# 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

View a complete list of group propertiesVIEW 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 is said to be a **group whose automorphism group is abelian** or a **group with abelian automorphism group** if 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 is a group whose automorphism group is abelian if and only if for every group containing as a normal subgroup, is also contained in the centralizer of derived subgroup of .

## 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