Efficient way to find normalizer

**jackie** · October 14th 2009, 02:26 PM

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.

**NonCommAlg** · October 14th 2009, 04:06 PM

Originally Posted by jackie

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: $(1 \ 2), \ (1 \ 3), \ (2 \ 3).$

**jackie** · October 14th 2009, 10:03 PM

Originally Posted by NonCommAlg

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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