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