Past exam of the mathematics course of the University of Cambridge / 2010 / ia / Paper 3 / 5D (Groups)

For a finite group action, the orbit-stabilizer theorem statesThe bijection $g{G}_x\mapsto gx$ explains the formula by identifying cosets with the group orbit.

The cube rotation group acts transitively on its six faces. A rotation fixing one face fixes its center and its outward normal, so its axis is the line joining that center to the opposite face's center. The allowed rotations are the four multiples of $\pi /2$ about this axis. Thus the stabilizer subgroup is $C_4$, andThe face-pair orbits of cube rotations are the ordered equal-face pairs, opposite-face pairs, and adjacent-face pairs. These relations are preserved by every cube rotation. The first two families are transitive because the face action is transitive and the opposite face is uniquely determined. For the third, first send the first face to a prescribed face, then use its four quarter-turns to send the neighboring second face to any chosen neighbor. Thus there are exactly three orbits, of sizesTheir sizes sum to $36=|X\times X|$.

A subgroup $N\le G$ is a normal subgroup if $gN{g}^{-1}=N$ for every $g\in G$.

Place the cube with its opposite-face axes along the coordinate axes. ConsiderThe three nonidentity elements are the half-turns about those axes. Multiplying two different ones gives the third, and each squares to the identity, so this is a Klein four-group of order four.

Every cube rotation permutes the three pairs of opposite faces. The associated group homomorphism $G\to S_3$ has exactly $N$ as its kernel: fixing all three face-axis pairs allows only independent sign changes of the three coordinate directions, with determinant one. Hence the kernel of the cube face-axis action is a normal subgroup, and