Group whose automorphism group is abelian

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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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Definition

QUICK PHRASES: abelian automorphism group, any two automorphisms commute

Symbol-free definition

A group is said to be a group whose automorphism group is abelian or a group with abelian automorphism group if its automorphism group is an abelian group or equivalently, if any two automorphisms of the group commute.

Definition with symbols

A group $G$ is said to be a group whose automorphism group is abelian or a group with abelian automorphism group if $\operatorname{Aut}(G)$ is an abelian group.

Formalisms

In terms of the supergroup property collapse operator

This group property can be defined in terms of the collapse of two subgroup properties in the following sense. Whenever the given group is embedded as a subgroup satisfying the first subgroup property (normal subgroup), in some bigger group, it also satisfies the second subgroup property (normal subgroup contained in centralizer of commutator subgroup), and vice versa.
View other group properties obtained in this way

A group $H$ is a group whose automorphism group is abelian if and only if for every group $G$ containing $H$ as a normal subgroup, $H$ is also contained in the centralizer of derived subgroup of $G$.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
cyclic group	generated by one element	cyclic implies abelian automorphism group	follows from abelian automorphism group not implies abelian	Locally cyclic group|FULL LIST, MORE INFO
locally cyclic group	every finitely generated subgroup is cyclic	locally cyclic implies abelian automorphism group	abelian and abelian automorphism group not implies locally cyclic	|FULL LIST, MORE INFO
group whose automorphism group is cyclic	automorphism group is a cyclic group	(follows from cyclic implies abelian)	follows from any example of a cyclic group whose automorphism group is not cyclic, e.g., cyclic group:Z8.

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
group whose inner automorphism group is central in automorphism group	inner automorphism group is in center of automorphism group			|FULL LIST, MORE INFO
group of nilpotency class two	inner automorphism group is abelian	aut-abelian implies class two	class two not implies aut-abelian	Group whose inner automorphism group is central in automorphism group|FULL LIST, MORE INFO
metabelian group	abelian normal subgroup with abelian quotient	(via class two)	(via class two)	Group of nilpotency class two|FULL LIST, MORE INFO
group whose automorphism group is nilpotent	automorphism group is nilpotent	follows from abelian implies nilpotent	nilpotent automorphism group not implies abelian automorphism group	|FULL LIST, MORE INFO

Related subgroup properties

• Aut-abelian normal subgroup is a normal subgroup of a group that is aut-abelian as a group.

Facts

• Derived subgroup centralizes normal subgroup whose automorphism group is abelian: Any normal subgroup whose automorphism group is abelian commutes with every element in the derived subgroup. Hence, it is contained in the centralizer of commutator subgroup.
• Finite abelian and abelian automorphism group implies cyclic