We know that $\mathbb{Q}\cong\mathbb{Z}\times\mathbb{Z}/\sim$, where the isomorphism is a ring isomorphism and the equivalence relation is defined as
$$(a,b)\sim(c,d)\Longleftrightarrow ad=bc$$
Then the relation is stated in terms of the integers' multiplication. So, if I want to give a presentation for the additive group of $\mathbb{Q}$, how should I modify the presentation
$$\mathbb{Z}\times\mathbb{Z}\cong\, \langle a,b|aba^{1}b^{-1} \rangle$$
as I can't use the multiplication? The equivalence relation that defines $\mathbb{Q}$ can be stated in additive terms as
$$(a,b)\sim(c,d)\Longleftrightarrow \begin{cases}a=nc\\b=nd\end{cases}$$
where of course $nx$ stands for $\sum_{i=1}^{n}x$, but I have no idea as how to insert it in the presentation. I'm in doubt that the additive group of $\mathbb{Q}$ is different from that of $\mathbb{Z}\times\mathbb{Z}$, but it seems to me that fractions should be identified on the (additive) group structure independently of the subsequent definition of the multiplication.
Best Answer
By the way, a presentation of $(\mathbb{Q},+)$ is
$$\langle x_1,x_2,\ldots \mid x_n^n=x_{n-1}, \ n \geq 2 \rangle.$$
To understand why, it is possible to consider the morphism induced by $x_n \mapsto \frac{1}{n!}$.
For more information, see Johnson's book, Presentations of groups, Chapter 5.7.