Cyclic Group/Examples/Subgroup of Multiplicative Group of Complex Numbers Generated by i
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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: $(2)$