Abelian fully invariant subgroup: Difference between revisions

Latest revision as of 00:33, 13 August 2013

This article describes a property that arises as the conjunction of a subgroup property: fully invariant 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 fully invariant subgroup or fully invariant abelian subgroup if $H$ is an abelian group as a group in its own right (or equivalently, is an abelian subgroup of $G$ ) and is also a fully invariant subgroup (or fully characteristic subgroup) of $G$ , i.e., for any endomorphism $\sigma$ of $G$ , we have $\sigma (H)\subseteq H$ .

Examples

Examples based on subgroup-defining functions and series

For a solvable group, the penultimate member of the derived series (i.e., the last member before reaching the trivial subgroup) is an abelian fully invariant subgroup.
For a nilpotent group, 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.

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:

Relation with other properties

Stronger properties

Property	Meaning	Intermediate notions
fully invariant subgroup of abelian group	the whole group is an abelian group and the subgroup is a fully invariant subgroup	\|FULL LIST, MORE INFO
characteristic direct factor of abelian group	the whole group is an abelian group and the subgroup is a characteristic subgroup as well as a direct factor	\|FULL LIST, MORE INFO
abelian verbal subgroup	the subgroup is abelian as well as a verbal subgroup	\|FULL LIST, MORE INFO
verbal subgroup of abelian group	the subgroup is a verbal subgroup and the whole group is an abelian group	\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
abelian characteristic subgroup	abelian and a characteristic subgroup -- invariant under all automorphisms	follows from fully invariant implies characteristic	follows from characteristic not implies fully invariant in finite abelian group	\|FULL LIST, MORE INFO
abelian normal subgroup	abelian and a normal subgroup -- invariant under all inner automorphisms	(via abelian characteristic, 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

@@ Line 3: / Line 3: @@
 ==Definition==
-A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed an '''abelian fully invariant subgroup''' or '''fully invariant abelian subgroup''' if <math>H</math> is an [[abelian group]] as a group in its own right (or equivalently, is an [[abelian subgroup]] of <math>G</math>) and is also a [[fully invariant subgroup]] (or fully characteristic subgroup) of <math>G</math>.
+A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed an '''abelian fully invariant subgroup''' or '''fully invariant abelian subgroup''' if <math>H</math> is an [[abelian group]] as a group in its own right (or equivalently, is an [[abelian subgroup]] of <math>G</math>) and is also a [[fully invariant subgroup]] (or fully characteristic subgroup) of <math>G</math>, i.e., for any [[endomorphism]] <math>\sigma</math> of <math>G</math>, we have <math>\sigma(H) \subseteq H</math>.
-==Facts==
+==Examples==
-* [[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.
+===Examples based on subgroup-defining functions and series===
+* For a [[solvable group]], the penultimate member of the [[derived series]] (i.e., the last member before reaching the trivial subgroup) is an abelian fully invariant subgroup.
+* For a [[nilpotent 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.
+{{subgroups satisfying group-subgroup property conjunction sorted by importance rank|fully invariant subgroup|abelian group}}
 ==Relation with other properties==
@@ Line 16: / Line 21: @@
 ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
 |-
-| [[Weaker than::fully invariant subgroup of abelian group]] || || || || {{intermediate notions short|abelian fully invariant subgroup|fully invariant subgroup of abelian group}}
+| [[Weaker than::fully invariant subgroup of abelian group]] || the whole group is an [[abelian group]] and the subgroup is a [[fully invariant subgroup]] || || || {{intermediate notions short|abelian fully invariant subgroup|fully invariant subgroup of abelian group}}
+|-
+| [[Weaker than::characteristic direct factor of abelian group]] || the whole group is an [[abelian group]] and the subgroup is a [[characteristic subgroup]] as well as a [[direct factor]] || || || {{intermediate notions short|abelian fully invariant subgroup|characteristic direct factor of abelian group}}
+|-
+| [[Weaker than::abelian verbal subgroup]] || the subgroup is abelian as well as a [[verbal subgroup]] || || || {{intermediate notions short|abelian fully invariant subgroup|abelian verbal subgroup}}
+|-
+| [[Weaker than::verbal subgroup of abelian group]] || the subgroup is a [[verbal subgroup]] and the whole group is an [[abelian group]] ||  || || {{intermediate notions short|abelian fully invariant subgroup|verbal subgroup of abelian subgroup}}
 |}
 ===Weaker properties===