Let $m<n$ be positive integers. Calculate $N_{S_n}(S_m)$. In particular, find when $N_{S_n}(S_m)=S_m.$
Hi. I don't know how to do this exercise. Some hint? only i have this:
Let $\sigma\in N_{S_n}(S_m)$ then $\sigma \tau\sigma^{-1}\in S_m$ for all $\tau\in S_m$. Now, I thought to use that $S_m=\langle (1\ 2),\ldots, (m-1\, m)\rangle$ but I don't see that it is very useful
Here are a couple of observations to help point you in the right direction.
It’s definitely the case that $N_{S_n}(S_m)\subsetneqq S_m$ when $n\ge m+2$: consider the transposition $(m+1\;m+2)$.
Now suppose that $n=m+1$ and $m>1$. If $\sigma\in S_n$ does not fix $n$, let $k=\sigma(n)$. There is a $\tau\in S_m$ such that $\tau(k)\ne k$, and hence $(\sigma^{-1}\tau\sigma)(n)=\sigma^{-1}(k)\ne n$, so that $\sigma^{-1}\tau\sigma\notin S_m$.