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Conjugacy classes in an

  • Thread starter pandasong
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  • #1
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(Moderator's note: thread moved from "Set Theory, Logic, Probability, Statistics")

the question is
if n is odd then there are exactly two conjugacy classes of n cycles in An each of which contains (n-1)!/2 elements.
also there is a hint says let An act on itself
i know the fact that since An acts on itself, x doesnt commute with any odd permutation. So it splits into two Ccl An (x) and CCl An (12)x(12).
But i cant figure it out how to calculate either the number of stabilizers or the number of orbits
 
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Answers and Replies

  • #2
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Your question is rather difficult to understand. eg. what do you mean by "CCl", "Ccl", (12)x(12)? And what is x?

I don't even think this is true! eg: in A3, the two 3-cycles: (123) and (132) commute, and so they are in the same conjugacy class.

For what it's worth, it might help you to note that if n is odd, then An contains every n-cycle, as these are even permutations. Also that the number of these n-cycles is: (n-1)! (as we can fix the 1 at the beginning of the cycle and then permute the other (n-1) numbers in every possible way).

Otherwise you will need to explain more.
 
  • #3
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Your question is rather difficult to understand. eg. what do you mean by "CCl", "Ccl", (12)x(12)? And what is x?

I don't even think this is true! eg: in A3, the two 3-cycles: (123) and (132) commute, and so they are in the same conjugacy class.

For what it's worth, it might help you to note that if n is odd, then An contains every n-cycle, as these are even permutations. Also that the number of these n-cycles is: (n-1)! (as we can fix the 1 at the beginning of the cycle and then permute the other (n-1) numbers in every possible way).

Otherwise you will need to explain more.
well the permutations you gave they are in ccl An
its what is written on my lecture notes actually i also dont know much about it but it has the same conj. class with ccl an
 

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