# 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

- 1964: Walter Ledermann:
*Introduction to the Theory of Finite Groups*(5th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Group Concept: Examples: $(2)$