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 = \set {\map f {x_1, x_2, \ldots, x_n}: f \text{ is a polynomial in $n$ variables} }$

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

Let $*: S_n \times F \to F$ be the mapping defined as:


 * $\forall \pi \in S_n, f \in F: \pi * \map f {x_1, x_2, \ldots, x_n} = \map f {x_{\map \pi 1}, x_{\map \pi 2}, \ldots, x_{\map \pi n} }$

Then $*$ is a group action.

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 $S_n$.

From Symmetric Group is Group:
 * $e * f = f$

thus fulfilling.

Then we have that:

thus fulfilling

Also see

 * Stabilizer of Polynomial