1. Let A be an abelian group, and let G = A x A (Cartesian product), which also is a group. Let 6: G→ A, o(a, b) = ab (product in A). Prove that is surjective. Prove that is a homomorphism. Find the kernel of o (and verify your answer). Where did you use that A is abelian? (a) (b) (d) 2. Let (G, *) and (G', $) be groups, and let : G→ G' be a homomorphism. Let K be a subgroup of G'. Let H = {ge G: 0(g) = K}. Prove that H is a subgroup of G. (Pay close attention to details. Please use and $ for the binary operations, not just multiplication.)

| Let o be the portion of the paraboloid given by: F(u, v) = u cos(v)i+ u sin(v)j + (1 – u²)k for which 1