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 $⟨g⟩=\{1,g,g^2,\dots \}$ of $G$ generated by $g$ in the above sense.