What is known about the quotient group $\mathbb{R} / \mathbb{Q}$? Let $G = \mathbb{R} / \mathbb{Q}$. Is this an interesting group to study?
Is it isomorphic to any more natural mathematical objects?
 A: Well, it is an abelian torsion-free with an uncountable ammount (cardinal) of elements and, thus,  it is also a non finitely-generated group. 
You may also be interested in the fact that 
$$\Bbb R/\Bbb Q\cong\left(\Bbb R/\Bbb Z\right)/\left(\Bbb Q/\Bbb Z\right)$$ 
so that $\,\Bbb R/\Bbb Q\,$ is a quotient of the circle group
A: Its structure is completely determined, being it a divisible torsion free abelian group; it's torsion free because, for $n\in\mathbb{N}$, $n>0$, and $x\in\mathbb{R}$,
$$
n(x + \mathbb{Q}) = \mathbb{Q}
$$
is equivalent to $nx\in\mathbb{Q}$, so to $x\in\mathbb{Q}$.
Thus $\mathbb{R}/\mathbb{Q}$ is a vector space of dimension $\mathfrak{c}=\lvert\mathbb{R}\rvert$ over $\mathbb{Q}$, by a cardinality argument. This implies, as observed in comments, that, as $\mathbb{Q}$-vector spaces and so as abelian groups,
$$
\mathbb{R}/\mathbb{Q}\cong\mathbb{R}.
$$
Finding a basis for it would imply finding a Hamel basis of $\mathbb{R}$ over $\mathbb{Q}$, because of the canonical $\mathbb{Q}$-linear mapping
$$
\pi\colon \mathbb{R}\to\mathbb{R}/\mathbb{Q}
$$
Just choose elements $x_\alpha\in\mathbb{R}$ that are mapped to a basis of $\mathbb{R}/\mathbb{Q}$; adding $1$ to this family would produce a Hamel basis for $\mathbb{R}$.
