Stabilizer of Polynomial

Theorem
Let $n \in \Z: n > 0$.

Let $\map f {x_1, x_2, \ldots, x_n}$ be a polynomial in $n$ variables $x_1, x_2, \ldots, x_n$.

Let $S_n$ denote the symmetric group on $n$ letters.

Let $\pi, \rho \in S_n$.

Let the group action $\pi * f$ be defined as the permutation on the polynomial $f$ by $\pi$.

Then the stabilizer of $f$ is the set of permutations on $n$ letters which fix $f$.

Proof
Follows directly from the definition of the stabilizer of $f$.