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.

Also known as
Some sources call this the period of the element.