Definition:Order of Group Element/Finite/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 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 $G$.
Also see
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 5.4$. Cyclic groups
- 1978: John S. Rose: A Course on Group Theory ... (previous) ... (next): $0$: Some Conventions and some Basic Facts