Order of Group Element/Examples/Imaginary Unit in Multiplicative Group of Complex Numbers

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: $\text{(ii)}$