## The Sym(n)-conjugacy class of an element of Alt(n) is an Alt(n)-conjugacy class if and only some representative commutes with an odd permutation

Let $\sigma \in A_n$. Show that all elements in the $S_n$-conjugacy class of $\sigma$ are conjugate in $A_n$ if and only if $\sigma$ commutes with an odd permutation.

We wish to show that $S_n \cdot \sigma = A_n \cdot \sigma$ if and only if some $\tau \in C_{S_n}(\sigma)$ is an odd permutation.

In this previous exercise, we saw that $S_n \cdot \sigma$ is a union of $[S_n : A_n C_{S_n}(\sigma)]$ distinct $A_n$-conjugacy classes; moreover, it is clear that $A_n \cdot \sigma \subseteq S_n \cdot \sigma$.

Note the following.

• $\sigma$ commutes with an odd permutation
• if and only if $C_{S_n}(\sigma) \not\leq A_n$
• if and only if $A_n C_{S_n}(\sigma) = S_n$
• if and only if $[S_n : A_n C_{S_n}(\sigma)] = 1$
• if and only if $S_n \cdot \sigma$ is a union of just one conjugacy class
• if and only if $S_n \cdot \sigma = A_n \cdot \sigma$.
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