Abelian characteristic subgroup: Difference between revisions

Latest revision as of 20:39, 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

A subgroup $H$ of a group $G$ is termed an abelian characteristic subgroup if $H$ is abelian as a group (i.e., $H$ is an abelian subgroup of $G$ and also, $H$ is a characteristic subgroup of $G$ , i.e., $H$ is invariant under all automorphisms of $G$ .

Examples

Here are some examples of subgroups in basic/important groups satisfying the property:

Here are some examples of subgroups in relatively less basic/important groups satisfying the property:

	Group part	Subgroup part	Quotient part
Center of dihedral group:D8	Dihedral group:D8	Cyclic group:Z2	Klein four-group

Here are some examples of subgroups in even more complicated/less basic groups satisfying the property:

Facts

Second half of lower central series of nilpotent group comprises abelian groups: In particular, this means that for a group $G$ of nilpotency class $c$ , all the subgroups $\gamma _{k}(G),k\geq (c+1)/2$ are abelian characteristic subgroups.
The center of any group is an abelian characteristic subgroup.

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 and the subgroup is characteristic	\|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
abelian subnormal subgroup	abelian and a subnormal subgroup			\|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 3: / Line 3: @@
 ==Definition==
-===Symbol-free definition===
+A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed an '''abelian characteristic subgroup''' if <math>H</math> is [[abelian group|abelian]] as a group (i.e., <math>H</math> is an [[abelian subgroup]] of <math>G</math> and also, <math>H</math> is a [[characteristic subgroup]] of <math>G</math>, i.e., <math>H</math> is invariant under all [[automorphism]]s of <math>G</math>.
-A [[subgroup]] of a [[group]] is termed an '''Abelian characteristic subgroup''' if it is [[Abelian group|Abelian]] as a group and [[characteristic subgroup|characteristic]] as a subgroup.
+==Examples==
-==Examples==
+{{subgroups satisfying group-subgroup property conjunction sorted by importance rank|characteristic subgroup|abelian group}}
+==Facts==
-{{group-subgroup property conjunction see examples|characteristic subgroup|abelian group}}
+* [[Second half of lower central series of nilpotent group comprises abelian groups]]: In particular, this means that for a group <math>G</math> of nilpotency class <math>c</math>, all the subgroups <math>\gamma_k(G), k \ge (c + 1)/2</math> are abelian characteristic subgroups.
+* The [[center]] of any group is an abelian characteristic subgroup.
 ==Relation with other properties==
@@ Line 15: / Line 18: @@
 ===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]] and the subgroup is characteristic || || || {{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::abelian subnormal subgroup]] || abelian and a [[subnormal subgroup]] || || || {{intermediate notions short|abelian subnormal 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]]
+* [[Group in which every abelian characteristic subgroup is central]]