Group whose automorphism group is abelian

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Definition

Symbol-free definition

A group is said to be aut-abelian 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 aut-abelian if $Aut(G)$ is an Abelian group.

Relation with other properties

Stronger properties

• Cyclic group: For proof of the implication, refer cyclic implies aut-abelian and for proof of its strictness (i.e. the reverse implication being false) refer aut-abelian not implies cyclic.

Weaker properties

• Group of nilpotency class two: For proof of the implication, refer aut-abelian implies class two and for proof of its strictness (i.e. the reverse implication being false) refer class two not implies aut-abelian.
• Metabelian group

Facts

Any aut-abelian normal subgroup commutes with every element in the commutator subgroup.