Positive Rational Numbers under Division do not form Group

Theorem

Let $\struct {\Q, /}$ denote the algebraic structure consisting of the set of rational numbers $\Q$ under the operation $/$ of division.

We have that $\struct {\Q, /}$ is not a group.

Proof

In order to be a group, it is necessary that $\struct {\Q, /}$ be an associative structure.

But consider the elements $2, 6, 12$ of $\Q$.

We have:

 $\ds \paren {12 / 6} / 2$ $=$ $\ds 2 / 1$ $\ds$ $=$ $\ds 1$

whereas:

 $\ds 12 / \paren {6 / 2}$ $=$ $\ds 12 / 3$ $\ds$ $=$ $\ds 4$

That is:

$\paren {12 / 6} / 2 \ne 12 / \paren {6 / 2}$

So it is not generally the case that for $a, b, c \in \Q$:

$\paren {a / b} / c = a / \paren {b / c}$

and so $/$ is not associative on $\Q$.

Hence $\struct {\Q, /}$ is not a group.

$\blacksquare$