Operation of $O_2$ on the plane. I am currently reading through M. Artin's Algebra 2.ed. In chapter 6, remarks similar to the following are often made with little explanation:
"Unless an origin is chosen, the orthogonal group $O_2$ doesn't operate on the plane $P$."
What exactly does this mean?
 A: If you consider the plane as a two-dimensional vector space, then $O_2$ acts on the plane by matrix multiplication in the obvious way.
If I just hand you an affine space which is isomorphic (as an affine space) to the plane, though, there's no natural choice of a vector space structure on it, so there's no natural action of $O_2$ on it.  If you pick a  point to be the origin, this completely determines a unique vector space structure compatible with the affine structure and now you have an action of $O_2$, but any such choice is just as good as any other and they give you different actions.
So, more precisely, there are infinitely many actions of $O_2$ on a plane with no fixed origin, none of which are natural, in contrast to the natural action of $O_2$ on a plane with a fixed origin given by matrix multiplication.
(Natural is a technical term essentially meaning that you can specify an object uniquely.  A precise definition is here: http://en.wikipedia.org/wiki/Natural_transformation)
