Order of an element

WARNING: POTENTIAL TERMINOLOGICAL CONFUSION: Please don't confuse this with order of a group

Definition

The order of an element $x$ in a group $G$ is the smallest positive integer $n$ for which $x^n$ is the identity element.

Such a $n$ may not always exist (if it exists, $x$ is said to be of finite order, or is termed a torsion element). It does exist when the group is finite. In fact, by Lagrange's theorem, the order of $x$ divides the order of $G$ (where order here means the total cardinality of the group).