Subsets of a Group and Lagrange's Theorem

In this chapter, we establish one of the most important theorems in finite group theory, i.e., Lagrange's Theorem. This theorem gives a relationship between the order of a finite group and the order of any subgroup (in particular, if $\vert G\vert < \infty$ and $H\subset G$ is a subgroup, then $\vert H\vert\ \mid \ \vert G\vert$). In order to establish Lagrange's theorem we first investigate subsets of a group and partitions of the group with respect to these subsets.