Why is group theory important in physics?

Question

I have read quite a bit about it and I have studied group theory in more than one math class. I assumed that the idea is basically you show that some set of objects are a group and then you can predict other properties of those objects. But I have never seen it stated that way or seen a simple example from physics that is an illustration of this idea -- frankly I may have not been able to penetrate far enough into the online explanations of physics and group theory to find this but if it is that way, I would put that in the introductory paragraph.

Answer 1

There are many examples of groups in physics. Some very important ones are rotation groups. This includes rotations in three-dimensional space as well as generalized rotations in four dimensional space-time. These form groups, because we can always undo a rotation using another rotation, there is an identity (rotation by angle 0), and we can define the product of two rotations as the result of first performing one rotation and then another.

More generally, we are concerned in physics with groups of symmetries (that is, transformations that do not effect the dynamics of our problem). Rotation groups are one possible example. Translation groups are another example. We can consider groups of continuous transformations, or groups made of discrete transformations (which might be important for crystals). We also end up with other kinds of symmetries, like gauge symmetries, in quantum field theory, which correspond with groups like $U(1)$ and $SU(3)$. By Noether's theorem, conservation laws in physics arise as a result of symmetry transformations, so understanding how symmetries work is very important.