TITLE: Permutation groups as symmetry groups of polynomials
QUESTION [15 upvotes]: I recently saw a question in a Text book, which asks to prove that "The group of symmetries of the polynomial $x_1x_2 + x_3x_4 + x_5x_6$ is a subgroup of $S_6$ of order $48$".
(By the group of Symmetries of this polynomial, we mean the stabilizer of the polynomial $x_1x_2+x_3x_4+x_5x_6$ in the action of the group $S_6$ on $\mathbb{Z}[x_1,x_2,\cdots,x_6]$ given by $\sigma.f(x_1,x_2,\cdots,x_6)=f(x_{\sigma(1)},x_{\sigma(1)},\cdots,x_{\sigma(6)})$.)
In view of this exercise, I would like to ask the following question:
Is every finite group realizable as the full stabilizer of a polynomial over $\mathbb{Z}$ in a certain number of indeterminates? If yes, then how can we construct that polynomial?

REPLY [9 votes]: If $X$ is a graph, define the polynomial $p_X$ to be the sum of the monomials $x_ix_j$, where $ij$ runs over the edges of $X$. (The polynomial in the question is the polynomial of a perfect matching.) Then the automorphism group of this polynomial (in the sense defined above) is the automorphism group of the graph $X$.
It has long been known that every finite group is the full automorphism group
of a finite graph. For background, see the wikipedia article on Frucht's theorem.