# Definition:Order of Group Element/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 $G$.

The **order of $x$ (in $G$)**, denoted $\order x$, is the order of the group generated by $x$:

- $\order x := \order {\gen x}$

## 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

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $25$. Cyclic Groups and Lagrange's Theorem - 1966: Richard A. Dean:
*Elements of Abstract Algebra*... (previous) ... (next): $\S 1.9$: Subgroups: Example $28$ - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 41$ - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): $\S 3.4$: Cyclic groups