Definition:Order of Group Element/Infinite

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

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

If there exists no $k \in \Z_{> 0}$ such that $x^k = e_G$, then $x$ is of infinite order, or has infinite order:


 * $\order x = \infty$

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