Rational Numbers under Multiplication do not form Group
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Theorem
The algebraic structure $\struct {\Q, \times}$ consisting of the set of rational numbers $\Q$ under multiplication $\times$ is not a group.
Proof
Aiming for a contradiction, suppose that $\struct {\Q, \times}$ is a group.
By the definition of the number $0 \in \Q$:
- $\forall x \in \Q: x \times 0 = 0 = 0 \times x$
Thus $0$ is a zero in the abstract algebraic sense.
From Group with Zero Element is Trivial, $\struct {\Q, \times}$ is the trivial group.
But $\Q$ contains other elements besides $0$.
From this contradiction it follows that $\struct {\Q, \times}$ is not a group.
$\blacksquare$
Also see
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 7$: Semigroups and Groups: Example $7.2$