1. Determine whether the binary operation * gives a group structure on the given set. If no group results, give the properties (Associativity, Existence of Identity Elements, Existence of Inverse Element) from definition of group, that does not hold and justify your answer. a. Let * be defined on Z by letting a * b = ab. b. Let * be defined on 2Z = {2n : n c Z} by letting a * b = a +b. c. Let * be defined on R+ by letting a *b = vab. d. Let * be defined on C by letting a * b = labl. 2. Let n E Z+ and nZ = {nm|me Z). Show that nZ is a group under the operation +. 3. Let S = R\\ {-1}. Define * on S by a * b = a + b + ab. a. Show that * gives a binary operation on S. b. Show that S is a group under the operation *. c. Find the solution of the equation 2 * r * 3 =7 in S. 4. Prove that if a = e for all a in a group G, then G is abelian. 5. Let G be a group. Prove that (ab)"1 = a-1b- if and only if G is abelian. 6. Prove that a nonempty set G, together with a binary operation * on G such that ar = b and ya = b have solutions in G for all a, b E G, is a group