What is so special about the Special Orthogonal group? I am trying to learn a bit about (geometrical Euclidean)  group theory and am now wondering "Wat is so special about the Special Orthogonal group?"
Or what is the difference between the groups $O(2)$ and $SO(2)?$ 
Or name some transformations belong to $O(2)$ but not to $SO(2)$ 
the same for the Linear Group  and the Special Linear group $SL(2)$.

In this context it just means each element has determinant 1.
