There is a question I would like to ask at MO, but it seems somewhat unorthodox compared most others, so I want to get some support (or discouragement) here first. What I want to do is to give a new definition, justify why it is a natural extension of a well known concept, and ask if someone has seen it or has something to say about it. I am trying to extend a theory that I understand to a setting outside my main expertise, and I therefore risk making a trivial question. I have put effort in trying to find research or notes about this new object but to no avail. I do have genuine research interest in this thing, so it is not only a matter of curiosity.
Would something like this make a good question? (I don't see why not, but I have not encountered such questions here yet, so I'm slightly worried.) Is there something that I should be particularly careful about when formulating a question of this kind? Are there examples of good and bad questions like this to learn from?
I chose to make this meta question quite general in order to focus discussion on the idea of definitions as questions rather than the particulars of the question I had in mind. But if you want a short description of my question, I can give one. The full question is lengthy, so I will not produce it here.
A short version of the question I had in mind:
Periodic geodesics on compact Lie groups can be described algebraically, without any reference to minimizing arc length: a periodic geodesic is a mapping $S^1\ni t\mapsto x\phi(t)\in G$ where $\phi:S^1\to G$ is a nontrivial homomorphism and $x\in G$. By analogue, we can define a geodesic on a finite group by replacing $S^1$ with a finite cyclic group. The geodesic flow on finite groups can be seen as a discrete time dynamical system. Have geodesics on finite groups been studied before, perhaps under another name? Does this structure look familiar to anyone? A problem in the field of inverse problems asks whether a function on a closed manifold is uniquely determined by its integrals over all closed geodesics. This problem has been studied quite a lot, also on Lie groups. A natural generalization of the question now asks whether a function on a finite group is uniquely determined by its sums over all geodesics. Has this problem been studied before? Is it known to have applications, abstract or concrete? I have obtained some results on this finite generalization. I am fairly confident that this problem is new to the inverse problems community, but I am not sure if it is well known in another field. It would be great if I could motivate the question or relate it to existing literature. Any ideas, references or analogues could be helpful in understanding and solving the problem.