Let G be a finite group of order 56. a) Determine the possible number of Sylow 7-subgroups in G. Provide a detailed justification using Sylow's theorems. b) Assume that G has exactly one Sylow 7-subgroup. Prove that this Sylow 7-subgroup is normal in G, and subsequently show that G contains a normal subgroup of order 8. c) Using the results from parts (a) and (b), classify all possible group structures for G. Discuss whether G is necessarily abelian or non-abelian based on your classification. Let G be a finite non-abelian simple group of order 60. a) Determine all possible isomorphism types of G. Justify your conclusion using the classification of simple groups of small order. b) Compute the automorphism group Aut(G) of G. Provide a detailed analysis of its structure, including any inner and outer automorphisms. c) Prove that every automorphism of G is inner or describe the existence of outer automorphisms. Use this to discuss the simplicity of Aut (G). d) Explore the action of Aut(G) on the set of Sylow subgroups of G. Determine whether this action is transitive and justify your answer.