# Definition:Order of Group Element/Infinite

## Definition

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

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

### Definition 1

$x$ **is of infinite order**, or **has infinite order** if and only if there exists no $k \in \Z_{> 0}$ such that $x^k = e_G$:

- $\order x = \infty$

### Definition 2

$x$ **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:

- $\order x = \infty$

### Definition 3

$x$ **is of infinite order**, or **has infinite order** if and only if the group $\gen x$ generated by $x$ is of infinite order.

- $\order x = \infty \iff \order {\gen x} = \infty$

## 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.