Properties of $H\lhd N(H)$ for $H\subset S_n$ Suppose we are given a subgroup $H$ of $G=S_n$. What are some techniques for studying the structure of $N_{G}(H)/H$, especially in the case that $H=\langle \sigma,\tau\rangle$?
Note: $\sigma,\tau$ are arbitrary permutations in $S_n$. If you feel you can make progress only by imposing extra restrictions on $\sigma,\tau$, that is fine - but I am ultimately interested in the case when $\sigma,\tau$ are "complicated".

Some concrete questions I am interested in:


*

*What is the most straightforward way to compute $[N_G(H) : H]$?

*Can we embed $N_G(H)/H$ in $S_n$?

*Can we make any statements about sets of transversals of $N_G(H)/H$?

*What are the subgroups $\langle\sigma,\tau\rangle\subset S_n$ that are self-normalizing?

*Can we characterize the self-normalizing subgroups of $S_n$?


The answer to Question 2 is no. I found a counterexample by brute force computation. Let $G=S_{12}$, $\sigma=(1, 4)(2, 3)(8, 10)(9, 11)$, $\tau= (5, 6)(7, 12)(8, 10)(9, 11)$ and $H=\langle \sigma, \tau \rangle$. Then $N_G(H)$ has order 3072, and the quotient $N/H$ has order 768. I just checked that $Q$ is not isomorphic to any of the 10723 subgroups (counting up to conjugacy) of $G$.
