91Ó°ÊÓ

Consider the polynomial \(p=\left(x_{1}-x_{2}\right)^{2}+\left(x_{3}-x_{4}\right)^{2}\). It is unaltered when the subscripts undergo any of the following permutations: $$ \begin{array}{lll} \left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 2 & 1 & 3 & 4 \end{array}\right) & \left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 1 & 2 & 4 & 3 \end{array}\right) & \left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 2 & 1 & 4 & 3 \end{array}\right) & \left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 3 & 4 & 1 & 2 \end{array}\right) \\ \left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 4 & 3 & 1 & 2 \end{array}\right) & \left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 3 & 4 & 2 & 1 \end{array}\right) & \left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 4 & 3 & 2 & 1 \end{array}\right) & \left(\begin{array}{llll} 1 & 2 & 3 & 4 \\ 1 & 2 & 3 & 4 \end{array}\right) \end{array} $$For example, the first of these permutations replaces \(p\) by $$ \left(x_{2}-x_{1}\right)^{2}+\left(x_{3}-x_{4}\right)^{2} $$ the second permutation replaces \(p\) by \(\left(x_{1}-x_{2}\right)^{2}+\left(x_{4}-x_{3}\right)^{2} ;\) and so on. The symmetries of a polynomial \(p\) are all the permutations of the subscripts which leave \(p\) unchanged. They form a group of permutations. List the symmetries of each of the following polynomials, and write their group table. $$ p=\left(x_{1}-x_{2}\right)\left(x_{2}-x_{3}\right)\left(x_{1}-x_{3}\right) $$