Consider $H = \{id, (1 2 3), (1 3 2)\}$ in $S_5$. How many elements are in its normalizer?
Observe that $H$ is generated by $h=(123)$, and $h^{-1}=(132)$, so the calculation boils down to finding how many $g\in S_5$ there are such that $ghg^{-1}=h$ or $ghg^{-1}=h^{-1}$. The former elements are exactly the elements in the centralizer of $h$, which I have found to be $6$, however I am finding it difficult to get anywhere from the latter relation. Can someone give me a hint? Thanks in advance!