Write out the multiplication table for all possible products of elements in the group \(S_{3}\) (permutations on three objects). Use this to demonstrate explicitly that \(S_{3}\) is a group, that it is non-abelian, and that it has two proper subgroups: the group \(\mathrm{S}_{2}\) of permutations on two objects, and a group (called the alternating group) \(\mathrm{A}_{3} \equiv\{e,(123),(321)\}\). Show that for an operator \(c_{3}\) that rotates a system by \(\frac{2 \pi}{3}\) about a given axis, the set \(\mathrm{C}_{3} \equiv\left\{1, c_{3}, c_{3}^{2}\right\}\) also constitutes an abelian group and it is isomorphic to A₃.