order (of a group)

The order of a group $G$ is the number of elements of $G$ , denoted $|G|$ ; if $|G|$ is finite, then $G$ is said to be a finite group.

The order of an element $g \in G$ is the smallest positive integer $n$ such that $g^n=e$ , where $e$ is the identity element; if there is no such $n$ , then $g$ is said to be of infinite order. By Lagrange's theorem, the order of any element in a finite group divides the order of the group.