Lemma 6

Let V be a finite dimensional vector space, G a finite group of automorphisms of V, and $v$ and $v'$ elements of V such that $v'\notin Gv$. There exists a G-invariant polynomial function $f$ on V such that $f(v')\not=f(v)$.

Indeed, for each $s\in G$ there exists a polynomial function $g_s$ on V equal to 1 at $v$ and to 0 at $sv'$. Then the function $g= 1-\prod_{s\in G}g_s$ is equal to 0 at

$v$ and to 1 on $Gv'$. The polynomial function $f=\prod_{t\in g}t.g$ is G-invariant, equal to 0 at $v$ and to 1 at $v'$.