Definition:Order of Group Element/Finite/Definition 1

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

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

$x$ is of finite order, or has finite order there exists $k \in \Z_{> 0}$ such that $x^k = e_G$.

Also see

 * Equivalence of Definitions of Finite Order Element