Permutation on Polynomial is Group Action

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

Let $$f \left({x_1, x_2, \ldots, x_n}\right)$$ 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 $$\pi * f$$ be the Definition:Permutation on a Polynomial$$f$$ by $$\pi$$.

Then:


 * 1) $$e * f = f$$;
 * 2) $$\pi \rho * f = \pi * \left({\rho * f}\right)$$;
 * 3) $$\forall \lambda \in \reals: \pi * \left({\lambda f}\right) = \lambda \left({\pi * f}\right)$$.

Thus this is an example of a group action where $$S_n$$ acts on the set of all polynomials in $$n$$ variables.

The stabilizer of a polynomial is the set of permutations which fix the given polynomial.