Positive Rational Numbers under Division do not form Group

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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:

\(\displaystyle \paren {12 / 6} / 2\) \(=\) \(\displaystyle 2 / 1\)
\(\displaystyle \) \(=\) \(\displaystyle 1\)

whereas:

\(\displaystyle 12 / \paren {6 / 2}\) \(=\) \(\displaystyle 12 / 3\)
\(\displaystyle \) \(=\) \(\displaystyle 4\)

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.

$\blacksquare$


Sources