After studying applications of Lagrange's theorem, we now return to cosets with a wider construction. A left coset multiplies a subgroup on the left of an element, and a right coset multiplies a subgroup on the right of an element. A double coset allows both actions at the same time. If $H$ and $K$ are subgroups of a group $G$, then the double coset $HaK$ consists of all elements obtained by multiplying $a$ on the left by elements of $H$ and on the right by elements of $K$. In this lecture, we define double cosets, compare them with ordinary cosets, and compute examples that show why the construction is useful.

Let $(G,\circ)$ be a group and let $H$ and $K$ be subgroups of $G$. For $a\in G$, the $\textbf{double coset}$ of $a$ with respect to $H$ and $K$ is the subset $$\begin{equation*} HaK=\{h\circ a\circ k:h\in H,\;k\in K\}. \end{equation*}$$ The element $a$ is called a representative of the double coset $HaK$.