Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 1 / 3F (Geometry of Group Actions)

A plane plane crystallographic group is a discrete subgroup of the Euclidean isometry group with compact quotient. Its translations form a normal rank-two Euclidean lattice $\Lambda$; this is the standard translation-lattice structure of a plane crystallographic group. Its point group is the finite group of linear parts, equivalently the quotient by those translations. Conjugating a translation by an isometry sends its translation vector to its image under the linear part, so every point-group element preserves $\Lambda$.

Choose a Euclidean lattice basis. A rotational linear part is then represented by an integer matrix $A$, since it sends the basis to Euclidean lattice vectors. In an orthonormal basis the same map has eigenvalues $e^{i\theta },e^{-i\theta }$, and its trace is independent of basis. ThusThese possibilities give rotation angles, modulo a full turn, $\pi$, $\pm 2\pi /3$, $\pm \pi /2$, $\pm \pi /3$, and $0$, respectively. Consequently the crystallographic restriction theorem isThe restriction comes from integrality of the Euclidean lattice action, not simply from finiteness of the point group.