Cyclic Group/Examples/Subgroup of Multiplicative Group of Complex Numbers Generated by i

Example of Cyclic Group

Consider the multiplicative group of complex numbers $\struct {\C_{\ne 0}, \times}$.

Consider the subgroup $\gen i$ of $\struct {\C_{\ne 0}, \times}$ generated by $i$.

Then $\gen i$ is an (finite) cyclic group of order $4$.

Proof

We have that $\gen i$ is subgroup generated by a single element of $\struct {\C_{\ne 0}, \times}$

By definition, $\gen i$ is a cyclic group.

By Example: Order of Imaginary Unit in Multiplicative Group of Complex Numbers, $i$ is of finite order $4$.

The result follows by definition of finite cyclic group.

$\blacksquare$

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 39$. Cyclic groups: $(2)$