The order of a group $G$ is the size of its underlying set. For example, if $G = (X, ∙)$ and $X$ has nine elements, we say that $G$ has order $9$ . If $X$ is infinite, we say $G$ is infinite; if $X$ is finite, we say $G$ is finite.

The order of an element $g \in G$ of a group is the smallest nonnegative integer $n$ such that $g^{n} = e$ , or $\infty$ if there is no such integer. The relationship between this usage of order and the above usage of order is that the order of $g \in G$ in this sense is the order of the Subgroup $⟨ g ⟩ = {1, g, g^{2}, \dots}$ of $G$ generated by $g$ in the above sense.

Order of a group