Normal subgroup whose automorphism group is abelian

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		Normal subgroup contained in centralizer of derived subgroup|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)		Normal subgroup contained in centralizer of derived subgroup, Normal subgroup whose inner automorphism group is central in automorphism group|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	Commutator-in-center subgroup, Normal subgroup contained in centralizer of derived subgroup, Normal subgroup whose inner automorphism group is central in automorphism group|FULL LIST, MORE INFO