Definition:Order of Group Element/Definition 1

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

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

The order of $x$ (in $G$), denoted $\order x$, is the smallest $k \in \Z_{> 0}$ such that $x^k = e_G$.

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 $\map {\operatorname {Ord} } x$.

Also see