Let ? be a set, and G a subgroup of the group Sym(?) of permutations of ?. Let ω ∈ ?, and let G_ω denote the stabilizer of ω in G.
(a) If H is a subgroup of G such that H acts transitively on ?, show that G = HG_ω.
(b) If G acts transitively on ?, show that G/G_ω ' ? as G-sets.