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Math Help - conjugacy classes of A5

  1. #1
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    conjugacy classes of A5

    "Prove that the 3-cycles in A_5 form a single conjugacy class"

    Okay, I understand what a conjugacy class is (subset in which every element is conjugate to every other element in the conjugacy class), but I'm wondering exactly how I should go about proving this? Do I have to brute force my way, or is there some theorem or something I am simply forgetting about?
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  2. #2
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    Quote Originally Posted by jmoney90 View Post
    "Prove that the 3-cycles in A_5 form a single conjugacy class"

    Okay, I understand what a conjugacy class is (subset in which every element is conjugate to every other element in the conjugacy class), but I'm wondering exactly how I should go about proving this? Do I have to brute force my way, or is there some theorem or something I am simply forgetting about?

    Theorem for you to find: a conjugacy class in S_n may remain a single conjugacy class or split in two conjugacy classes in A_n: it will split in two conjugacy classes iff there is no odd permutation with whom some representative of the conjugacy class commutes.

    Tonio
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  3. #3
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    Quote Originally Posted by tonio View Post
    Theorem for you to find: a conjugacy class in S_n may remain a single conjugacy class or split in two conjugacy classes in A_n: it will split in two conjugacy classes iff there is no odd permutation with whom some representative of the conjugacy class commutes.

    Tonio
    Ahh, that makes things a lot easier. THank you!
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    Thumbs up

    Quote Originally Posted by jmoney90 View Post
    Ahh, that makes things a lot easier. THank you!

    Any time.

    Tonio
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  5. #5
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    Quote Originally Posted by tonio View Post
    Any time.

    Tonio
    Okay, just to make sure I have the right idea, I could take a 3-cycle such as (3 4 5) and then multiply it with (1 2) and it would commute since both cycles share no common numbers, and thus I prove that the 3-cycles make a conjugacy class in A_5 as per the theorem you provided, correct?
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  6. #6
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    Quote Originally Posted by jmoney90 View Post
    Okay, just to make sure I have the right idea, I could take a 3-cycle such as (3 4 5) and then multiply it with (1 2) and it would commute since both cycles share no common numbers, and thus I prove that the 3-cycles make a conjugacy class in A_5 as per the theorem you provided, correct?

    Yup...the interesting thing, though, is that you'd prove the theorem that allows you to deduce that, imo.

    Tonio
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