Definition:Order of Group Element

Definition
The order $\left|{x}\right|$ of an element $x$ of a group $G$ is the smallest $k \in \N: k > 1$ such that $x^k = e_G$, where $e_G$ is the identity of $G$.

If there is no such $k$, then $x$ is said to be of infinite order, or has infinite order.

Otherwise it is of finite order, or has finite order.

Some sources call this the period of the element.