find the centralizer Z(a) of the element a=(134) in S5
Let be the conjugacy class of and let be the centralizer of that element. Then, the number of elements of is equal to the index of in . Now the number of conjugacy classes of is equal to . Therefore, the index of in is equal to . This tells us that has six elements. Let us see if we can figure them out through reasoning as opposed to mindless computation of all 120 elements in . First, certainly and so are all elements of the centralizer. Second, is a disjoint cycle from therefore . Thus, we see that all commute with . We found six elements, therefore, those elements mentioned above is the centralizer.