Permutation on Polynomial is Group Action

Theorem
Let $$n \in \mathbb{Z}: n > 0$$.

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

Then $$\pi \wedge f$$ is the polynomial obtained by applying the permutation $$\pi$$ to the subscripts on the variables of $$f$$, where:


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

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

This is called the the permutation of the polynomial $$f$$ by $$\pi$$, or the $$f$$-permutation by $$\pi$$.

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