Locally cyclic implies abelian automorphism group

Statement
Any locally cyclic group is a group whose automorphism group is abelian, i.e., it has an abelian automorphism group.

Related facts

 * Cyclic implies abelian automorphism group
 * Abelian and abelian automorphism group not implies locally cyclic
 * Abelian automorphism group not implies abelian
 * Finite abelian and abelian automorphism group implies cyclic