Intermediately characteristic subgroup: Difference between revisions

Latest revision as of 13:47, 1 June 2020

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Definition

Symbol-free definition

A subgroup of a group is said to be intermediately characteristic if it is characteristic not only in the whole group but also in every intermediate subgroup.

Definition with symbols

A subgroup $H$ of a group $G$ is said to be intermediately characteristic if for any intermediate subgroup $K$ (such that $H \leq K \leq G$ ), $H$ is characteristic in $K$ .

Formalisms

In terms of the intermediately operator

This property is obtained by applying the intermediately operator to the property: characteristic subgroup
View other properties obtained by applying the intermediately operator

The subgroup property of being intermediately characteristic can be obtained by applying the intermediately operator to the subgroup property of being characteristic.

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
order-unique subgroup	finite subgroup and no other subgroup of the same order	(via isomorph-free)	(via isomorph-free)	\|FULL LIST, MORE INFO
isomorph-free subgroup	no other isomorphic subgroup	(obvious)	any non-co-Hopfian group, such as the group of integers, as a subgroup of itself	\|FULL LIST, MORE INFO
isomorph-containing subgroup	contains every isomorphic subgroup			\|FULL LIST, MORE INFO
homomorph-containing subgroup	contains every homomorphic image			\|FULL LIST, MORE INFO
subisomorph-containing subgroup	contains every subgroup isomorphic to a subgroup of it			\|FULL LIST, MORE INFO
intermediately fully invariant subgroup	fully invariant in every intermediate subgroup			\|FULL LIST, MORE INFO
intermediately injective endomorphism-invariant subgroup	injective endomorphism-invariant in every intermediate subgroup	(obvious)		\|FULL LIST, MORE INFO
transfer-closed characteristic subgroup	intersection with any subgroup is characteristic in that subgroup	transfer-closed characteristic implies intermediately characteristic	intermediately characteristic not implies transfer-closed characteristic	\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
intersection of finitely many intermediately characteristic subgroups	intersection of finitely many intermediately characteristic subgroups	(obvious)	follows from intermediate characteristicity is not finite-intersection-closed	\|FULL LIST, MORE INFO
sub-intermediately characteristic subgroup	there is a chain of subgroups from the subgroup to the whole group, with each member intermediately characteristic in its successor	(obvious))	follows from intermediate characteristicity is not transitive	\|FULL LIST, MORE INFO
characteristic subgroup	invariant under all automorphisms	(obvious)	characteristicity does not satisfy intermediate subgroup condition	\|FULL LIST, MORE INFO
normal subgroup	invariant under all inner automorphisms (conjugations)	(via characteristic)	(via characteristic)	Characteristic subgroup\|FULL LIST, MORE INFO

Related properties

Image-closed characteristic subgroup

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	intermediate characteristicity is not transitive		It is possible to have groups $H \leq K \leq G$ such that $H$ is intermediately characteristic in $K$ and $K$ is intermediately characteristic in $G$ but $H$ is not intermediately characteristic in $G$ .
quotient-transitive subgroup property	Yes	intermediate characteristicity is quotient-transitive		Suppose we have groups $H \leq K \leq G$ such that $H$ is intermediately characteristic in $G$ and $K / H$ is intermediately characteristic in $G / H$ . Then, $K$ is intermediately characteristic in $G$ .
intermediate subgroup condition	Yes	(obvious)	0	Suppose $H \leq K \leq G$ are groups such that $H$ is intermediately characteristic in $G$ . Then, $H$ is intermediately characteristic in $K$ .
finite-intersection-closed subgroup property	No	intersection of two isomorph-free subgroups need not be intermediately characteristic		It is possible to have a group $G$ and intermediately characteristic subgroups $H, K$ of $G$ such that $H \cap K$ is not intermediately characteristic in $G$ (i.e., there exists $L \leq G$ containing $H \cap K$ as a non-characteristic subgroup). In fact, we can choose an example where both $H$ and $K$ are isomorph-free in $G$ .
strongly join-closed subgroup property	Yes	intermediate characteristicity is strongly join-closed		Suppose $H_{i}, i \in I$ is a family of intermediately characteristic subgroups of a group $G$ . Then, thee join of subgroups $⟨ H_{i} ⟩_{i \in I}$ is also intermediately characteristic in $G$ .

