The following provides an alternative way to establish Lagrange's Theorem. Let G be a group of order n, and let H be a subgroup of G of order m.
(a) Define the relation R on G as follows: If a, b ∈ G, then a R b if a-1b ∈ H. Prove that R is an equivalence relation on G.
(b) For a, b ∈ G, prove that a R b if and only if aH = bH.
(c) If a ∈ G, prove that [a], the equivalence class of a under R, satisfies [a] = aH.
(d) For each a ∈ G, prove that |aH| = |H|.
(e) Now establish the conclusion of Lagrange's Theorem, namely that |H| divides |G|.