Cosets, orbits and Lagrange's Theorem

The standard proof of Lagrange's Theorem for a subgroup \(H \le G\) involves establishing a bijection \(h \leftrightarrow gh\) between \(H\) and each coset of \(H\) in \(G\), and concluding that since \(\left|H\right| = \left|gH\right|\) and the cosets partition \(G\), \(\left|G\right|\) divides \(\left|H\right|\).

There is an equivalent formulation using orbits. Let \(H\) act on \(G\) by right multiplication. Then the orbit of any \(g \in G\) under \(H\) is defined to be \(\{gh \mid h \in H\}\), which is just the left coset \(gH\). By proving that each orbit has \(\left|H\right|\) elements and the orbits partition \(G\), we arrive at essentially the same proof.