# Definition:Order of Group Element/Finite

< Definition:Order of Group Element(Redirected from Definition:Finite Order Element)

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## 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 finite order**, or **has finite order** if and only if there exists $k \in \Z_{> 0}$ such that $x^k = e_G$.

### Definition 2

$x$ **is of finite order**, or **has finite order** if and only if there exist $m, n \in \Z_{> 0}$ such that $m \ne n$ but $x^m = x^n$.

## Also known as

An **element of finite order** of $G$ is also known as a **torsion element** of $G$.