Symmetry Group Freedom: Choosing How Groups Act on Coordinates

[kent davidge]

One could argue that this question should be posted on the maths forum, but I see it so frequently in physics that I find it more productive to ask it here.

In a symmetry group, do we have freedom of choice of how the group is going to act in the coordinates? Or is the way the group act on the coordinates imposed on us by the group itself?

An example of what I mean: do I have the freedom to identify a group of order 2 as the responsible for rotations around say, X axis, or this same group I could say that is the group of reflections in say, XY plane?

 

[strangerep]

One writes out the transformation between coordinates, and then derives generators from that. E.g., the coordinate transformations corresponding to rotations around the X axis are in terms of an angle $\theta$. Then by taking a derivative wrt $\theta$ at $\theta=0$ one can derive differential operators which generate the finite transformations. (I can post a bit more detail if you need it).

Try this exercise for both your cases. You'll find that the ordinary rotation group is very different from the group of reflections.