Definition:Order of Group Element

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.