After counting quotient groups, we now study when a quotient group is abelian. The quotient $G/H$ is abelian when the failure of elements of $G$ to commute is absorbed by $H$. This failure is measured by commutators. Thus abelian quotient groups do not require $G$ itself to be abelian; they require all commutators to fall inside the normal subgroup being factored out. This lesson is a first step toward the commutator subgroup and abelianisation, although we keep the discussion at the undergraduate quotient-group level.

Let $(G,\circ)$ be a group and let $x,y\in G$. The \textbf{commutator} of $x$ and $y$ is the element $$\begin{equation*} x\circ y\circ x^{-1}\circ y^{-1}. \end{equation*}$$