Having studied normal subgroups, we now use them to build new groups from old groups. The quotient group is formed by taking all cosets of a normal subgroup and multiplying those cosets as if they were elements. This construction is one of the central ideas of abstract algebra because it allows us to collapse a subgroup to the identity and study the remaining structure. Students often remember the formula $(aH)(bH)=(a\circ b)H$ but forget the real issue: this formula must be independent of the chosen representatives. In this lesson, quotient group construction is developed carefully from coset multiplication to the group axioms.

Let $(G,\circ)$ be a group and let $H$ be a subgroup of $G$. The \textbf{quotient set} of $G$ by $H$ is the set of all left cosets of $H$ in $G$, denoted by $$\begin{equation*} G/H=\{aH:a\in G\}. \end{equation*}$$