Past exam of the mathematics course of the University of Cambridge / 2015 / ia / Paper 3 / 5D / a / Solution

The identity permutation fixes every multivariate polynomial. With permutation composition $(\sigma \tau )(i)=\sigma (\tau (i))$, substitution givesThus this is a left variable-permutation action on polynomials. The substitution acts on variable names; using the displayed definition avoids reversing the composition order.

For the given multivariate polynomial, the group orbit corresponds to the three partitions of four indices into two unordered pairs:These three polynomials are distinct, and every pairing occurs under a permutation. The orbit-stabilizer theorem givesThe eight elements of the stabiliser subgroup independently interchange the members of each pair, and may also interchange the two pairs.