Definition:Order of Group Element/Definition 3
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Definition
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 largest $k \in \Z_{>0}$ such that:
- $\forall i, j \in \Z: 0 \le i < j < k \implies x^i \ne x^j$
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 $\map {\operatorname {Ord} } x$.