The euclidean plane isometry  consists of 5 motions: Identity Reflection Rotation Translation Glide Reflection Suppose $G<E_2$ is a discrete group. First consider the translation subgroup of $G$. It holds $G\cap \mathbb{R}^2 < E_2$. We have $E_2 = \mathbb{R}^2 \rtimes O_2 $. $E_2 … Continue reading Lecture Notes 5.11. and 6.11.2012