centralizer of permutations in Symmetric group I would greatly appreciate it if you kindly give me some advice.
Let $a$ and $b$ be two distinct permutations in $\text{Sym}(n)$. When $C_{S_n}(a)=C_{S_n}(b)$?
I guess either $a$ and $b$ be in the same conjugacy class with the cyclic type $(x)(y)(z\;w)a_0$ where $(z\;w)$ is a transposition and $a_0$ is a product of cycles of size at least three,
or $a=(x)(y)a_1$ and $b=(x\;y)b_1$ where $a_1$ and $b_1$ are conjugate. Moreover, $a_1$ and $b_1$ are the product of cycles of size at least three.
Thanks in advance for your consideration.
 A: This seems tricky to get right, so let me have a go. I hope someone will correct any inaccuracies!
For a permutation $g$ on a finite set $\Omega$ and $k \ge 0$, let $\Omega_k(g)$ denote the subset of $\Omega$ consisting of the support of all cycles of $g$ of length $k$. So, for fixed $g$, the $\Omega_k(g)$ partition $\Omega$.
For example. if $g = (3,4)(5,6)(7,8)(9,10,11,12,13)(14,15,16,17,18)(19,20,21,22,23,24)$ with $\Omega = \{1, \ldots,24\}$, then $\Omega_k(g) = \{1,2\}$, $\{3,4,5,6,7,8\}$, $\{9,10,11,12,13,14,15,16,17,18\}$, $\{19,20,21,22,23,24\}$, for $k=1,2,5,6$, respectively, and all other $\Omega_k(g)$ are empty.
Then, for $a,b \in S := {\rm Sym}(\Omega)$, we have $C_S(a) = C_S(b)$ if and only if the following two conditions hold:
(i) $\Omega_k(a)=\Omega_k(b)$ for all $k \ge 3$, and either this holds also for $k=1,2$, or $\Omega_1(a) = \Omega_2(b)$ with $|\Omega_1(a)|=2$ and/or  $\Omega_2(a) = \Omega_1(b)$ with $|\Omega_2(a)|=2$.
(ii) For all $ k \ge 0$, the restrictions of $a$ and $b$ to $\Omega_k(a)$ are powers of each other, except when $k=1$ or 2 in the exceptional cases of Condition (i).
