= Solution
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. Thus
$$
2\cos\theta=\operatorname{tr}A\in\mathbb Z\cap[-2,2]=\{-2,-1,0,1,2\}.
$$
These possibilities give rotation angles, modulo a full turn, $\pi$, $\pm2\pi/3$, $\pm\pi/2$, $\pm\pi/3$, and $0$, respectively. Consequently the <crystallographic restriction theorem> is
$$
\boxed{\text{rotation order }\in\{1,2,3,4,6\}.}
$$
The restriction comes from integrality of the <Euclidean lattice> action, not simply from finiteness of the <point group>.
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