph-containing, if finite || {{intermediate notions short|injective endomorphism-invariant subgroup|isomorph-containing subgroup}}
+| [[Stronger than::intermediately injective endomorphism-invariant subgroup]] || injective endomorphism-invariant in every intermediate subgroup || || || {{intermediate notions short|intermediately injective endomorphism-invariant subgroup|isomorph-containing subgroup}}
+|-
+| [[Stronger than::intermediately characteristic subgroup]] || characteristic in every intermediate subgroup || || || {{intermediate notions short|intermediately characteristic subgroup|isomorph-containing subgroup}}
+|-
+| [[Stronger than::normal subgroup]] || invariant under all inner automorphisms || || || {{intermediate notions short|normal subgroup|isomorph-containing subgroup}}
+|-
+| [[Stronger than::normal-isomorph-containing subgroup]] || contains any normal subgroup of the whole group isomorphic to it || (obvious) || any proper nontrivial subgroup of a finite simple non-abelian group || {{intermediate notions short|normal-isomorph-containing subgroup|isomorph-containing subgroup}}
+|}
 ==Metaproperties==