Should I care if some system forms a group? I am currently studying a discrete dynamical system where the function composition on the set of periodic points forms a finite group.

From a dynamical systems perspective, is there any reason to care about this property?  Would it be worth continuing to study the properties of this group?
Assuming that you have checked that each element does have an inverse, note that:


  The set if iterates on the set of periodic points is finite if and only if there is a maximal period.


So it is fair to say that nothing else can be learned.
Assuming that you have checked that each element does have an inverse, note that:


  The set if iterates on the set of periodic points is finite if and only if there is a maximal period.


So it is fair to say that nothing else can be learned.