1. a) State and prove Ca b). Prove that the set I(G) of all inner automorphisms of a group G is a normal 3subgroup of A(G), the group of all automorphisms of G and that I(G) isisomorphic to G/Z, where Z is the centre of G.State and prove Cauchy's theorem for finite groups. 46(4 + 5 + 5)2. a State and prove Sylow's theorem.Sob) If G is a group of order 231, show that 11 - Sylow subgroup of G is contained inthe centre of GoSyl0w 30 thootine i 155)c) Define a solvable group. If G is a group and H is a subgroup of G; then show thatH is solvable. Based on problem (58)(4 +6+4)3. a Tf in a ring R, x = x for all x then show that 2x = 0 and x + y = 0 imply x=y.b) Prove that the ideal S of the ring I of all integers is maximal if and only if S isgenerated by some prime integer.refer parec) Find the units in the ring of Gaussian integers and show that they form amultiplicative abelian group. refer Page 15(3 +4+7)​