Find the centralizer of ( 1 2.....n) in Sn
The given cycle has (n-1)! conjugates: All conjugates have the same cycle structure, and there are only (n-1)! distinct n-cycles that may be in the conjugacy class because the n! total permutations that may be written as a conjugate cycle decompose into equivalence classes of identical cycles of common class size n. Thus, the index of the normalizer is (n-1)!, so the normalizer has order n. But the normalizer of an element is the centralizer, so the order of the centralizer is n, and the n powers of the given cycle are the only commuting elements.