The previous lecture defined the double coset $HaK$ and showed how it extends ordinary left and right cosets. We now prove the structural reason double cosets organize a group: they are equivalence classes for a natural relation. Two elements $a$ and $b$ are related when $b$ can be obtained from $a$ by multiplying on the left by an element of $H$ and on the right by an element of $K$. This relation partitions the group into double cosets. In this lecture, we prove the equivalence relation carefully and identify each equivalence class with a double coset.

Let $(G,\circ)$ be a group and let $H$ and $K$ be subgroups of $G$. The $\textbf{double coset relation}$ on $G$ determined by $H$ and $K$ is the relation $\rho$ defined by $$\begin{equation*} a\rho b \iff b=h\circ a\circ k \end{equation*}$$ for some $h\in H$ and some $k\in K$.