Order of Group Element/Examples/Rotation Through nth Part of Full Angle

Example of Order of Group Element
Let $G$ denote the group of isometries in the plane under composition of mappings.

Let $r$ be the rotation of the plane about a given point $O$ through an angle $\dfrac {2 \pi} n$, for some $n \in \Z_{> 0}$.

Then $r$ is the generator of a subgroup $\gen r$ of $G$ which is of order $n$.