Definition:Order of Group Element/Finite/Definition 1

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 exists $k \in \Z_{> 0}$ such that $x^k = e_G$.

Also known as

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

Also see

• Equivalence of Definitions of Finite Order Element

Sources

• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): order: 6. (of a group or element)