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Thread: 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 $\displaystyle <(123)>$ in $\displaystyle S_4$. So, basically I need to find all the elements in $\displaystyle S_4$ that commutes with $\displaystyle (123)$. I know two trivial ones are $\displaystyle (1)$ and $\displaystyle (123)$. But do I really need to list all the elements in $\displaystyle S_4$ and check which one commutes with $\displaystyle (123)$.
    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 $\displaystyle <(123)>$ in $\displaystyle S_4$. So, basically I need to find all the elements in $\displaystyle S_4$ that commutes with $\displaystyle (123)$. I know two trivial ones are $\displaystyle (1)$ and $\displaystyle (123)$. But do I really need to list all the elements in $\displaystyle S_4$ and check which one commutes with $\displaystyle (123)$.
    Any help is really appreciated.
    that part in red is wrong! the normalizer of $\displaystyle <\sigma>,$ where $\displaystyle \sigma=(1 \ 2 \ 3),$ in $\displaystyle S_4$ is the set of all $\displaystyle \tau \in S_4$ such that $\displaystyle \tau \sigma \tau^{-1} \in <\sigma>.$ so either $\displaystyle \tau = (1)$ or $\displaystyle \tau \sigma = \sigma \tau$ or $\displaystyle \tau \sigma = \sigma^2 \tau.$

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

    need to check 3 possible values of $\displaystyle \tau$ only: $\displaystyle (1 \ 2), \ (1 \ 3), \ (2 \ 3).$
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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 $\displaystyle <\sigma>,$ where $\displaystyle \sigma=(1 \ 2 \ 3),$ in $\displaystyle S_4$ is the set of all $\displaystyle \tau \in S_4$ such that $\displaystyle \tau \sigma \tau^{-1} \in <\sigma>.$ so either $\displaystyle \tau = (1)$ or $\displaystyle \tau \sigma = \sigma \tau$ or $\displaystyle \tau \sigma = \sigma^2 \tau.$

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

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