Lagrange’s Theorem
Recall that for any cyclic group $\mathbb{Z}/n\mathbb{Z}$, the cyclic group generated by a single element $k$ has order that divides $n$. Indeed, recall that the order of $k$ is given by
\[\begin{align*} \mathrm{ord}(k) = \frac{n}{\gcd(k, n)}. \end{align*}\]This result can be made considerably more general, and that result is very naturally expressed in terms of group actions.
Theorem (Lagrange’s Theorem): Let $G$ be a finite group and $H \le G$. Then
\[|G| = |H|\cdot|G/H|\]where $|G/H|$ is the set of cosets of $H$.
Proof. Let $H$ act on $G$ by left multiplication. Notice that $H$ partitions $G$ into orbits. Therefore, $G$ is the disjoint union \(G = \bigsqcup_x Hx\) of orbits (where each $x$ is a representor of a distinct orbit). If we fix $x \in G$, then the map
\[\begin{align*} f:H &\to Hx \\ h &\mapsto hx \end{align*}\]is a bijection. Indeed, $h_1 x = h_2x$ implies $h_1 = h_2$ by the cancellation property in $G$ and surjectivity is implied by the definition of $Hx$ being an orbit (the action of $H$ is transitive on its orbits). Thus, $|H| = |Hx|$. Since $|G/H|$ counts the number of distinct orbits the action of $H$ on $G$, the claim follows.