# 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