Hi I need a correct answer to these two questions.

Questions are about Groups and Symmetry

2. As demonstrated in class the symmetry group of a cube, 8 (0), consists of the following 48 matrices :|:1 0 0 0 :|:1 0 0 0 i1 0 i1 0 , 0 0 :l:1 , :tl 0 0 , U U :|:1 :lzl 0 U 0 2H 0 i1 0 0 0 0 i1 0 21:1 0 0 0 il , 0 21:1 0 , 3:1 0 0 , 0 i1 0 :lzl 0 U U 0 :lzl together with usual matrix multiplication. As covered in Problem Set 2, all of the following are homomorphisms: o (I) : 8(0) > 8(0) dened by @((a,j)) = ((1% o det : 8(0) > {1, 1}, where det(A) is the determinant of A. o p: 8(0) ) {1, 1} dened by p(A) = det(@(A)) o q : 8(0) > {1, 1} dened by q(A) = det(A(A)) o f : 8(0) > 8(0) dened by f(A) = p(A) A, which is actually an automorphism of 8(0). (a) Find all the normal subgroups of 8(0). (You may assume that of the 98 subgroups of 8(0), only nine of them are normal subgroups.) (b) What is the subgroup lattice of all normal subgroups of 8 (0)? ((3) Except for the subgroup {I}, indicate in terms of 84, 144,22 or external direct products of two or more of these groups, to which group each normal subgroup of 8(0) is isomorphic. (d) For each normal subgroup H of 8 (0), nd a homomorphism g : 8(0) > 8(0) such that H = ker(g). In each case, what is im (9) and what is it isomorphic to? (e) To which group is Inn (8(0)) isomorphic? (f) Explain why the automorphism f : 8(0) > 8(0) is not an inner automorphism of 8(0). 3. Prove that for any group G, Inn (0)