Abelian characteristic subgroup: Difference between revisions

Revision as of 17:17, 12 August 2013

This article describes a property that arises as the conjunction of a subgroup property: characteristic subgroup with a group property (itself viewed as a subgroup property): Abelian group
View a complete list of such conjunctions

Definition

Symbol-free definition

A subgroup of a group is termed an Abelian characteristic subgroup if it is Abelian as a group and characteristic as a subgroup.

Examples

VIEW: subgroups satisfying this property | subgroups dissatisfying property characteristic subgroup | subgroups dissatisfying property abelian group
VIEW: Related subgroup property satisfactions | Related subgroup property dissatisfactions

Relation with other properties

Stronger properties

Property	Meaning	Intermediate notions
cyclic characteristic subgroup	cyclic group and a characteristic subgroup of the whole group	\|FULL LIST, MORE INFO
characteristic central subgroup	central subgroup (i.e., contained in the center) and a characteristic subgroup of the whole group	\|FULL LIST, MORE INFO
abelian critical subgroup		\|FULL LIST, MORE INFO
characteristic subgroup of center	characteristic subgroup of the center	\|FULL LIST, MORE INFO
characteristic subgroup of abelian group	the whole group is an abelian group	\|FULL LIST, MORE INFO
maximal among abelian characteristic subgroups		\|FULL LIST, MORE INFO
abelian fully invariant subgroup	abelian and a fully invariant subgroup -- invariant under all endomorphisms	\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
abelian normal subgroup	abelian and a normal subgroup -- invariant under all inner automorphisms	follows from characteristic implies normal	follows from normal not implies characteristic in the collection of all groups satisfying a nontrivial finite direct product-closed group property	\|FULL LIST, MORE INFO
class two characteristic subgroup	characteristic subgroup that is also a group of nilpotency class two			\|FULL LIST, MORE INFO
nilpotent characteristic subgroup	characteristic subgroup that is also a nilpotent group	follows from abelian implies nilpotent	follows from nilpotent not implies abelian	\|FULL LIST, MORE INFO
solvable characteristic subgroup	characteristic subgroup that is also a solvable group	(via nilpotent)	(via nilpotent)	\|FULL LIST, MORE INFO

Related group properties

Group in which every abelian characteristic subgroup is central

@@ Line 15: / Line 15: @@
 ===Stronger properties===
-* [[Weaker than::Cyclic characteristic subgroup]]
+{| class="sortable" border="1"
-* [[Weaker than::Characteristic central subgroup]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-* [[Weaker than::Abelian critical subgroup]]
+|-
-* [[Weaker than::Maximal among abelian characteristic subgroups]]
+| [[Weaker than::cyclic characteristic subgroup]] || [[cyclic group]] and a [[characteristic subgroup]] of the whole group|| || || {{intermediate notions short|abelian characteristic subgroup|cyclic characteristic subgroup}}
+|-
+| [[Weaker than::characteristic central subgroup]] || [[central subgroup]] (i.e., contained in the [[center]]) and a [[characteristic subgroup]] of the whole group || || || {{intermediate notions short|abelian characteristic subgroup|characteristic central subgroup}}
+|-
+| [[Weaker than::abelian critical subgroup]] || || || || {{intermediate notions short|abelian characteristic subgroup|abelian critical subgroup}}
+|-
+| [[Weaker than::characteristic subgroup of center]] || [[characteristic subgroup]] ''of'' the [[center]] || || || {{intermediate notions short|abelian characteristic subgroup|characteristic subgroup of center}}
+|-
+| [[Weaker than::characteristic subgroup of abelian group]] || the whole group is an [[abelian group]] || || || {{intermediate notions short|abelian characteristic subgroup|characteristic subgroup of abelian group}}
+|-
+| [[Weaker than::maximal among abelian characteristic subgroups]] || || || || {{intermediate notions short|abelian characteristic subgroup|maximal among abelian characteristic subgroups}}
+|-
+| [[Weaker than::abelian fully invariant subgroup]] || abelian and a [[fully invariant subgroup]] -- invariant under all [[endomorphism]]s || || || {{intermediate notions short|abelian characteristic subgroup|abelian fully invariant subgroup}}
+|}
 ===Weaker properties===
-* [[Stronger than::Abelian normal subgroup]]
+{| class="sortable" border="1"
-* [[Stronger than::Class two characteristic subgroup]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-* [[Stronger than::Nilpotent characteristic subgroup]]
+|-
+| [[Stronger than::abelian normal subgroup]] || abelian and a [[normal subgroup]] -- invariant under all [[inner automorphism]]s || follows from [[characteristic implies normal]] || follows from [[normal not implies characteristic in the collection of all groups satisfying a nontrivial finite direct product-closed group property]] || {{intermediate notions short|abelian normal subgroup|abelian characteristic subgroup}}
+|-
+| [[Stronger than::class two characteristic subgroup]] || characteristic subgroup that is also a [[group of nilpotency class two]] || || || {{intermediate notions short|class two characteristic subgroup|abelian characteristic subgroup}}
+|-
+| [[Stronger than::nilpotent characteristic subgroup]] || characteristic subgroup that is also a [[nilpotent group]] || follows from [[abelian implies nilpotent]] || follows from [[nilpotent not implies abelian]] || {{intermediate notions short|nilpotent characteristic subgroup|abelian characteristic subgroup}}
+|-
+| [[Stronger than::solvable characteristic subgroup]] || characteristic subgroup that is also a [[solvable group]] || (via nilpotent) || (via nilpotent) || {{intermediate notions short|solvable characteristic subgroup|abelian characteristic subgroup}}
+|}
 ===Related group properties===
 * [[Group in which every abelian characteristic subgroup is central]]