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It is known that two permutations in $S_n$ are conjugate if and only if they have same cycle decomposition shape.

Given $\sigma\in A_n$, it is also well known whether conjugacy class of $\sigma$ in $A_n$ is same as that in $S_n$; this can be done just looking at cycle structure of $\sigma$.

Q. Given $\sigma$ and $\tau$ in $A_n$ and have same cycle structure. How do we decide whether $\sigma$ and $\tau$ are conjugate in $A_n$?

For example, $(123)$ and $(132)$; in $A_4$ they are not conjugate but in $A_5$ they are!

$(12345)$ and $(21345)$ are not conjugate in $A_5$ but in bigger groups they become conjugate. In general, what is the criteria to decide whether given two permutations of $A_n$ with same cycle structure are conjugate in $A_n$ or not?

Edit: (With Derek's comment):

Hypothesis 1. Given $\sigma,\tau \in A_n$. Suppose, they have same cycle structure.

Hypothesis 2. $\sigma=\sigma_1 \circ \cdots \sigma_2$ be complete decomposition (i.e. including single cycles) such that all $\sigma_i$'s are cycles of odd length (possible singleton) and distinct.

(1) We know by hyp.2 that $\sigma^{S_n}$ (conj. class of $\sigma$ in $S_n$) is union of two conj. classes in $A_n$, say $C_1,C_2$.

(2) Since $\sigma$ and $\tau$ have same cycle structure, conclusion (3) holds for $\tau$ also.

(3) By we write $\sigma^{S_n}=C_1\cup C_2$; by hypothesis 1, $\tau^{S_n}=C_1\cup C_2$

(6) My question is now, how can we decide whether $\sigma$ and $\tau$ lie in same $C_i$'s or distinct $C_i$'s?

p Groups
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  • "It is known that two permutations in Sn are conjugate if and only if they have same cycle decomposition." Do you mean cycle decomposition shape? – user217285 Dec 27 '16 at 06:34
  • The question and answer at http://math.stackexchange.com/questions/144686/conjugacy-classes-in-a-n?noredirect=1&lq=1 explain this. (This question might also be a duplicate of that) – user399601 Dec 27 '16 at 06:35
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    Also $(12345)$ and $(12345)^{-1}$ are conjugate by $(1)(25)(34)$ which is in $A_5$ unless I am missing something – user399601 Dec 27 '16 at 06:43
  • @user399601, the criteria given in answer to your link seems not so well applicable. For example, (123) and (132) in $A_5$; look them in $S_4$, and a conjugating element can be (1)(23), which is odd permutation. But (1)(23)(45) is also a conjugating element; in general, the conjugating element can be odd, can be even for same pair of given permutations; but how do we decide it from cycle structure whether conjugating element can be even permutation? – p Groups Dec 27 '16 at 06:57
  • @pGroups The argument in the link only applies when the conjugacy class in $S_n$ splits into two classes of $A_n$, for which there is a simple criterion based on the cycle type – user399601 Dec 27 '16 at 07:00
  • This question is answered in several previous questions, including http://math.stackexchange.com/questions/1260330 I would vote for this being closed as a duplicate (but I won't do so because when I vote for a duplicate it takes effect immediately). – Derek Holt Dec 27 '16 at 08:47
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    To decide whether two elements of $S_n$ in a class that splits in $A_n$ are conjugate in $A_n$, you just find a conjugating element and see whether it is odd or even. – Derek Holt Dec 27 '16 at 09:14
  • Consider $(123)$ and $(132)$ in $S_5$; if I take conjugating element $(23)$, then we can't deduce that they are conjugate in $S_5$ but not in $A_5$; in other words, how can we decide possiblity of gettting conjugating element to be an even permutations? is it by Trial and Error? – p Groups Dec 27 '16 at 09:17
  • But we know from the criterion that the class of $(1,2,3)$ is not split in $A_5$, so we don't need to look for a conjugating element in this case. We only need to do that for classes that split in $A_n$. – Derek Holt Dec 27 '16 at 09:19
  • Also discussed here, and in a special case here. I would rather let somebody else pick the best duplicate target. – Jyrki Lahtonen Dec 27 '16 at 13:03

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