Paper 3, Section I, D

(a) Let $G$ be the group of symmetries of the cube, and consider the action of $G$ on the set of edges of the cube. Determine the stabilizer of an edge and its orbit. Hence compute the order of $G$.

(b) The symmetric group $S_{n}$ acts on the set $X=\{1, \ldots, n\}$, and hence acts on $X \times X$ by $g(x, y)=(g x, g y)$. Determine the orbits of $S_{n}$ on $X \times X$.

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