# centralizer

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.