Isomorph-containing subgroup: Difference between revisions

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Definition

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

Equivalent definitions in tabular format

Below are many equivalent definitions of isomorph-containing subgroup.

No.	Shorthand	A subgroup of a group is isomorph-containing in it if...	A subgroup $H$ of a group $G$ is called an isomorph-containing subgroup of $G$ if ...
1	contains all isomorphic subgroups	it contains any subgroup of the whole group isomorphic to it.	for any subgroup $M$ of $G$ such that $H$ and $M$ are isomorphic groups, $M \leq H$ .
2	weakly closed in any ambient group	it is weakly closed in any ambient group of the whole group.	for any group $L$ containing $G$ , $H$ is weakly closed in $G$ with respect to $L$ .

This definition is presented using a tabular format. |View all pages with definitions in tabular format

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

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	Difficulty level (0-5)	Statement with symbols
transitive subgroup property	No	isomorph-containment is not transitive		It is possible to have groups $H \leq K \leq G$ such that $H$ is an isomorph-containing subgroup of $K$ and $K$ is an isomorph-containing subgroup of $G$ , but $H$ is not an isomorph-containing subgroup of $G$ .
trim subgroup property	Yes		0	In any group $G$ , the trivial subgroup ${e}$ and the whole group $G$ are both isomorph-containing subgroups of $G$ .
intermediate subgroup condition	Yes		1	If $H \leq K \leq G$ are groups such that $H$ is isomorph-containing in $G$ , then $H$ is also isomorph-containing in $K$ .

@@ Line 3: / Line 3: @@
 ==Definition==
+{{quick phrase|[[quick phrase::contains all isomorphic subgroups]], [[quick phrase::weakly closed in any ambient group]]}}
-A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed an '''isomorph-containing subgroup''' if, whenever <math>K \le G</math> is a subgroup of <math>G</math> isomorphic to <math>H</math>, <math>K \le H</math>.
+===Equivalent definitions in tabular format===
+Below are many '''equivalent''' definitions of isomorph-containing subgroup.
+{| class="sortable" border="1"
+! No. !! Shorthand !! A [[subgroup]] of a [[group]] is isomorph-containing in it if... !! A subgroup <math>H</math> of a group <math>G</math> is called an isomorph-containing subgroup of <math>G</math> if ...
+|-
+| 1 || contains all isomorphic subgroups || it contains any subgroup of the whole group isomorphic to it. || for any subgroup <math>M</math> of <math>G</math> such that <math>H</math> and <math>M</math> are isomorphic groups, <math>M \le H</math>.
+|-
+| 2 || weakly closed in any ambient group || it is weakly closed in any ambient group of the whole group. || for any group <math>L</math> containing <math>G</math>, <math>H</math> is [[defining ingredient::weakly closed subgroup|weakly closed]] in <math>G</math> with respect to <math>L</math>.
+|}
+{{tabular definition format}}
+===Equivalence of definitions===
+{{further|[[Isomorph-containing iff weakly closed in any ambient group]]}}
+==Examples==
+{{subgroup property see examples|isomorph-containing subgroup}}
 ==Relation with other properties==
 ===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]]
+! 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::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::Intermediately I-characteristic subgroup]]
+{| class="sortable" border="1"
-* [[Stronger than::I-characteristic subgroup]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-* [[Stronger than::Intermediately characteristic subgroup]]
+|-
-* [[Stronger than::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::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==
-{{trim}}
+{{wikilocal-section}}
+Here is a summary:
-{{intsubcondn}}
+{| class="sortable" border="1"
+!Metaproperty name !! Satisfied? !! Proof !! Difficulty level (0-5) !! Statement with symbols
+|-
+|[[satisfies metaproperty::transitive subgroup property]] || No || [[isomorph-containment is not transitive]] || {{#show: isomorph-containment is not transitive | ?Difficulty level}} || It is possible to have groups <math>H \le K \le G</math> such that <math>H</math> is an isomorph-containing subgroup of <math>K</math> and <math>K</math> is an isomorph-containing subgroup of <math>G</math>, but <math>H</math> is not an isomorph-containing subgroup of <math>G</math>.
+|-
+|[[satisfies metaproperty::trim subgroup property]] || Yes || || 0 || In any group <math>G</math>, the trivial subgroup <math>\{ e \}</math> and the whole group <math>G</math> are both isomorph-containing subgroups of <math>G</math>.
+|-
+|[[satisfies metaproperty::intermediate subgroup condition]] || Yes || || 1 || If <math>H \le K \le G</math> are groups such that <math>H</math> is isomorph-containing in <math>G</math>, then <math>H</math> is also isomorph-containing in <math>K</math>.
+|}