Originally Posted by

**dwsmith** $\displaystyle (\mathbb{Q}, +)$ by $\displaystyle \left |\frac{a}{b}\right | < 1, \ a,b\in\mathbb{Z}$ and $\displaystyle b\neq 0$ is this a group.

Identity is 0.

Associative is true.

Inverse (I am not sure but I don't think this works).

If we let $\displaystyle -\frac{a}{b}$ be the inverse, we have a problem since

$\displaystyle \displaystyle\left |\frac{a}{b}\right |=\begin{cases}\frac{a}{b}, \ \frac{a}{b}\geq 0\\-\frac{a}{b}, \ \frac{a}{b}< 0\end{cases}$

So is this not a group or is this approach to the inverse wrong?