Wallpaper group

Example

A wallpaper group is a crystallographic group of dimension $2$: a discrete group of isometries of the plane $\R^2$ with compact quotient. A concrete one, called $p4$, is the group $\Gamma$ of all maps $$v \mapsto R^k v + t, \qquad k \in \set{0, 1, 2, 3}, \ t \in \Z^2,$$ where $R(x, y) = (-y, x)$ is the rotation by a quarter turn about the origin. The set $\Gamma$ is a group, because $R$ maps $\Z^2$ onto itself, so composites and inverses of such maps have the same form. It is discrete, because its elements have only four linear parts and their translation parts lie in $\Z^2$. The quotient is compact, because every orbit meets the unit square $[0, 1]^2$, already under the translations alone. The translation subgroup is $\Z^2$, of index $4$, as part 1 of the Bieberbach theorems predicts.