This exercise determines the conjugate classes of S_n for every integer n ≥ 1. 

a. Show that if a = (a₁, a₂ , · · · ·, a_m) is a cycle in S_n and τ is any element of S_n then τστ^-1 = (ra₁, ra₂,· · · ·,ra_m).

b. Argue from (a) that any two cycles in S_n of the same length are conjugate. 

c. Argue from (a) and (b) that a product of s disjoint cycles in S_n of lengths r_i for i = 1, 2, · · · ·, s is conjugate to every other product of s disjoint cycles of lengths r_i in S_n. 

d. Show that the number of conjugate classes in S_n is p(n), where p(n) is the number of ways, neglecting the order of the summands, that n can be expressed as a sum of positive integers. The number p(n) is the number of partitions of n. 

e. Compute p(n) for n = 1, 2, 3, 4, 5, 6, 7.