Fundemental relation between group symmetries and periodicity?

[DJsTeLF]

Fundemental relation between group symmetries and periodicity??

My question is simply: Is there a fundamental relation between group symmetries and periodicity?

I been studying group theory within my recent studies of QFT and the Standard Model and the aforementioned question occurred to me so I figured one of the more mathematical physicists on here might be able / willing to answer / elaborate on in what circumstances the answer is yes.

 

[DJsTeLF]

Ok I've given this a little more thought and realized I really should elaborate on what I meant and was hoping to hear in response..

Say for example one considers a distrete group operation like C, P or T conjugation and that the system in question is symmetric for each. This could be pictured (as I tend to do with things in my 'minds eye') as the flipping (or rotation of pi radians) of a 1D line of unit length, centred at the origin, through (around) a 2nd dimension.

Similarly for continuous groups, particularly that of SO(2) rotations in a 2D manifold, one can picture a unit circle analogy but this time with group elements consisting of the infinestimal angles about each of the axis. I.e. it would map out a unit 3-sphere.

In the disctrete case the line will be periodically equivalent to its initial orientation whereas in the continuous case this period would tend to zero.

I guess I could therefore phrase a few more specific questions as follows:

1/ In both circumstances discussed above one can unitilise an extra-dimension to visulatize the 'shape' that results from the group operations in question. Is there any sense in doing this and what would the implication be for other types of groups such as SU(2) for example?

2/ Somewhat related to the first question can anyone suggest a method of visualizing group operations for the dimensions that we can picture in our minds eye that could then be extended to hypersurfaces where the need arises?

3/ I am aware EM Lagrangian (without any of the complications from unifying it with the weak force) is U(1) symmetric for a phase-factor of the form e^i*theta since absolute phase cannot be observed. Does it make any sense however to contrast this with the phase of a sine-wave that would only be symmetric under a tranformation of a complete period of its oscillation?