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.

The definition uses left cosets, but the number of left cosets is equal to the number of right cosets. Hence the index is also the number of distinct right cosets. A common mistake is to confuse $[G:H]$ with $|H|$. The number $|H|$ counts elements inside the subgroup, while $[G:H]$ counts cosets of the subgroup in the whole group.

The index of a subgroup becomes visible when the group is split into cosets. We use $\mathbb{Z}_n$ under addition because its cosets are easy to list and compare. Change the modulus and the subgroup generator, then observe how the same subgroup size produces equal-sized coset blocks. Notice that the index is the number of blocks, not the number of elements in one block.

Each coset has the same number of elements as the subgroup. When the subgroup gets larger, fewer cosets are needed to cover the same finite group. This is the counting idea that later becomes Lagrange's theorem.

The index can be computed directly in modular examples by finding the subgroup and then counting its cosets. Use $\mathbb{Z}_n$ and a generator for the subgroup to compute $|H|$ and $[G:H]$. The calculator lists the generated subgroup first, then reports the number of cosets. This reinforces the difference between subgroup size and index.