5.
(a) Let ? = (1 2 3 4 5 6) in S6. Show that G = {?, ?, ?^2, ?^3,?^4, ?^5} is a group using the operation of S6. Is G abelian? Howmany elements ? of G satisfy ?^2 = ?? ?^3 = ?? ? is the identitypermutation.
(b) Show that (1 2) is not a product of 3-cycles. Must bewritten as a proof!
(c) If a^4 = 1 and ab = b(a^2) in a group, show that a = 1. Mustbe written as a proof!
(d) Show that a group G is abelian if and only if (gh)^2 =(g^2)(h^2) for all g and h in G. Must be written as a proof!
