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Math Help - Efficient way to find normalizer

  1. #1
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    Efficient way to find normalizer

    Can someone help me do this problem efficiently? I am supposed to find the normalizer of <(123)> in S_4. So, basically I need to find all the elements in S_4 that commutes with (123). I know two trivial ones are (1) and (123). But do I really need to list all the elements in S_4 and check which one commutes with (123).
    Any help is really appreciated.
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  2. #2
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    Quote Originally Posted by jackie View Post
    Can someone help me do this problem efficiently? I am supposed to find the normalizer of <(123)> in S_4. So, basically I need to find all the elements in S_4 that commutes with (123). I know two trivial ones are (1) and (123). But do I really need to list all the elements in S_4 and check which one commutes with (123).
    Any help is really appreciated.
    that part in red is wrong! the normalizer of <\sigma>, where \sigma=(1 \ 2 \ 3), in S_4 is the set of all \tau \in S_4 such that \tau \sigma \tau^{-1} \in <\sigma>. so either \tau = (1) or \tau \sigma = \sigma \tau or \tau \sigma = \sigma^2 \tau.

    since for all i=1,2,3, we have \sigma(i) \neq i, \ \sigma^2(i) \neq i, we must have \tau(4)=4, for all \tau in the normalizer of <\sigma>. obviously (1), \ \sigma, \ \sigma^2 are in the normalizer. so you actually

    need to check 3 possible values of \tau only: (1 \ 2), \ (1 \ 3), \ (2 \ 3).
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  3. #3
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    Quote Originally Posted by NonCommAlg View Post
    that part in red is wrong! the normalizer of <\sigma>, where \sigma=(1 \ 2 \ 3), in S_4 is the set of all \tau \in S_4 such that \tau \sigma \tau^{-1} \in <\sigma>. so either \tau = (1) or \tau \sigma = \sigma \tau or \tau \sigma = \sigma^2 \tau.

    since for all i=1,2,3, we have \sigma(i) \neq i, \ \sigma^2(i) \neq i, we must have \tau(4)=4, for all \tau in the normalizer of <\sigma>. obviously (1), \ \sigma, \ \sigma^2 are in the normalizer. so you actually

    need to check 3 possible values of \tau only: (1 \ 2), \ (1 \ 3), \ (2 \ 3).
    Thanks a lot for your help NonComm. I think I mixed up the definitions. So if the question was: Let S_4 act on itself by conjugation. Find the centralizer of (123). Then I need to look for elements that commute with (123) right?
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