# Cyclic Group/Examples

## Examples of Cyclic Groups

### Subgroup of $\struct {\R_{\ne 0}, \times}$ Generated by $2$

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

Consider the subgroup $\gen 2$ of $\struct {\R_{\ne 0}, \times}$ generated by $2$.

Then $\gen 2$ is an infinite cyclic group.

### Subgroup of $\struct {\C_{\ne 0}, \times}$ Generated by $i$

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 $4$.