Let $H=\langle (12345)\rangle$. Find the normalizer of H in $S_5$.
First I noticed that two cycles are conjugate iff they have the same cycletype and we know that given $g\in S_5$ and $x=(x_1\ldots x_k)\in S_5$, a k-cycle, its conjugate is $gxg^{-1}=(g(x_1)\ldots g(x_k))$, but I do not see how all the information above helps me to determine the normalizer of H. Any tips or maybe a solution on how to find all $g\in S_5$ such that $gHg^{-1}=H$?