Positive Rational Numbers under Division do not form Group

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Theorem

Let $\struct {\Q_{>0}, /}$ denote the algebraic structure consisting of the set of (strictly) positive rational numbers $\Q_{>0}$ under the operation $/$ of division.


We have that $\struct {\Q_{>0}, /}$ is not a group.


Proof

In order to be a group, it is necessary that $\struct {\Q_{>0}, /}$ be an associative structure.

But consider the elements $2, 6, 12$ of $\Q_{>0}$.

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 $a, b, c \in \Q_{>0}$:

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

and so $/$ is not associative on $\Q_{>0}$.

Hence $\struct {\Q_{>0}, /}$ is not a group.

$\blacksquare$


Sources