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**NonCommAlg** 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: $\displaystyle (1 \ 2), \ (1 \ 3), \ (2 \ 3).$