# Centralizer of a transposition in permutation group $S_n$

let $\pi$ be a transposition in permutation group $S_n$ that $n>1$. Is the centralizer of $\pi$ on $S_n$ i.e $C_{S_n}(\pi)$ isomorphic with $\mathbb Z_2\times S_{n-2}$ or $\mathbb Z_2\times S_{n-1}$?

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Hint: Assume for convenience that $\pi$ is the permutation $(1 \ 2)$. Show that if $\sigma \in S_n$ satisfies $\sigma(1) \notin \{1, 2\}$ or $\sigma(2) \notin \{1, 2\}$ then $\sigma$ does not commute with $\pi$.
A permutation $\sigma$ is in the centralizer of $(1 \ 2)$ if and only if $\sigma(1), \sigma(2) \in \{1, 2\}$. –  Jim Mar 11 '13 at 20:56