Function restriction formalism chart: Difference between revisions

Latest revision as of 03:47, 20 August 2021

Subgroup property	Function restriction expression	Endo-invariance	Balanced	Invariance	Left-inner	Quotient-hereditary invariance
Normal subgroup	Inner $\to$ Aut, Inner $\to$ Function	Yes	No	Yes	Yes	Yes
Characteristic subgroup	Aut $\to$ Aut, Aut $\to$ Function	Yes	Yes	Yes	No	No
Strictly characteristic subgroup	Surj. End $\to$ Function, Surj. End. $\to$ End.	Yes	No	Yes	No	Yes
Fully invariant subgroup	End. $\to$ End.	Yes	Yes	Yes	No	Yes
Injective endomorphism-invariant subgroup	Inj. End. $\to$ End., Inj. end. $\to$ Inj. End.	Yes	Yes	Yes	No	No
Central factor	Inner $\to$ Inner	No	Yes	No	Yes	No
Transitively normal subgroup	Normal $\to$ Normal, Inner $\to$ Normal	No	Yes	No	Yes	No
Conjugacy-closed normal subgroup	Class $\to$ Class, Inner $\to$ Class	No	Yes	No	Yes	?
Retraction-invariant subgroup	Retraction $\to$ Retraction, Retraction $\to$ Function	Yes	Yes	Yes	No	Yes
Powering-invariant subgroup	Rational power map $\to$ Rational power map, Rational power map $\to$ Function	No	Yes	Yes	No	?
Local powering-invariant subgroup	Local powering $\to$ Local powering, Local powering $\to$ Function	No	Yes	Yes	No	?

Note:

Endo-invariance implies join-closed. For full proof, refer: Endo-invariance implies join-closed
Balanced implies transitive. Conversely, any function-restriction-expressible subgroup property that is transitive, must be balanced. For full proof, refer: Balanced implies transitive
Invariance implies strongly intersection-closed. For full proof, refer: Invariance implies intersection-closed
Left-inner implies left-extensibility-stable, that in turn implies intermediate subgroup condition. For full proof, refer: Left-extensibility-stable implies intermediate subgroup condition
Quotient-hereditary invariance implies quotient-transitive

@@ Line 1: / Line 1: @@
 {| class="wikitable" border="1"
-! Subgroup property !! Expression !! Endo-invariance !! Balanced !! Invariance !! Left-inner !! Quotient-hereditary invariance
+! [[Subgroup property]] !! [[Function restriction expression]] !! [[Endo-invariance property|Endo-invariance]] !! [[Balanced subgroup property (function restriction formalism)|Balanced]] !! [[Invariance property|Invariance]] !! [[Left-inner subgroup property|Left-inner]]!! Quotient-hereditary invariance
 |-
-|[[Normal subgroup]] || Inner <math>\to</math> Aut, Inner <math>\to</math> Function || Yes || No || Yes || Yes || Yes |-
+|[[Normal subgroup]] || [[Inner automorphism|Inner]] <math>\to</math> [[Automorphism|Aut]], [[Inner automorphism|Inner]] <math>\to</math> Function || Yes || No || Yes || Yes || Yes
-|[[Characteristic subgroup]] || Aut <math>\to</math> Aut, Aut <math>\to</math> Function || Yes || Yes || Yes || No || No |-
+|-
-|[[Strictly characteristic subgroup]] || Surj. End <math>\to</math> Function, Surj. End. <math>\to</math> End. || Yes || No || Yes || No || Yes |- |}
+|[[Characteristic subgroup]] || [[Automorphism|Aut]] <math>\to</math> [[Automorphism|Aut]], [[Automorphism|Aut]] <math>\to</math> Function || Yes || Yes || Yes || No || No
+|-
+|[[Strictly characteristic subgroup]] || [[Surjective endomorphism|Surj. End]] <math>\to</math> Function, Surj. End. <math>\to</math> [[Endomorphism|End.]] || Yes || No || Yes || No || Yes
+|-
+|[[Fully invariant subgroup]] || [[Endomorphism|End.]] <math>\to</math> [[Endomorphism|End.]] || Yes || Yes || Yes || No || Yes
+|-
+|[[Injective endomorphism-invariant subgroup]] || [[Injective endomorphism|Inj. End.]]<math>\to</math> [[Endomorphism|End.]], Inj. end. <math>\to</math> Inj. End. || Yes || Yes || Yes || No || No
+|-
+|[[Central factor]] || [[Inner automorphism|Inner]] <math>\to</math> [[Inner automorphism|Inner]] || No || Yes || No || Yes || No
+|-
+|[[Transitively normal subgroup]] || [[Normal automorphism|Normal]] <math>\to</math> [[Normal automorphism|Normal]], [[Inner automorphism|Inner]] <math>\to</math> [[Normal automorphism|Normal]] || No || Yes || No || Yes || No
+|-
+|[[Conjugacy-closed normal subgroup]] || [[Class automorphism|Class]] <math>\to</math> [[Class automorphism|Class]], [[Inner automorphism|Inner]] <math>\to</math> [[Class automorphism|Class]] || No || Yes || No || Yes || ?
+|-
+|[[Retraction-invariant subgroup]] || [[Retraction]] <math>\to</math> [[Retraction]], [[Retraction]] <math>\to</math> Function || Yes || Yes || Yes || No || Yes
+|-
+| [[Powering-invariant subgroup]] || Rational power map <math>\to</math> Rational power map, Rational power map <math>\to</math> Function || No || Yes || Yes || No || ?
+|-
+| [[Local powering-invariant subgroup]] || Local powering <math>\to</math> Local powering, Local powering <math>\to</math> Function || No || Yes || Yes || No || ?
+|}
+Note:
+* [[Endo-invariance property|Endo-invariance]] implies [[join-closed subgroup property|join-closed]]. {{proofat|[[Endo-invariance implies join-closed]]}}
+* [[Balanced subgroup property|Balanced]] implies [[transitive subgroup property|transitive]]. Conversely, any function-restriction-expressible subgroup property that is transitive, must be balanced. {{proofat|[[Balanced implies transitive]]}}
+* [[Invariance property|Invariance]] implies [[strongly intersection-closed subgroup property|strongly intersection-closed]]. {{proofat|[[Invariance implies intersection-closed]]}}
+* [[Left-inner subgroup property|Left-inner]] implies [[left-extensibility-stable subgroup property|left-extensibility-stable]], that in turn implies [[intermediate subgroup condition]]. {{proofat|[[Left-extensibility-stable implies intermediate subgroup condition]]}}
+* Quotient-hereditary invariance implies [[quotient-transitive subgroup property|quotient-transitive]]