After centre and centralisers, we now pass from special subgroups to the way a subgroup sits inside the whole group. The main idea in this lecture is the coset of a subgroup. A coset is obtained by multiplying every element of a subgroup by one fixed element of the group. This construction matters because it is the first systematic method for dividing a group into equal-sized parts. In later lectures, cosets will lead to index, Lagrange's theorem, quotient groups, and normal subgroups. In this class note, we define left and right cosets, compute first examples, and see why nonabelian groups require us to distinguish the two sides.

Let $(G,\circ)$ be a group and let $H$ be a subgroup of $G$. For $a\in G$, the $\textbf{left coset}$ of $H$ in $G$ determined by $a$ is the subset $$\begin{equation*} aH=\{a\circ h:h\in H\}. \end{equation*}$$ The element $a$ is called a representative of the left coset $aH$.