TITLE: Understanding $D_n ⊂ O_2(R) ⊂ E(R^2 ) ⊂ S(R^2 )$ and their elements
QUESTION [0 upvotes]: I'm trying to get a grip of these groups:  $D_n ⊂ O_2(R) ⊂ E(R^2 ) ⊂ S(R^2 )$
where $D_n$ is best understood by me, but how about these other ones? Can you people give me an intuïtive explanation if their elements, order of their elements and so on?
Specifically $O_2(R) , E(R^2)$. Now I'm told that $O_2(R)$ can have elements of any finite order, can you give me an example? And what is $E(R^2)$ actually

REPLY [1 votes]: The orthogonal group $O_2(\Bbb R)$ is the group of all linear transformations $\Bbb R^2 \to \Bbb R^2$ that preserve length; these are precisely the rotations about the origin (which themselves comprise the special orthgoonal group $SO_2(R)$) and the reflections across a line through the origin. One can show that if we identify a linear transformation $\Bbb R^2 \to \Bbb R^2$ with its matrix (w.r.t. the standard representation), a matrix $A$ is in $O_2(\Bbb R)$ iff $A^T A = I_2$.
A rotation by an angle $\alpha$ followed by a rotation by an angle $\beta$ is the same thing as a rotation by the angle $\alpha + \beta$, so for positive integers $k$, the rotation by $\frac{2 \pi}{k}$ radians has order $k$ (in fact, such an element and any reflection together generate a copy of the dihedral group $D_{2k}$ of order $2k$). Rotations by an irrational multiple of $\pi$ have infinite order. All reflections have order $2$.
The group $E_2(\Bbb R)$ is the Euclidean group, which consists of all affine transformations that preserve length. These are precisely the affine transformations ${\bf x} \mapsto A {\bf x} + b$, where $A \in O_2(\Bbb R)$ and $b \in \Bbb R^2$. The composition rule shows that $E_2(\Bbb R)$ is a semidirect product: $E_2(\Bbb R) = O_2(\Bbb R) \ltimes \Bbb R^2$.
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