Paper 3, Section II, 7E7 \mathrm{E}

Groups | Part IA, 2017

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

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

where nn is the number of distinct orbits of XX under the action of GG.

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

Show that

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

and deduce that

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

(b) Let HH be the group of rotational symmetries of the cube. Show that HH 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.]

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