# Definition:Order of Group Element/Finite

## 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 $G$.