It seems that I can generalize a result from compact, connected Lie groups to finite groups, but in order to do so, I need to have some kind of geodesics on finite groups. Below is a proposition for the definition of a geodesic. My main question is whether such geodesics have been studied or used.

Definition and justification: If $G$ is a compact, connected Lie group with a bi-invariant metric, we can construct all closed geodesics as follows. Take any nontrivial homomorphism $\phi:S^1\to G$ and an element $x\in G$. Then $S^1\ni t\mapsto x\phi(t)\in G$ is a closed geodesic.

The same construction can be done for finite groups if the circle $S^1$ is replaced with a cyclic group $C_n$. More explicitly, a geodesic of length $n$ on a finite group $G$ is a mapping $C_n\ni t\mapsto x\phi(t)\in G$, where $x\in G$ and $\phi:C_n\to G$ is a nontrivial homomorphism.

Reasons why this feels like a good definition:

• Every subgroup is totally geodesic in both cases (Lie and finite).
• On every nontrivial group there exists at least one geodesic in both cases.
• Geodesics are invariant under left and right translations in both cases.
• A finite abelian group is a product of cyclic groups, and a compact, connected, abelian Lie group is a product of copies of $S^1$ (a torus).

Motivation: It is a typical problem in integral geometry (a subfield of inverse problems) to ask whether a function on a closed manifold is determined by its integrals over closed geodesics. In the case of compact, connected Lie groups, this is possible if and only if the group is not the trivial group, $S^1$ nor $S^3$.

I wanted to consider the corresponding problem on finite groups, where of course the integral is replaced with a sum. That is, is a function on a finite group determined by its sums over all geodesics? It seems that I can give a decent answer, but not a complete classification of groups where the answer is affirmative. My current classification covers, for example, all Abelian, symmetric, alternating, dihedral and dicyclic groups.

Questions: In order of descending importance, this is what I would like to know:

1. Have geodesics in finite groups been studied before, perhaps under another name?
2. Has this "integral geometry problem" in finite groups been studied or does it have applications (concrete or abstract)?
3. Do these geodesics have some length minimization property analogously to the continuous case?

If you can answer any of the questions with a modified definition of geodesics, please do so. I am looking for a reasonable definition — or several if there are several — and I certainly do not want to forbid other definitions than mine. A possible variant is described below.

An answer about any particular finite group is most welcome. I am not asking for you to solve my "discrete integral geometry problem" here, since I already have a rather comprehensive answer; I want to know if this problem and the related geodesics are already known. References to any existing answers are, of course, most welcome. (If someone wants to know what I know about the problem, please contact me personally. There is too much to be included here.)

Norms on finite groups seem to have been studied, but I have found no study of geodesics.

Geodesics as dynamical systems: (For those prefer this point of view.) Geodesics on Lie groups (and all other Riemannian manifolds) can be realized as continuous time dynamical systems. There is a Hamiltonian flow on the cotangent bundle $T^*M$ (or the cosphere bundle $S^*M$) so that the projections of flow lines to $M$ are the geodesics.

Similarly, let $G$ be a finite group and consider the discrete time dynamical system on $G\times G$ that sends $(a,b)$ to $(b,ba^{-1}b)$. Note that a system starting outside (resp. on) the diagonal $\Delta\subset G\times G$ stays outside (resp. on) the diagonal. By finiteness any "discrete flow line" of this system on $G\times G\setminus\Delta$ is periodic. The projections of such flow lines to the first component of $G\times G$ are in one-to-one correspondence with discrete geodesics in the sense defined above.

A variant: Peter Michor suggested a different formulation in the comments below. I defined geodesics to be translations of cyclic subgroups, but it also makes sense to define geodesics as translations of maximal cyclic subgroups. In this setting a "Lie subgroup" $H$ of the finite group $G$ is such a subgroup that any maximal cyclic subgroup on it is also maximal in $G$. Then there are also non-Lie subgroups but all Lie subgroups are totally geodesic, analogously with the positive dimensional Lie groups. Answers to my questions with this kind of geodesics are also very welcome. (I have not looked at the integral geometry problem in this setting yet.)

Notes: A recent meta discussion showed green light to asking a question of this kind. There is an earlier question about geodesics on graphs, but it does not answer my question.