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Solve each problem:Problem 1: Let H be a subgroup of group (G, °). a.) Show that NG(H) / CG(H) is isomorphic to a subgroup of Aut(H). b.) Show that CG(H) = NG(H) if |H| = 2. Problem 2: Let G be a group of order |G| = 992 = 25 · 31. a.) Show that at least one of its Sylow p-subgroups is normal. b.) If G is Abelian, find all prime-power decompositions and invariant decompositions. Problem 3: Prove that if G / Z(G) is cyclic, then G is Abelian. Problem 4: Let G be a group of order 99. a.) Prove that there exists a subgroup of H of order 3. (Should be a one-sentence proof.) b.) Prove there is a unique subgroup K of G such that K / H ˜ Z3 and G / K ˜ Z11. Problem 5: Characterize those integers n such that the only Abelian groups of order n are cyclic. Problem 6: Express U(165) as an internal direct product of proper subgroups in 4 different ways. Problem 7: Suppose (R, +, °) is a ring in which a ° a = 0 implies a = 0. Show that R has no nonzero nilpotent elements. Problem 8: Let A and B be ideals of a ring (R, +, °). If A n B = {0}, show that a ° b = 0 whenever a ? A and b ? B. Problem 9: Show that x4 + 1 is irreducible over Q but reducible over R. What is the smallest extension field containing Q for which x4+ 1 is reducible? Notation: NG(H) denotes the normalizer of H in G. CG(H) denotes the centralizer of H in G. Aut(H) denotes the class of all automorphisms of H. Z(G) denotes the center of G. U(165) denotes the group of units for 165.