"On autocommutators and the autocommutator subgroup in infinite abelian groups"

t is well known that the set of commutators in a group usually does not form a subgroup. A similar phenomenon occurs for the set of autocommutators. There exists a group of order 64 and nilpotency class 2, where the set of autocommutators does not form a subgroup, and this group is of minimal order with this property. However, for finite abelian groups, the set of autocommutators is always a subgroup.We will show in this paper that this is no longer true for infinite abelian groups.We characterize finitely generated infinite abelian groups in which the set of autocommutators does not form a subgroup and show that in an infinite abelian torsion group the set of commutators is a subgroup. Lastly,we investigate torsion-free abelian groups with finite automorphism group andwe studywhether the set of autocommutators forms a subgroup in those groups.