# 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 $\left\vert{x}\right\vert$, is the largest $k \in \Z_{\gt 0}$ 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 $\map {\operatorname {Ord} } x$.