Most groups I encountered so far in classical mechanics an special relativity (Galilei-group, Poincare-group, Lorentz-group and so on) described transformations of the system that was to look at. This is somehow intuitive to me, because a transformation succeeded by another transformation is of course a transformation again.
However (and this is something I can't really wrap my head around): In QM, when talking about representations, it seems to me that choosing a representation maps group elements to operators, that is, to observables of the system. Which is of course less intuitive to me (given two positions, why would they have any "group" structure? Of course, if I imagine them to be vectors, I could just add them, but is this what is meant by position becoming a group element now?)
Which is true now? Group elements corresponding to transformations, or to observables? Is there an intuitive approach in how to to group elements are related to observables in EVERY theory that is based on groups?
Edit: I know that in QM, Observables are seen as Operators of an Algebra that act on the Hilbert Space. And of course, those Operators do form a group as well (because linear operators do form the general linear group). But this group is not always the same group that "the theory is built uppon"