Order of a group

The order $|G|$ of a group $G$ is the size of its underlying set. For example, if $G=(X,\bullet)$ 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 $\langle g \rangle = \{ 1, g, g^2, \dots \}$ of $G$ [generating_set generated by] $g$ in the above sense.