State and prove Lagrange’s Theorem for finite groups: if \(G\) is a finite group and \(H \le G\) is a subgroup, then \(|H|\) divides \(|G|\) and \(|G| = [G : H] \cdot |H|\).
Solution
The left cosets of \(H\) in \(G\) partition \(G\). Indeed, define \[ x\sim y\iff x^{-1}y\in H. \] This is an equivalence relation, and the equivalence class of \(g\in G\) is exactly \[ gH=\{gh:h\in H\}. \] Hence \[ G=\bigsqcup_{i=1}^{[G:H]} g_iH \] for any choice of left-coset representatives \(g_i\).
For each \(g\in G\), left multiplication \[ H\to gH,\qquad h\mapsto gh \] is a bijection, so every left coset has cardinality \(|H|\). Therefore \[ |G| =\sum_{i=1}^{[G:H]}|g_iH| =[G:H]|H|. \] In particular, \(|H|\) divides \(|G|\).