Let Sn be the group of permutations of {1,...,n}, and suppose n is even, n > 4. Let g = (12) ∈ Sn, and h = (12)(34)... (n−1 n)∈ Sn.
(i) Compute the centraliser of g, and the orders of the centraliser of g and of the centraliser of h.
(ii) Now let n = 6. Let G be the group of all symmetries of the cube, and X the set of faces of the cube. Show that the action of G on X makes G isomorphic to the centraliser of h in S6. [Hint: Show that −1 ∈ G permutes the faces of the cube according to h.] Show that G is also isomorphic to the centraliser of g in S6.
My thoughts: Since for symmetric group, elements conjugate to each other iff they have the same cycle type, so by combinatorics we can have the size of conjugacy classes of g and h, thus the size of centralisers of g. But I get stuck on the second part of the question. Because we know G is isomorphic to S4 * C2 which is isomorphic to S4* , now we need to prove S4 * is isomorphic to the centralisers of h. We know they have the same size but I can't find an appropriate group homomorphism.
It would be greatly appreciated if anyone can give me any suggestions!