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 $G$.
Definition 1
$x$ is of infinite order, or has infinite order, if and only if there exists no $k \in \Z_{> 0}$ such that $x^k = e_G$:
- $\order x = \infty$
Definition 2
$x$ is of infinite order, or has infinite order if and only if the powers $x, x^2, x^3, \ldots$ of $x$ are all distinct:
- $\order x = \infty$
Definition 3
$x$ is of infinite order, or has infinite order if and only if the group $\gen x$ generated by $x$ is of infinite order.
- $\order x = \infty \iff \order {\gen x} = \infty$
Also known as
Some sources refer to the order of an element of a group as its period.
Also denoted as
The order of an element $x$ in a group is sometimes seen as $\map o x$.
Some sources render it as $\map {\operatorname {Ord} } x$.
Hence, in the context of an element of infinite order, the notation $\map o x = \infty$ can sometimes be seen.