Definition:Order of Group Element

Definition
Let $G$ be a group whose identity is $e_G$.

Let $x \in G$ be an element of $G$.

The order $\left|{x}\right|$ of $x$ is the smallest $k \in \N: k > 1$ such that $x^k = e_G$.

That is, such that:
 * $\forall i, j \in \Z: 0 \le i < j < k \implies x^i \ne x^j$

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.

Also denoted as
The order of an element $g \in G$ is sometimes seen as $o \left({g}\right)$.

Some sources render it as $\operatorname{Ord} \left({g}\right)$.

Also see

 * Equal Powers of Group Element implies Finite Order