Definition:Order of Group Element/Infinite/Definition 2

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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 if and only if the powers $x, x^2, x^3, \ldots$ of $x$ are all distinct:

$\order x = \infty$

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.

Also see