I would like to know how is possible to calculate the order of the normalizer of $H=\langle s\rangle$ in $S_n$ where $s$ is an assigned permutation of $S_n$.
I know that finding the order of the centralizer is relatively easy using the following "propositions":
1) Two permutations are conjugate IFF they have the same cycle structure.
2) $\frac{|G|}{|cl(x)|}=|C_{G}(x)|$.
but for the normalizer what do I have to do? there is a standard method? I thought to use the theorem $N/C$ and in particular the fact that $\frac{|N_{G}(H)|}{|C_{G}(H)|}$ must divide $|Aut(H)|$ but sometimes this process is not enough to reach a solution.
For instance if $s=(1,2,3,4)(5,6,7,8)(9,10)(11,12)$ in $S_{12}$ then $|cl(s)|=\frac{12!}{2^8}$ and $|C_{G}(H)|=2^8$. Now $|Aut(H)|=2$ so by theorem $N/C$ i only can conclude $|N_{S_{12}}(H)|=2^8$ or $=2^9$.
I really hope you can help me I've my algebra exam in less then a week!!
P.S I'm sorry for my English I hope everything is at least understandable