Permutation on Polynomial is Group Action

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

Let $F_n$ be the set of all polynomials in $n$ variables $x_1, x_2, \ldots, x_n$:
 * $F = \left\{{f \left({x_1, x_2, \ldots, x_n}\right): f \text{ is a polynomial in $n$ variables} }\right\}$.

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

Then $S_n: F \to F$ is a group action on the set of all polynomials in $n$ variables.

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

Let $\pi * f$ be the permutation on the polynomial $f$ by $\pi$.

Let $e \in S_n$ be the identity of $S_n$.

By From Group of Permutations is Group:
 * $\quad e * f = f$

thus fulfilling axiom GA-1 of the definition of a Group Action.

Then we have that:

thus fulfilling axiom GA-2 of the definition of a Group Action.

Also see

 * Stabilizer of Polynomial