Consider the symmetric group $S_n$ and let $H$ be the subgroup that fixes $n$ (so its isomorphic to $S_{n-1}$)
Show the representation defined by $S_n$ acting on $S_n \backslash H$ is isomorphic to the $n$-dimensional permutation representation of $S_n$.
So here's what I got,
We already know that $\vert S_n / H \vert = n$
So I want to go by saying $\exists$ two group homomorphisms $\rho_1,\rho_2$ s.t.
$\rho_1: S_n \rightarrow \mathrm{Aut}(k^n)$ (Where Aut($k^n$) is really GL$_n(k)$, $k$ is some field.
And
$\rho_2: S_n \rightarrow \mathrm{Aut}(S_n / H)$
Then could I let $f: k^n \rightarrow S_n /S_{n-1}$
be defined by taking $\overline{v} \in k^n$ and mapping it to left coset $\overline{v}S_{n-1}$ and show this map is well defined and that
$f(\rho_1(\sigma)\overline{v})=\rho_2(\sigma)f(\overline{v})$ for $\sigma \in S_n$, $\overline{v} \in k^n$
Then do I just need to find a vector space isomorphism that preserves the $G$-action? Are my maps correct? Am I good so far/going the right direction? Thats really all I want to know. (Im taking rep theory and have not taken group/ring theory in about a decade)