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There is basic result about splitting of a conjugacy class (of even permutations) of $S_n$ in the subgroup $A_n$, mentioned in almost all books and notes on the permutation groups.

But, is there any result, which tells us, when two even permutations are conjugate in $A_n$?

Maths Rahul
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    @SouravGhosh: Yes, but the question asks when two even permutations with same cycle structure are conjugate in $A_n$. – spin Aug 10 '22 at 08:52
  • Does this answer your question? Splitting of conjugacy class in $A_n$ – kabenyuk Aug 10 '22 at 13:20
  • Most of the questions of similar type ask: whether a single conjugacy class in $S_n$ of even permutations remains a conjugacy class in $A_n$ or breaks into two classes. My question here is different: given two even permutations, how can we decide if they are conjugate in $A_n$ or not? In books, I don't find my question; and also, I did not get reason of "closing". – Maths Rahul Aug 12 '22 at 09:15

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Suppose you have $\sigma \in A_n$. The following are equivalent:

  • The $S_n$-conjugacy class of $\sigma$ splits into two $A_n$-conjugacy classes.
  • $C_{A_n}(\sigma) \leq A_n$.
  • In the cycle decomposition of $\sigma$, the cycle lengths are odd and distinct.

Consider then $\sigma,\sigma' \in A_n$ which are conjugate in $S_n$, and such that the conjugacy class is split. There exists $\tau \in S_n$ such that $\tau \sigma \tau^{-1} = \sigma'$. We have $C_{S_n}(\sigma) \leq A_n$, so $\sigma, \sigma'$ are conjugate in $S_n$ if and only if $\tau \in A_n$.

So the question is: what is the parity of a conjugating element?

Write $\sigma = \sigma_1 \cdots \sigma_k$ and $\sigma' = \sigma_1' \cdots \sigma_k'$ as a product of disjoint cycles, with $\sigma_i,\sigma_i' \in S_{n_i}$ some $n_i$-cycles, with $n_i$ odd.

There exists $\tau_i \in S_{n_i}$ such that $\tau_i \sigma_i \tau_i^{-1} = \sigma_i'$. Then $\tau = \tau_1 \cdots \tau_k$ is a conjugating element, with $sgn(\tau) = sgn(\tau_1) \cdots sgn(\tau_k)$.

So conjugacy of $\sigma,\sigma' \in A_n$ is reduced to the following question:

Let $n$ be odd, and let $\sigma, \sigma' \in A_n$ be $n$-cycles. When are $\sigma,\sigma'$ conjugate in $A_n$?

Say $\sigma = (i_1 \cdots i_n)$ and $\sigma' = (i_1' \cdots i_{n}')$. A conjugating element is $$\tau = \begin{pmatrix} i_1 & \cdots & i_n \\ i_1' & \cdots & i_{n}' \end{pmatrix}$$ and $\sigma,\sigma'$ are conjugate in $A_n$ if and only if $\tau$ is an even permutation.

spin
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