# Infinite noncyclic groups with abelian automorphism group

Is there any infinite noncyclic group whose automorphism group is abelian..can we find a sufficient condition for infinite group to have an abelian automorphism group Thank you

• Group of integers? Jun 9 '19 at 17:43
• @Mare obviously not. A group with abelian automorphism group has to be 2-step nilpotent.
– YCor
Jun 9 '19 at 17:48
• Every cyclic group has abelian automorphism group..i am seeking for a non cyclic infinite group or a sufficient condition that makes an infinite group to be a Miller group Jun 9 '19 at 17:49
• Additive group of rationals has abelian automorphism group too, and is not cyclic. Jun 9 '19 at 18:06
• @YCor So presumably you consider abelian groups to be $2$-step nilpotent? (Perhaps $n$-step nilpotent does not mean the same as nilpotent of class $n$.) Jun 9 '19 at 19:06

There is an infinite abelian group $$A$$ with $$Aut(A) = G$$ for a finite abelian group $$G$$ iff $$G$$ is of even order and is a direct product of cyclic groups of orders 2, 3, and 4 with the property that if $$G$$ has an element of order 12 it also has an element of order 2 that is not a sixth power.
Examples of torsion-free groups $$A$$ with $$Aut(A)=G$$ for $$G$$ as above are constructed in [Fuchs: Infinite abelian groups II, Chap. XVI, Sect. 116] in examples 1, 2 and Theorem 116.2.