Tower Law for Subgroups/Proof 1

Proof
Let $p = \index G H$, $q = \index H K$.

By hypothesis these numbers are finite.

Therefore, there exist $g_1, \ldots, g_p \in G$ such that $G$ is a disjoint union: $\displaystyle G = \bigsqcup_{i \mathop = 1}^p g_i H$

Similarly, there exist $h_1,\ldots,h_q \in H$ such that $H$ is a disjoint union: $\displaystyle H = \bigsqcup_{j \mathop = 1}^q h_j K$

Thus:

This expression for $G$ is the disjoint union of $p q$ cosets.

Therefore the number of elements of the coset space is:
 * $\index G K = p q = \index G H \index H K$