Multiplicative Group of Complex Numbers is not Direct Product of Reals with Reals
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Theorem
Let $\struct {\C_{\ne 0}, \times}$ be the multiplicative group of complex numbers.
Let $\struct {\R_{\ne 0}, \times}$ be the multiplicative group of real numbers.
Then the direct product $\struct {\R_{\ne 0}, \times} \times \struct {\R_{\ne 0}, \times}$ is not isomorphic with $\struct {\C_{\ne 0}, \times}$.
Proof
Let $\tuple {a, b}$ and $\tuple {c, d}$ be pairs of non-zero real numbers:
- $\tuple {a, b} \in \R_{\ne 0} \times \R_{\ne 0}$
- $\tuple {c, d} \in \R_{\ne 0} \times \R_{\ne 0}$
Then by definition of group direct product:
- $\tuple {a, b} \times \tuple {c, d} = \tuple {a \times c, b \times d}$
However, by interpreting $\tuple {a, b}$ and $\tuple {c, d}$ as complex numbers:
- $\tuple {a, b} \times \tuple {c, d} = \tuple {a \times c - b \times d, b \times c + a \times d}$
by definition of complex multiplication.
Hence the result.
$\blacksquare$
Sources
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Direct Products