Ashow that ab and ba are conjugate for any a b isin g b

Question

Elements x and y of a group G are called conjugate if y = z-1xz for some z ∈ G.

(a) Show that ab and ba are conjugate for any a, b ∈ G.

(b) Define a relation R on G by (x, y) ∈ R if and only if x and y are conjugate inG. Show that R is an equivalence relation. The equivalence classes for this relation arecalled conjugacy classes.

(c) How many conjugacy classes does the symmetric group G = S3 have? Write outthe elements of each conjugacy class