Let R be the set of all polynomials with integer coefficients in the independent variables x_1, x_2, x_3, x_4, ie. the members of R are finite sums of elements of the form $\displaystyle ax_1^{r_1}x_2^{r_2}x_3^{r_3}x_4^{r_4}$ where a is any integer and r_1, ..., r_4 are nonnegative integers. Each $\displaystyle \sigma \in S_4$ gives a permutation of {x_1, ..., x_4} by defining $\displaystyle \sigma \cdot x_i = x_{\sigma (i)}$. This may be extended to a map from R to R by defining $\displaystyle \sigma \cdot p( x_1, x_2, x_3, x_4 ) = p( x_{\sigma (1)}, x_{\sigma (2)}, x_{\sigma (3)}, x_{\sigma (4)}$ for all $\displaystyle p( x_1, x_2, x_3, x_4 ) \in R$.

Exhibit all permutations in S_4 that stabilize the element $\displaystyle x_1 x_2 + x_3 x_4$ and prove that they form a subgroup isomorphic to the dihedral group D_8.