Centralizers in symmetric groups

For a group $G$ and an element $g \in G$ , the centralizer of $g$ in $G$ is the subgroup

C_{G} (g) = {h \in G : h g h^{- 1} = g} .

We say $g$ and $g^{'}$ are conjugate in $G$ if there exists an element $h \in G$ such that $g^{'} = h g h^{- 1}$ .

Suppose $S_{n}$ is a symmetric group with $n \geq 4$ , and $σ$ is one of the $(n - 2)$ -cycles in $S_{n}$ . (There are $\frac{n!}{2 (n - 2)}$ such cycles.)

Prove that $[S_{n} : C_{S_{n}} (σ)] = [A_{n} : C_{A_{n}} (σ)]$ .
Determine whether all $(n - 2)$ -cycles are conjugate in $A_{n}$ .

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For a group $G$ and an element $g\in G$, the {\bfseries centralizer} of $g$ in $G$ is the subgroup
\[
	C_G(g)=\{h\in G:hgh^{-1}=g\}.
\]
We say $g$ and $g’$ are {\bfseries conjugate in $G$} if there exists an element $h\in G$ such that $g’=hgh^{-1}$.

Suppose $S_n$ is a symmetric group with $n\geq 4$, and $\sigma$ is one of the $(n-2)$-cycles in $S_n$. (There are $\frac{n!}{2(n-2)}$ such cycles.)
\begin{enumerate}[label=\alph*)]
	\item Prove that $[S_n:C_{S_n}(\sigma)]=[A_n:C_{A_n}(\sigma)]$.
	\item Determine whether all $(n-2)$-cycles are conjugate in $A_n$.
\end{enumerate}