Effect of property operators

Right transiter

It turns out that any intermediately characteristic subgroup of a transfer-closed characteristic subgroup is again intermediately characteristic. This follows from some simple reasoning and the fact that characteristicity is itself transitive. Further information: Intermediately characteristic of transfer-closed characteristic implies intermediately characteristic

Hence, the right transiter of the property of being intermediately characteristic is weaker than the property of being transfer-closed characteristic.

@@ Line 11: / Line 11: @@
 ===Definition with symbols===
-A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is said to be '''intermediately characteristic''' if forany intermediate subgroup <math>K</math> (such that <math>H \le K \le G</math>), <math>H</math> is characteristic in <math>K</math>.
+A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is said to be '''intermediately characteristic''' if for any intermediate subgroup <math>K</math> (such that <math>H \le K \le G</math>), <math>H</math> is characteristic in <math>K</math>.
 ==Formalisms==
@@ Line 27: / Line 27: @@
 ===Stronger properties===
-* [[Weaker than::Isomorph-free subgroup]]
+{| class="sortable" border="1"
-* [[Weaker than::Order-unique subgroup]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-* [[Weaker than::Transfer-closed characteristic subgroup]]: {{proofofstrictimplicationat|[[Transfer-closed characteristic implies intermediately characteristic]]|[[intermediately characteristic not implies transfer-closed characteristic]]}}
+|-
+| [[Weaker than::order-unique subgroup]] || finite subgroup and no other subgroup of the same order || (via isomorph-free) || (via isomorph-free) || {{intermediate notions short|intermediately characteristic subgroup|order-unique subgroup}}
+|-
+| [[Weaker than::isomorph-free subgroup]] || no other isomorphic subgroup || (obvious) || any non-co-Hopfian group, such as the [[group of integers]], as a subgroup of itself|| {{intermediate notions short|intermediately characteristic subgroup|isomorph-free subgroup}}
+|-
+| [[Weaker than::isomorph-containing subgroup]] || contains every isomorphic subgroup || || || {{intermediate notions short|intermediately characteristic subgroup|isomorph-containing subgroup}}
+|-
+| [[Weaker than::homomorph-containing subgroup]] || contains every homomorphic image || || || {{intermediate notions short|intermediately characteristic subgroup|homomorph-containing subgroup}}
+|-
+| [[Weaker than::subisomorph-containing subgroup]] || contains every subgroup isomorphic to a subgroup of it || || || {{intermediate notions short|intermediately characteristic subgroup|subisomorph-containing subgroup}}
+|-
+| [[Weaker than::intermediately fully invariant subgroup]] || fully invariant in every intermediate subgroup || || || {{intermediate notions short|intermediately characteristic subgroup|intermediately fully invariant subgroup}}
+|-
+| [[Weaker than::intermediately injective endomorphism-invariant subgroup]] || [[injective endomorphism-invariant subgroup|injective endomorphism-invariant]] in every intermediate subgroup || (obvious) || || {{intermediate notions short|intermediately characteristic subgroup|intermediately injective endomorphism-invariant subgroup}}
+|-
+| [[Weaker than::transfer-closed characteristic subgroup]] || intersection with any subgroup is characteristic in that subgroup || [[transfer-closed characteristic implies intermediately characteristic]] || [[intermediately characteristic not implies transfer-closed characteristic]] || {{intermediate notions short|intermediately characteristic subgroup|transfer-closed characteristic subgroup}}
+|}
 ===Weaker properties===
-* [[Stronger than::Characteristic subgroup]]
+{| class="sortable" border="1"
-* [[Stronger than::Potentially characteristic subgroup]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-* [[Stronger than::Normal subgroup]]
+|-
+| [[Stronger than::intersection of finitely many intermediately characteristic subgroups]] || intersection of finitely many intermediately characteristic subgroups  || (obvious) || follows from [[intermediate characteristicity is not finite-intersection-closed]] || {{intermediate notions short|intersection of finitely many intermediately characteristic subgroups|intermediately characteristic subgroup}}
