How many elements in S8 commute with the permutation 123 456

How many elements in S_8 commute with the permutation (123) (45678)? How many elements are there in S_8 with order 15?

Solution

(a)

There are three ways to match the cycle (123) with itself and 5 ways to match the cycle (45678) with itself, so there should be 15 commuting permutations.

(b)

If = where and are disjoint cycles, then || = lcm(||, ||). Also order of a k-cycle is k. Therefore the only possible disjoint cycle decompositions for a permutation S₈ with || = 15 is (5, 3). The number of such permutations is C(8,5)*(5!/5)* C(3,3)*(3!/3) = 2688.