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    centralizer

    find the centralizer Z(a) of the element a=(134) in S5
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    Quote Originally Posted by wonjujin View Post
    find the centralizer Z(a) of the element a=(134) in S5
    Let [(134)] be the conjugacy class of (134) and let C(134) be the centralizer of that element. Then, the number of elements of [(134)] is equal to the index of C(134) in S_5. Now the number of conjugacy classes of (134) is equal to 2! \cdot {5\choose 3} = 20. Therefore, the index of C(134) in G is equal to \tfrac{5!}{20} = 6. This tells us that C(134) has six elements. Let us see if we can figure them out through reasoning as opposed to mindless computation of all 120 elements in S_5. First, certainly \left< (134)\right> \subset C(134) and so \text{id},(134),(134)^2 are all elements of the centralizer. Second, (25) is a disjoint cycle from (134) therefore (25)(134) = (134)(25). Thus, we see that  \text{id},(134),(134)^2,(25),(25)(134),(25)(134)^2 all commute with (134). We found six elements, therefore, those elements mentioned above is the centralizer.
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