Proposition

Prove that $f \in k[x, y, z]$ is symmetric if and only if $f(x, y, z) = f(y, x, z) = f(y, z, x)$.

Solution

Consider $S \subset S_3$ be the set of all permutations under which $f$ is invariant. It suffices to show $S = S_3$. Since $S$ has to be a subgroup of $S_3$, $\abs{S} \mid 6$. $f(x, y, z) = f(y, x, z) = f(y, z, x)$ implies that $S$ contains two nontrivial elements, which are not the inverse of each other. Therefore, $S$ contains at least three nontrivial elements with the identity element so $S = S_3$.