If $G$ is a locally cyclic group , then is $\operatorname{Aut}(G)$ abelian?

Question

Let $G$ be a locally cyclic group, then is it true that $\operatorname{Aut}(G)$ is abelian ? I know that $G$ has to be abelian but I cannot decide for $\operatorname{Aut}(G)$.

Answer 1

I don't know if there's a nice way to do this. Here's a not nice way.

Start with the observations 1) a locally cyclic group is either torsion or torsion free, and 2) every automorphism of a group lifts to an automorphism of its injective hull. The point of 2) is that it means we can get our result by proving automorphism groups of injective hulls of locally cyclic groups are abelian.

Consider the torsion-free case. Then $G$ embeds into $\mathbb{Q}$ (this is a nice argument and not difficult: pick $g \in G\setminus \{e\}$ and map it to 1, then for any $h \in G$ we have $nh=mg$ for some $n,m\in\mathbb{Z}$ by local cyclicity so map $h\mapsto m/n$, the details are given here http://groupprops.subwiki.org/wiki/Equivalence_of_definitions_of_locally_cyclic_aperiodic_group). Its injective hull is then $\mathbb{Q}$, and so its automorphism group $\mathbb{Q}^*$ is abelian.

Now let $G$ be torsion. Then $G$ is a direct product of locally cyclic $p$-groups, one for each prime $p$ (contained in http://groupprops.subwiki.org/wiki/Equivalence_of_definitions_of_locally_cyclic_periodic_group). Its injective hull is then a restricted direct product of Prüfer groups $\mathbb{Z}(p^\infty)$, at most one for each prime. The automorphism group of the injective hull is the product of the automorphism groups of the factors, which are the invertible $p$-adic integers, so abelian.

This problem is Exercise 113.2 p.254 in Fuchs' Infinite Abelian Groups vol. 2.