After studying subgroups inside quotient groups, we now count quotient groups in the finite case. Since the elements of $G/H$ are cosets of $H$, the order of $G/H$ is exactly the number of cosets of $H$ in $G$. Thus the order of a quotient group is the index of $H$ in $G$. When $G$ is finite, Lagrange's theorem converts this index into the formula $|G/H|=|G|/|H|$. This formula is simple, but students often misuse it by counting elements of $H$ instead of counting cosets.

Let $(G,\circ)$ be a finite group and let $H\trianglelefteq G$. The \textbf{order of the quotient group} $G/H$ is the number of cosets of $H$ in $G$, denoted by $$\begin{equation*} |G/H|. \end{equation*}$$