Let G be a group. An automorphism of G is a group isomorphism from G to G. Let Aut(G) be the set of all automorphisms of G. (a) Prove that (Aut(G), ā) is a group with function composition as the operation. (b) For any g ā G, define Ģg : G ā G such that x ā gxgā»Ā¹ for all x ā G. Prove that Ģg ā Aut(G). (c) Define f : G ā Aut(G) such that g ā Ģg (where Ģg is defined in (b)). Then f is a group homomorphism. (d) Prove that the quotient group G/Z(G) is isomorphic to a subgroup of Aut(G) using the First isomorphism theorem.