Additive Group of Rational Numbers is not Isomorphic to Multiplicative Group of Rational Numbers
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Theorem
Let $\struct {\Q, +}$ be the additive group of rational numbers.
Let $\struct {\Q_{\ne 0}, \times}$ be the multiplicative group of rational numbers.
Then $\struct {\Q_{\ne 0}, \times}$ is not isomorphic to $\struct {\Q, +}$.
Proof
A direct application of Additive Group and Multiplicative Group of Field are not Isomorphic.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 6$: Isomorphisms of Algebraic Structures: Exercise $6.10$