Multiplicative Group of Positive Rationals is Non-Cyclic

Theorem
Let $\struct {\Q_{>0}, \times}$ be the multiplicative group of positive rational numbers.

Then $\struct {\Q_{>0}, \times}$ is not a cyclic group.

Proof
$\struct {\Q_{>0}, \times}$ is cyclic.

Then $\struct {\Q_{>0}, \times}$ has a generator $x$ such that $x > 1$.

It would follow that:
 * $\Q = \set {\ldots, x^{-2}, x^{-1}, 1, x, x^2, \ldots}$

where the elements are arranged in ascending order.

But then consider $y \in \Q: y = \dfrac {1 + x} 2$.

So $1 < y < x$ and so $y \notin \Q$.

From this contradiction it is concluded that there can be no such single generator of $\struct {\Q_{>0}, \times}$.

Therefore, by definition, $\struct {\Q_{>0}, \times}$ is not a cyclic group.