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 real 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.

$\blacksquare$


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