# 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 $S_n$.

$e * f = f$

thus fulfilling Group Action Axiom $GA \, 1$.

Then we have that:

 $\displaystyle \paren {\pi \circ \rho} * f$ $=$ $\displaystyle \map \pi {\rho * f}$ $\displaystyle$ $=$ $\displaystyle \pi * \paren {\rho * f}$

thus fulfilling Group Action Axiom $GA \, 2$

$\blacksquare$