After studying cosets of subgroups, the next natural step is to count how many distinct cosets a subgroup produces. This number is called the index of the subgroup. The idea is simple but powerful: a subgroup may be smaller than the whole group, but its cosets spread through the group in equal-sized non-overlapping blocks. The index measures how many such blocks are needed. In this lecture, we define the index of a subgroup, compute it in familiar examples, and prepare the counting language needed for Lagrange's theorem.

Let $(G,\circ)$ be a group and let $H$ be a subgroup of $G$. The $\textbf{index}$ of $H$ in $G$ is the number of distinct left cosets of $H$ in $G$. It is denoted by $$\begin{equation*} [G:H]. \end{equation*}$$ If the number of distinct left cosets is finite, then $[G:H]$ is that positive integer. If the number of distinct left cosets is infinite, then $[G:H]$ is infinite.