Definition:Order of Group Element/Infinite/Definition 3

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 the group $\gen x$ generated by $x$ is of infinite order.


 * $\order x = \infty \iff \order {\gen x} = \infty$

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