Permutation_on_Polynomial_is_Group_Action_v2

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.