Problem 3.

(1) Prove that (ac)(bd) = (abc)(abd) in Sn for distinct a, b, c, d {1, 2, 3, , n}.

(2) Prove that the product of every pair of transpositions in Sn can be expressed as a product of at most two cycles of length 3.

(3) Prove that any permutation in the alternating group An with n 3 is a product of cycles of length 3.