Definition:Order of Group Element/Infinite/Definition 2
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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 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
Sources
- 1978: John S. Rose: A Course on Group Theory ... (previous) ... (next): $0$: Some Conventions and some Basic Facts