+|-
+| [[Stronger than::sub-intermediately characteristic subgroup]] || there is a chain of subgroups from the subgroup to the whole group, with each member intermediately characteristic in its successor || (obvious)) || follows from [[intermediate characteristicity is not transitive]] || {{intermediate notions short|sub-intermediately characteristic subgroup|intermediately characteristic subgroup}}
+|-
+| [[Stronger than::characteristic subgroup]] || invariant under all automorphisms || (obvious) || [[characteristicity does not satisfy intermediate subgroup condition]] || {{intermediate notions short|characteristic subgroup|intermediately characteristic subgroup}}
+|-
+| [[Stronger than::normal subgroup]] || invariant under all inner automorphisms (conjugations) || (via characteristic) || (via characteristic) || {{intermediate notions short|normal subgroup|intermediately characteristic subgroup}}
+|}
 ===Related properties===
@@ Line 43: / Line 67: @@
 ==Metaproperties==
-{{intransitive}}
+{{wikilocal-section}}
-{{proofat|[[Intermediate characteristicity is not transitive]]}}
-{{trim}}
+Here is a summary:
-{{quot-transitive}}
-{{proofat|[[Intermediate characteristicity is quotient-transitive]]}}
+{| class="sortable" border="1"
-{{further|[[Intermediately operator preserves quotient-transitivity]], [[Characteristicity is quotient-transitive]]}}
+!Metaproperty name !! Satisfied? !! Proof !! Difficulty level (0-5) !! Statement with symbols
-{{intsubcondn}}
+|-
+| [[dissatisfies metaproperty::transitive subgroup property]] || No || [[intermediate characteristicity is not transitive]] || {{#show: intermediate characteristicity is not transitive | ?Difficulty level}} || It is possible to have groups <math>H \le K \le G</math> such that <math>H</math> is intermediately characteristic in <math>K</math> and <math>K</math> is intermediately characteristic in <math>G</math> but <math>H</math> is not intermediately characteristic in <math>G</math>.
+|-
+| [[satisfies metaproperty::quotient-transitive subgroup property]] || Yes || [[intermediate characteristicity is quotient-transitive]] || {{#show: intermediate characteristicity is quotient-transitive| ?Difficulty level}} || Suppose we have groups <math>H \le K \le G</math> such that <math>H</math> is intermediately characteristic in <math>G</math> and <math>K/H</math> is intermediately characteristic in <math>G/H</math>. Then, <math>K</math> is intermediately characteristic in <math>G</math>.
+|-
+| [[satisfies metaproperty::intermediate subgroup condition]] || Yes || (obvious) || 0 || Suppose <math>H \le K \le G</math> are groups such that <math>H</math> is intermediately characteristic in <math>G</math>. Then, <math>H</math> is intermediately characteristic in <math>K</math>.
+|-
+| [[dissatisfies metaproperty::finite-intersection-closed subgroup property]] || No || [[intersection of two isomorph-free subgroups need not be intermediately characteristic]] || {{#show:intersection of two isomorph-free subgroups need not be intermediately characteristic| ?Difficulty level}} || It is possible to have a group <math>G</math> and intermediately characteristic subgroups <math>H, K</math> of <math>G</math> such that <math>H \cap K</math> is not intermediately characteristic in <math>G</math> (i.e., there exists <math>L \le G</math> containing <math>H \cap K</math> as a non-characteristic subgroup). In fact, we can choose an example where both <math>H</math> and <math>K</math> are isomorph-free in <math>G</math>.
+|-
+| [[satisfies metaproperty::strongly join-closed subgroup property]] || Yes || [[intermediate characteristicity is strongly join-closed]] || {{#show:intermediate characteristicity is strongly join-closed| ?Difficulty level}} || Suppose <math>H_i, i \in I</math> is a family of intermediately characteristic subgroups of a group <math>G</math>. Then, thee [[join of subgroups]] <math>\langle H_i \rangle_{i \in I}</math> is also intermediately characteristic in <math>G</math>.
+|}
 ==Effect of property operators==