Group theoretical characterization of $\mathbb Q$ and $\mathbb Q^\star$
Matthew Martinez 
$\mathbb Z$ can be characterized in group-theoretic language as the infinite cyclic group. But what kind of a group is $\mathbb Q$ under addition? What about $\mathbb Q^\star$? What kind of subgroup structure do they have, and is there a canonical way to single them out as groups?
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$\begingroup$The additive group $\mathbb Q$ is the minimal non-trivial torsion-free divisible group.
$\endgroup$ 12 $\begingroup$There is a presentation $\mathbb{Q} = \langle (x_n)_{n \geq 1} : (n \cdot x_{nk} = x_k)_{n,k \geq 1} \rangle$, where $x_n$ corresponds to $\frac{1}{n}$. This may also be seen as the colimit of the diagram $(\mathbb{N}^+,|) \to \mathsf{Ab}$ which maps each $k \geq 1$ to $\mathbb{Z}$ and a divisibility relation $k|m$ to multiplication by $\frac{m}{k} : \mathbb{Z} \to \mathbb{Z}$. Actually, this way arbitrary localizations may be constructed (see for example Eisenbud's book on commutative algebra), and $\mathbb{Q}$ is the localization of the $\mathbb{Z}$-module $\mathbb{Z}$ at $\mathbb{Z} \setminus \{0\}$.
The non-trivial subgroups of $\mathbb{Q}$ contain some integer $\neq 0$, which induces an automorphism of $\mathbb{Q}$ so that this integer may be assumed to be $=1$. The classification thus amounts to a classification of the subgroups of $\mathbb{Q}/\mathbb{Z}$. This is well-known, they are of the form $\bigoplus_{p} A_p$, where for each prime $p$ either $A_p$ is either $\langle \frac{1}{p^n} \bmod \mathbb{Z} \rangle \cong \mathbb{Z}/p^n$ for some $n$ or $\langle \frac{1}{p^n} \bmod \mathbb{Z} : n \geq 1 \rangle \cong \mathbb{Z}/p^\infty$.
As for $\mathbb{Q}^*$, we may use prime factorizations to obtain a direct sum decomposition $\{\pm 1 \} \times \bigoplus_{p} \mathbb{Z}$. I doubt that we can classify the subgroups (without going into set theory ...). At least, we have some canonical subgroups, namely the ones induced by the summands. The subgroups containing $\pm 1$ and $\bigoplus_{p} 2\mathbb{Z}$ correpsond to sub-vector spaces of $\bigoplus_p \mathbb{F}_2$, which are already horrible.
$\endgroup$ 5 $\begingroup$If you characterize $\mathbb Q$,
you can see that $Aut(\mathbb Q)\cong \mathbb Q^* $ so you can see $\mathbb Q^*$ as a automorphism group of $\mathbb Q$.
And for $\mathbb Q$, there is an easy way by using "Rings";
$\mathbb Q$ is the field of fraction of $\mathbb Z$.
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