1. 
(i)
P1 := [8, 5, 6, 4, 3, 9, 7, 2, 1]
P2 := [2, 4, 1, 5, 8, 9, 7, 3, 6]
MulPers(P1, P2) = [3, 8, 9, 5, 1, 6, 7, 4, 2]
MulPers(P2, P1) = [5, 4, 8, 3, 2, 1, 7, 6, 9]
No, they are not the same.

(ii)
P := [3, 6, 4, 7, 1, 2, 8, 9, 5]
InvPer(P) = [5, 6, 1, 3, 9, 2, 4, 7, 8]

(iii)
P := [1, 3, 6, 8, 5, 4, 7, 9, 2]
PtoC(p) = [[1], [5], [7], [9, 2, 3, 6, 4, 8]]

(iv)
P := [6, 5, 8, 3, 7, 1, 4, 2, 9]
inv(P) = 20
maj(P) = 16


2
InvGF:=proc(n,q) local P, p, s:
s:=0:
P:=permute(n):
for p in P do
  s:= s + q^(inv(p)):
end do:
s:
end:

MajGF:=proc(n,q) local P, p, s:
s:=0:
P:=permute(n):
for p in P do
  s:= s + q^(maj(p)):
end do:
s:
end:

Yes, they are the same.


3.
InvGF(5,3)=251680
InvGF(4,3)=2080
InvGF(3,3)=52
InvGF(2,3)=4

InvGF(5,3)/InvGF(4,3)=121
InvGF(4,3)/InvGF(3,3)=40
InvGF(3,3)/InvGF(2,2)=13

InvGF(4,2)=315
InvGF(3,2)=21
InvGF(2,2)=3

InvGF(4,2)/InvGF(3,2)=15
InvGF(3,2)/InvGF(2,2)=7