The derived series of a group G is defined recursively by G₀ = G and G_i + 1 = [G_i,G_i] where [H,H] denotes the commutator subgroup of a group H. The derived series reaches G_i = {e} for some i if and only if G is solvable. For G = S₄, then G₁ = A₄, G₂ = V₄, and G₃ = {e}.