Paper 3, Section II, $7 \mathrm{E}$

(a) Let $G$ be a finite group acting on a finite set $X$. State the Orbit-Stabiliser theorem. [Define the terms used.] Prove that

$\sum_{x \in X}|\operatorname{Stab}(x)|=n|G|$

where $n$ is the number of distinct orbits of $X$ under the action of $G$.

Let $S=\{(g, x) \in G \times X: g \cdot x=x\}$, and for $g \in G$, let $\operatorname{Fix}(g)=\{x \in X: g \cdot x=x\}$.

Show that

$|S|=\sum_{x \in X}|\operatorname{Stab}(x)|=\sum_{g \in G}|\operatorname{Fix}(g)|$

and deduce that

$n=\frac{1}{|G|} \sum_{g \in G}|\operatorname{Fix}(g)|$

(b) Let $H$ be the group of rotational symmetries of the cube. Show that $H$ has 24 elements. [If your proof involves calculating stabilisers, then you must carefully verify such calculations.]

Using $(*)$, find the number of distinct ways of colouring the faces of the cube red, green and blue, where two colourings are distinct if one cannot be obtained from the other by a rotation of the cube. [A colouring need not use all three colours.]

*Typos? Please submit corrections to this page on GitHub.*