Exercise 4.3.20 ( All elements in the conjugacy class of $\sigma$ in $S_n$ are conjugate in $A_n$ if and only if $\sigma$ commutes with an odd permutation)

Question

Let $\sigma \in A_n$. Show that all elements in the conjugacy class of $\sigma$ in $S_n$ (i.e., all elements of the same cycle type as $\sigma$) are conjugate in $A_n$ if and only if $\sigma$ commutes with an odd permutation. [Use the preceding exercise.]

Richard Ganaye · Accepted Answer

\begin{proof} Suppose that $\sigma \in A_n$ commutes with an odd permutation $	au$. Then $	au \in C_{S_n}(\sigma)$ and $	au 
ot \in A_n$. Therefore
$$A_n C_{S_n}(\sigma) 
e A_n:$$
indeed, $	au = 1 \cdot 	au \in A_n C_{S_n}(\sigma)$, but $	au 
ot \in A_n$.

Since $A_n$ is a maximal subgroup of $S_n$, and $A_n \leq A_n C_{S_n}(\sigma) \leq S_n$, where $A_n C_{S_n}(\sigma) 
e A_n$, we obtain
$$A_n C_{S_n} = S_n.$$
If we apply the result of Exercise 19, with $G = S_n$, $H = A_n \unlhd S_n$ and $x = \sigma$, then the number $k$ of orbits of $A_n$ acting by conjugation on  conjugacy class $\mathcal K$  of $\sigma$ is given by
$$k = |G : H C_G(x)| = |S_n : A_nC_{S_n}(\sigma)| = 1.$$
Since $k = 1$ the orbits of $\sigma$ for the actions of $G$ and of $H$ are the same:
$$\{\lambda \sigma \lambda^{-1} \mid \lambda \in A_n\} = \{\lambda \sigma \lambda^{-1}\mid  \lambda\in S_n\},$$
so all elements in the conjugacy class of $\sigma$ in $S_n$ are conjugate in $A_n$.

\bigskip
Conversely, suppose that all elements in the conjugacy class of $\sigma$ in $S_n$ are conjugate in $A_n$, so that $k = 1$, where $k$ is the number of conjugacy classes of $\sigma$ for the action of $A_n$ in the conjugacy class $\mathcal K$ of $\sigma$ for the action of $S_n$. By Exercise 19,
$$k = 1 = |S_n : A_nC_{S_n}(\sigma)|,$$
therefore
$$A_n C_{S_n}(\sigma) = S_n.$$
Hence $C_{S_n}(\sigma) 
ot \subseteq A_n$ (otherwise $A_n = S_n$), so there exists a permutation $	au \in S_n - A_n$ such that $	au \in C_{S_n}(\sigma) $, i.e., $	au$ is an odd permutation which commutes with $\sigma$.

\bigskip
In conclusion, all elements in the conjugacy class of $\sigma$ in $S_n$ are conjugate in $A_n$ if and only if $\sigma$ commutes with an odd permutation.
\end{proof}

Exercise 4.3.20 ( All elements in the conjugacy class of $\sigma$ in $S_n$ are conjugate in $A_n$ if and only if $\sigma$ commutes with an odd permutation)

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Exercise 4.3.20 ( All elements in the conjugacy class of $\sigma$ in $S_n$ are conjugate in $A_n$ if and only if $\sigma$ commutes with an odd permutation)

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