Permutations with a fixed element How would you understand the following set of permutations
$$S' :=\{ \pi\in S_n\mid \pi(n)=n\}$$
is the subset of all the permutations that fix the $n$-element?
Is it simply $S_{n-1}$? For example, consider $S_3$, then the subset contains all the permutation that fix the $3$. So the $S'=\{ (213) \}$?

The fixed points of a function $f$ are precisely those elements $x$ in the domain of $f$ such that $f(x)=x$. Since each permutation is a function and your set $S'$ is, by definition, the set of all permutations $\pi$ such that $\pi(n)=n$ (and $S'$ contains no other elements), it follows that $S'$ is precisely the set of all permutations that fix that $n$ element.
Note that $S'\cong S_{n-1}$.
By the way, your example is mistaken. Note that $(213)\notin S'$ when $n=3$ because it does not fix $3$; it sends $3$ to $2$.
