# 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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VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## 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