I'm reading this paper on torsion-free abelian groups and I'm trying to work out all the details I found at the first page.
Following Fuchs' book on "Infinite Abelian Groups", I was able to explain rigorously to myself why the space of all torsion-free abelian groups of rank $1\le r\le n$, where $n$ is a positive integer, can be identified with the space $S(\Bbb{Q}^n)$ of all non-zero subgroups of $\Bbb{Q}^n$ with $1\le r\le n$ linearly independent elements ($15$-th row of the first page starting after Introduction).
Recall that there is a bijection between the power set $\mathcal{P}(\Bbb{Q}^n)$ and the space $2^{\Bbb{Q}^n}$ of all characteristic functions on $\Bbb{Q}^n$, then we can pull-back on $\mathcal{P}(\Bbb{Q}^n)$ the product topology in such a way that the topology on $\mathcal{P}(\Bbb{Q}^n)$ has the basic open sets of the form $\{A\in\mathcal{P}(\Bbb{Q}^n\mid F_1\subseteq A,F_2\cap A=\emptyset)\}$, where $F_1,F_2$ are finite subsets of $\Bbb{Q}^n$.
(1) Now, in order to prove that $S(\Bbb{Q}^n)$ is a Borel subset of the power set $\mathcal{P}(\Bbb{Q})$ (row $19$), I have used the Lopez-Escobar Theorem. This result states that, if $L$ is a countable first-order language and $Mod(L)$ is the (Polish) space of countable (hence with universe $\Bbb{N}$) $L$-structures, then invariant Borel subsets of $Mod(L)$ are exactly those of the form $Mod(\sigma)=\{x\in Mod(L)\mid x\models \sigma\}$, for some sentence $\sigma$ in the infinitary logic $L_{\omega_1\omega}$. Note: After the comments of @Ycor, I understand that, probably, I can show directly and in a simpler way that $S(\Bbb{Q}^n)$ is a Borel subset of $\mathcal{P}(\Bbb{Q}^n)$. So the $\boldsymbol{1}$-st question is: how can I prove it?
(2) The author claims that "the natural action of $GL_n(\Bbb{Q})$ on the vector space $\Bbb{Q}^n$ induces a corresponding Borel action on $S(\Bbb{Q}^n)$". My $\boldsymbol{2}$-nd question is: how can I prove the action is Borel?
Using definitions, I tried to prove that the preimage under the function $GL_n(\Bbb{Q})\times S(\Bbb{Q}^n)\to S(\Bbb{Q}^n)$ defined by$((q_{ij})_{i,j},A)\mapsto (q_{ij})_{i,j}\cdot A$ of an open subset of $S(\Bbb{Q}^n)$ is Borel, but I don't get it. Moreover, I showed that the natural action is actually continuous, but I'm not sure it can help.
(3) Finally (row $25$), the author states that if $A,B\in S(\Bbb{Q}^n)$, then $A\cong B$ iff there exists $\varphi\in GL_n(\Bbb{Q})$ s.t. $\varphi(A)=B$". As I understand it, I have only to prove that $Aut(\Bbb{Q}^n)=GL_n(\Bbb{Q})$. Clearly, $\supseteq$ holds. My Last question is: what about $\subseteq$?
I'm a beginner and this is the first paper I read, so try to give an explicit answer, whenever possible. Thank you in advance for your help.