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 $G$.