A question about the action of $S_n$ on $K[x_1,...,x_n]$ Let ${K}$be the field ($\,Char\,K\not=0)$. Let $n\in \mathbb{Z}^{+}$. $S_n$ acts on $K[x_1,...,x_n]$in the following way:
If $p\in K[x_1,...,x_n]$ and $\sigma\in S_n$, then $\sigma p$  is the polynomial $p(x_{\sigma(1)},x_{\sigma(2)},...,x_{\sigma (n)})$. 
Question: Let $H$ be a subgroup of $S_n$, must there exist a polynomial $p\in K[x_1,...,x_n]$ such that $stab(p)=H$ ?
(Where $stab(p)$ is defined to be $\{\sigma\in S_n|\sigma p=p\})$
Thank you

Edit: I also want to know the answer to the question in the case that our field $K$ is $\mathbb{C}$

Yes.
Let $f(x_1,\ldots, x_n)=\prod_{k=1}^n x_k^k$. Then 
$p=\sum_{h\in H} h(f)$ has the desired property. (This works for all fields, also those of nonzero characteristic)
