Prove or disprove.Normal Subgroup.

Question

If $H=\{\sigma\in S_n: \sigma (n)=n\}$, then H is a normal subgroup of $S_n$ for $n\geq3$.

How to solve this problem.If we have to disprove it then give an example.

Answer 1

Without (many) words:

$$n=4:\;\;\;\;(14)(123)(14)=(234)\notin H$$

Answer 2

A subgroup is normal iff it is invariant under conjugation.

Conjugation in $S_n$ is just renaming. More precisely, the permutation $\tau \sigma \tau^{-1}$ does exactly what $\sigma$ does, except that the numbers are renamed via $\tau$ (or $\tau^{-1}$, depending on which side you compose).

Your $H$ is not invariant under all possible renamings, only under those that fix $n$. Hence, $H$ is not normal. (Hence, also, the counterexample given by @DonAntonio.)