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): aut-abelian group
View a complete list of such conjunctions

Statement

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

• $H$ is a normal subgroup of $G$.
• $H$ is an aut-abelian group: 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 aut-abelian	aut-abelian not implies cyclic	|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Subgroup contained in centralizer of commutator subgroup	contained in the centralizer of commutator subgroup	aut-abelian normal subgroup is contained in centralizer of commutator subgroup		|FULL LIST, MORE INFO
Normal subgroup contained in centralizer of commutator subgroup	contained in the centralizer of commutator subgroup	aut-abelian normal subgroup is contained in centralizer of commutator subgroup		|FULL LIST, MORE INFO
Commutator-in-center subgroup	its commutator with whole group is in its center	(via normal subgroup contained in centralizer of commutator 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	aut-abelian implies class two	class two not implies aut-abelian	|FULL LIST, MORE INFO