Definition:Order of Group Element/Infinite/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 infinite order, or has infinite order there exists no $k \in \Z_{> 0}$ such that $x^k = e_G$:


 * $\order x = \infty$

Also denoted as
Hence, in the context of an element of infinite order, the notation $\map o x = \infty$ can sometimes be seen.

Also see

 * Equivalence of Definitions of Infinite Order Element