Explain how the idea of "partition" is used in the proof of lagrange's theorem?

Question

I know partiion is a collection of subsets of $S$ that are non empty, disjoint and their union is $S$; and Lagrange's Theorem: If $G$ is a finite group and $H$ is a subgroup of $G$, then $H$ is a subgroup of $G$, then $\lvert H\rvert$ divides $\lvert G\rvert$. Moreover, the number of distinct left (right) costs of $H$ in $G$ is $\lvert G\rvert/\lvert H\rvert$. can someone Explain how the idea of "partition" is used in the proof of lagrange's theorem.

Answer 1

A partition just means that you are splitting up $G$ into disjoint subsets whose union is all of $G$. In the proof of Lagrange's theorem, these disjoint subset are the (left) cosets of $H$. Moreover, each coset of $H$ has the same number of elements, namely $|H|$.

So if you have $N$ items and you divide the $N$ items up into subsets where each subset has $M$ items, then $M$ must divide $N$. Also, the number of subsets is $N/M$. In the case of groups, this gives $|H|$ divides $|G|$, and the number of cosets is $|G|/|H|$.