4. Let G = D8 = {a, b: a^4 = b^2 = 1, bab = a^-1}, and H = Q8 = {c, d: c^4 = 1, c^2 = d^2, dcd = c^-1}.
(a) Let x, y be the permutations in S4 which are given by x = (1 2), y = (3 4), and let K be the subgroup (x, y) of S4. Show that both the functions φ: G → K and ψ: H → K, defined by φ: a^r b^s → x^r y^s, ψ: c^r d^s → x^r y^s (0 ≤ r ≤ 3, 0 ≤ s < 1), are homomorphisms. Find Ker φ and Ker ψ.
(b) Let X, Y be the 2 x 2 matrices which are given by X = [i -2; i 6], Y = [i 2; i -6], and let L be the subgroup (X, Y) of GL(2, C). Show that just one of the functions λ: G → L and μ: H → L, defined by λ: a^r b^s → X^r Y^s, μ: c^r d^s → X^r Y^s (0 ≤ r ≤ 3, 0 ≤ s < 1), is a homomorphism. Prove that this homomorphism is an isomorphism.