Order of Group Element/Examples/Imaginary Unit in Multiplicative Group of Complex Numbers
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Examples of Order of Group Element
Consider the multiplicative group of complex numbers $\struct {\C_{\ne 0}, \times}$.
The order of $i$ in $\struct {\C_{\ne 0}, \times}$ is $4$.
Proof
We have:
\(\ds i^1\) | \(=\) | \(\ds 1\) | ||||||||||||
\(\ds i^2\) | \(=\) | \(\ds -1\) | ||||||||||||
\(\ds i^3\) | \(=\) | \(\ds -i\) | ||||||||||||
\(\ds i^4\) | \(=\) | \(\ds 1\) |
Hence the result by definition of order of group element.
$\blacksquare$
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 38$. Period of an element: Illustrations: $\text{(ii)}$