Let S, be the symmetric group of degree n where n > 5. Given any non-trivial alpha,beta is an element of S-n, we prove that the product alpha(Sn)beta(Sn) of the conjugacy classes alpha(Sn) and beta(Sn) is never a conjugacy class. Furthermore, if n is odd and not a multiple of three, then alpha(Sn)beta(Sn) is the union of at least three distinct conjugacy classes. We also describe the elements alpha,beta is an element of S-n in the case when alpha(Sn)beta(Sn) is the union of exactly two distinct conjugacy classes.
Journal of Group Theory
(2008). Symmetric Groups and Conjugacy Classes. Journal of Group Theory, 11(3), 371-379.
Available at: https://aquila.usm.edu/fac_pubs/1759