# Find a torsion free, non cyclic, abelian group $A$ such that $\operatorname{Aut}(A)$ has order 2

Is there any chance to find a torsion free, non cyclic, abelian group $A$ such that $\operatorname{Aut}(A)=\mathbb Z_2$? ($\mathbb Z_2$ is the cyclic group of order $2$)

Notation

$\operatorname{Aut}(A)$ is the group of autmorphisms of $A$.

#### Solutions Collecting From Web of "Find a torsion free, non cyclic, abelian group $A$ such that $\operatorname{Aut}(A)$ has order 2"

Yes. Let $A=\left\{ \tfrac ab : a,b \in \mathbb{Z}, b \textrm{ is square-free} \right\}$.

If $A \leq \mathbb{Q}$ then its ring of endomorphisms is a subring of $\mathbb{Q}$. Its automorphisms are the invertible elements, those $\tfrac ab \in \mathbb{Q}^\times$ such that multiplication by them is both injective (automatic) and surjective (special). An abelian group $A$ is said to be $n$-divisible for an element $n \in \mathbb{Z}\setminus\{0\}$ when $nA = A$, that is, when multiplication by $n$ is surjective. Suppose $\tfrac ab A = A$ with $\gcd(a,b)=1$. Find $u,v$ such that $au+bv=1$, then for $x \in A$, we have $\tfrac 1b x = \tfrac{au+bv}{b} = u \tfrac ab x + v x \in A$, so $A$ is $b$-divisible. By symmetry, since $\tfrac ba = {\left(\tfrac ab\right)}^{-1} \in \newcommand{\Aut}{\operatorname{Aut}}\Aut(A)$ as well, $A$ is $a$-divisible.

In other words, $\Aut(A) = \left\{ \tfrac ab \in \mathbb{Q}^\times : A \text{ is$a$-divisible and$b$-divisible } \right\}$.

Every abelian group is $1$-divisible and $-1$-divisible, so $Z_2 \cong \{1,-1\} \leq \Aut(A)$. Most torsion-free rank 1 groups are only $1$ and $-1$ divisible, but they might also be $p$-divisible for some primes $p$. If they are, then their automorphism group is bigger. Explicitly, $\Aut(A) \cong Z_2 \times F$, where $F$ is the free abelian group with basis those primes $p$ such that $A$ is $p$-divisible.

For the specific group $A=\left\{ \tfrac ab : a,b \in \mathbb{Z}, b \textrm{ is square-free} \right\}$ that I mention, $A$ is not $p$-divisible for any prime $p$ (since the denominators would eventually be divisible by $p^2$).

Most torsion-free rank-one $A$ work, you just need to make sure they aren’t $p$-divisible for any prime $p$.

Your earlier question ($\Aut(A)$ has a unique element of order 2) was satisfied by all torsion-free rank-one groups.