(1) (a) Determine the number of elements of order 6 in Z36 ⊕ Z9.
(b) Determine the number of cyclic subgroups of order 6 in Z36 ⊕ Z9.
(2) Let G = Z4 ⊕ U(4), H = ⟨(2, 3)⟩, and K = ⟨(2, 1)⟩. Show that although H ≈ K, the quotient groups G/H and G/K are not isomorphic.
(3) What is the order of 14 + ⟨8⟩ in the quotient group Z24/⟨8⟩?
(4) Let G = Z4 ⊕ Z4, and H = {(0, 0), (2, 0), (0, 2), (2, 2)}, and K = ⟨(1, 2)⟩. Is G/H isomorphic to Z4 or Z2 ⊕ Z2? Is G/K isomorphic to Z4 or Z2 ⊕ Z2?