$(123)(456)$ and $(531)(264)$ are conjugate in $A_6$, and $(12345)(678)$ and$(43786)(215)$ are not conjugate in $A_8$.

Question

Show that $(123)(456)$ and $(531)(264)$ are conjugate in $A_6$, and show that $(12345)(678)$ and$(43786)(215)$ are not conjugate in $A_8$.

I only found that $(123)(456)$ and $(531)(264)$ are conjugate in $S_6$, with (25), but I can't really find something in $A_6$ that makes them conjugate. Also don't know how to show that $(12345)(678)$ and$(43786)(215)$ are not conjugate in $A_8$.

Answer 1

Just to avoid confusion an odd cycle is an even permutation, for example $$(123)=(12)(13)$$ and a even length cycle is odd $$(1234)=(12)(13)(14).$$ Now if $g\in S_n$, then the number of elements conjugate to $g$ is $[S_n:C(g)]$ where $C(g)$ is the centraliser of $g$ in $S_n$. If $g\in A_n$ then the number of elements conjugate to $g$ in $A_n$ is $[A_n:A_n\cap C(g)]$.

There are two possibilities: 1)$C(g)\neq A_n\cap C(g)$, that is that $g$ is centralized by an odd element and we see that $$\frac{[S_n:C(g)]}{[A_n:A_n\cap C(g)]}=1$$ and the conjugacy class of $g$ is the same in $A_n$ as it is in $S_n$.

2) that $C(g)=A_n\cap C(g)$, and we have $$\frac{[S_n:C(g)]}{[A_n:A_n\cap C(g)]}=2$$ so the conjugacy class of $g$ is splits into two pieces in $A_n$.

Now if a permutation contains an even cycle (odd permutation) then it will be centralised by that cycle, and thus we have the first case.

If it has two or more odd cycles of the same length then it is also centralised by an odd permutation, namely the permutation which switches the cycles. For example $(123)(456)$ commutes with $(14)(25)(36)$. Thus the conjugacy classes are also the same in $A_n$ and $S_n$, and this is your first example. (more about this later).

If the permutation consists of odd cycles of different length then it only commutes with an even permutation, since that permutation will preserve the cycle structure and a cycle only commutes with its own powers. Thus the conjugacy class in this case splits in two. And we see in this situation that if two permutations are conjugate by an odd permutation they are not conjugate in $A_n$.

So lets look at the examples. The second example, $(12345)(678)$ and $(43786)(215)$ are conjugate by $$\begin{pmatrix}1&2&3&4&5&6&7&8\\ 4&3&7&8&6&2&1&3 \\ \end{pmatrix}=(14856237)$$ and odd permutation, thus they are not conjugate in $A_8$.

Now we know that $(123)(456)$ and $(531)(264)$ are conjugate in $A_6$, but if we want a permutation we could try

$$\begin{pmatrix}1&2&3&4&5&6\\ 5&3&1&2&6&4\end{pmatrix}=(1564231)$$ also odd but there is another possibility, that is

$$\begin{pmatrix}1&2&3&4&5&6\\ 2&6&4&5&3&1\end{pmatrix}=(126)(345)$$ which is even.

Answer 2

Can you find a permutation outside $A_6$ that centralises $(1\,2\,3)(4\,5\,6)$?

The centraliser of $\sigma=(1\,2\,3\,4\,5)(6\,7\,8)$ is the cyclic group generated by $\sigma$, so lies within $A_8$. Therefore if there is an odd permutation conjugating this to a given permutation, all permutations which do so are odd.