Normal subgroup whose automorphism group is abelian: Difference between revisions

Latest revision as of 19:41, 20 June 2013

This article describes a property that arises as the conjunction of a subgroup property: normal subgroup with a group property (itself viewed as a subgroup property): group whose automorphism group is abelian
View a complete list of such conjunctions

Statement

A subgroup $H$ of a group $G$ is termed an normal subgroup whose automorphism group is abelian of $G$ if it satisfies both these conditions:

$H$ is a normal subgroup of $G$ .
$H$ is a group whose automorphism group is abelian: the automorphism group $\operatorname {Aut} (H)$ is an abelian group.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
cyclic normal subgroup	cyclic group and normal subgroup	cyclic implies abelian automorphism group	abelian automorphism group not implies abelian	\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
subgroup contained in centralizer of derived subgroup	contained in the centralizer of derived subgroup	derived subgroup centralizes normal subgroup whose automorphism group is abelian		\|FULL LIST, MORE INFO
normal subgroup contained in centralizer of derived subgroup	contained in the centralizer of derived subgroup	derived subgroup centralizes normal subgroup whose automorphism group is abelian		\|FULL LIST, MORE INFO
normal subgroup whose inner automorphism group is central in automorphism group	normal subgroup that is also a group whose inner automorphism group is central in automorphism group			\|FULL LIST, MORE INFO
commutator-in-center subgroup	its commutator with whole group is in its center	(via normal subgroup contained in centralizer of derived subgroup)		\|FULL LIST, MORE INFO
commutator-in-centralizer subgroup	its commutator with whole group centralizes it
hereditarily 2-subnormal subgroup	every subgroup of it is 2-subnormal
class two normal subgroup	normal subgroup and has nilpotency class at most two	abelian automorphism group implies class two	class two not implies abelian automorphism group	\|FULL LIST, MORE INFO

@@ Line 1: / Line 1: @@
-{{group-subgroup property conjunction|normal subgroup|aut-abelian group}}
+{{group-subgroup property conjunction|normal subgroup|group whose automorphism group is abelian}}
 ==Statement==
-A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed an '''aut-abelian normal subgroup''' of <math>G</math> if it satisfies ''both'' these conditions:
+A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed an '''normal subgroup whose automorphism group is abelian''' of <math>G</math> if it satisfies ''both'' these conditions:
 * <math>H</math> is a [[normal subgroup]] of <math>G</math>.
-* <math>H</math> is an [[aut-abelian group]]: the [[automorphism group]] <math>\operatorname{Aut}(H)</math> is an [[abelian group]].
+* <math>H</math> is a [[group whose automorphism group is abelian]]: the [[automorphism group]] <math>\operatorname{Aut}(H)</math> is an [[abelian group]].
 ==Relation with other properties==
 ===Stronger properties===
+{| class="sortable" border="1"
-* [[Weaker than::Cyclic normal subgroup]]: {{proofofstrictimplicationat|[[cyclic implies aut-abelian]]|[[aut-abelian not implies cyclic]]}}
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Weaker than::cyclic normal subgroup]] || cyclic group and normal subgroup || [[cyclic implies abelian automorphism group]] || [[abelian automorphism group not implies abelian]] || {{intermediate notions short|normal subgroup whose automorphism group is abelian|cyclic normal subgroup}}
+|}
 ===Weaker properties===
-* [[Stronger than::Class two normal subgroup]]: {{proofofstrictimplicationat|[[aut-abelian implies class two]]|[[class two not implies aut-abelian]]}}
+{| class="sortable" border="1"
-* Subgroup contained in the [[centralizer of commutator subgroup]]: {{proofat|[[commutator subgroup centralizes aut-abelian normal subgroup]]}}
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Stronger than::subgroup contained in centralizer of derived subgroup]] || contained in the [[centralizer of derived subgroup]] || [[derived subgroup centralizes normal subgroup whose automorphism group is abelian]] || || {{intermediate notions short|subgroup contained in centralizer of derived subgroup|normal subgroup whose automorphism group is abelian}}
+|-
+| [[Stronger than::normal subgroup contained in centralizer of derived subgroup]] || contained in the [[centralizer of derived subgroup]] || [[derived subgroup centralizes normal subgroup whose automorphism group is abelian]] || || {{intermediate notions short|normal subgroup contained in centralizer of derived subgroup|normal subgroup whose automorphism group is abelian}}
+|-
+| [[Stronger than::normal subgroup whose inner automorphism group is central in automorphism group]] || [[normal subgroup]] that is also a [[group whose inner automorphism group is central in automorphism group]] || || || {{intermediate notions short|normal subgroup whose inner automorphism group is central in automorphism group|normal subgroup whose automorphism group is abelian}}
+|-
+| [[Stronger than::commutator-in-center subgroup]] || its [[commutator of two subgroups|commutator with whole group]] is in its [[center]] || (via normal subgroup contained in centralizer of derived subgroup) || || {{intermediate notions short|commutator-in-center subgroup|normal subgroup whose automorphism group is abelian}}
+|-
+| [[Stronger than::commutator-in-centralizer subgroup]] || its commutator with whole group centralizes it || || ||
+|-
+| [[Stronger than::hereditarily 2-subnormal subgroup]] || every subgroup of it is 2-subnormal || || ||
+|-
+| [[Stronger than::class two normal subgroup]] || normal subgroup and has nilpotency class at most two || [[abelian automorphism group implies class two]] || [[class two not implies abelian automorphism group]] || {{intermediate notions short|class two normal subgroup|normal subgroup whose automorphism group is abelian}}
+|}