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.
We require $n>1$ in order to exclude singletons as geodesics.
It does not matter whether we multiply by $x$ from the left or from the right.
Left and right translations (cosets) give the same geodesics since if $\phi:C_n\to G$ is a nontrivial homomorphism, so is $t\mapsto x\phi(t)x^{-1}$.
Reasons why this feels like a good definition:
- Every subgroup is totally geodesic in both cases (Lie and finite).
- In 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:
- Have geodesics in finite groups been studied before, perhaps under another name?
- Has this "integral geometry problem" in finite groups been studied or does it have applications (concrete or abstract)?
- 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.
One 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 who 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.
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.
Best Answer
First, my apologies for this late answer, I only found the question today. Below, I probably recall too many things, but I felt it could put some context around the short answer to question 1 saying: yes, under the names "geodesic in an involutory quandle", or "cycle in a symmetric set".
1) Recall that a homogeneous symmetric space (I borrow the terminology from Loos [2]) is a homogeneous space $G/H$ of a Lie group $G$, where $H$ is an open subgroup of the fixed points subgroup $G^\sigma$ of an involution $\sigma$ of $G$ ($\sigma\in Aut(G)$ and $\sigma^2=Id$).
A homogeneous symmetric space has a canonical connection $\nabla$ for which the geodesics are of the kind $t\mapsto g\cdot \exp(tX) H$ with $g\in G$ and $X\in \mathfrak{p}$. Here $\mathfrak p=\{X\in Lie(G) \mid Lie(\sigma)(X)=-X\}$.
Every Lie group $K$ (compact or not) is a homogeneous symmetric space, with $G=K\times K$, $\sigma(k,k')=(k',k)$ and $H=diag(K)$.
2) There is an intrinsic presentation of symmetric spaces due to Loos: a symmetric space is a smooth manifold with a smooth product law $(x,y)\mapsto x\bullet y=s_xy$ such that
The maps $s_x:M\to M$ are called the symmetries. A morphism of symmetric spaces $M\to N$ is a smooth map $\phi:M\to N$ such that $\phi(s_xy)=s_{\phi(x)}\phi(y)$. Any homogeneous symmetric space $G/H$ is a symmetric space in this sense, with $s_{gH}(g'H) = g\sigma(g^{-1}g')H$. Conversely, any connected symmetric space $M$ (with a choice of base point $o\in M$) is a homogeneous symmetric space $G/H$, with $G$ the subgroup of $Aut(M)$ generated by $\{s_xs_y\mid x,y\in M\}$ (it is a finite-dimensional Lie group), involution $\sigma(g)=s_ogs_o$, and $H=Stab_G(o)$.
In this context, it can be seen that a geodesic in $M$ is simply a morphism of symmetric spaces from the real line $\mathbb R$ to $M$. Here, $\mathbb R$ has the symmetries $s_xy=2x-y$.
3) If we remove axiom 4. in the definition of symmetric space (and forget about smoothness), we get a purely algebraic object which appears under various names in the literature: kei (Takasaki [5]), symmetric set (Nobusawa [3], for finite sets), involutory quandle (Joyce [1]), symmetric groupoid (Pierce [4]).
In analogy with the smooth case, one may define a geodesic in $M$ as a morphism of such spaces, from the integers $\mathbb Z$ to $M$. Here, $\mathbb Z$ has the symmetries $s_xy=2x-y$.
Joyce gives an abstract definition of involutory quandle with geodesics [1, p. 30] and shows that any involutory quandle can be seen as an involutory quandle with geodesics, essentially by defining the geodesics as is done above.
Obviously, a geodesic is determined by the images of 0 and 1 (the point-and-tangent-vector datum determining a geodesic in the smooth case is replaced by a pair of points in the discrete case). Nobusawa [3, pp. 570-571] calls these geodesics cycles (symmetric subspaces generated by two points).
As in the smooth case, any (finite) group $G$ can be seen as a (finite) symmetric set by setting $s_gh=gh^{-1}g$. In the finite group case, the geodesics then coincide with your definition (except that singletons are not excluded).
4) That said, to my knowledge, these geodesics have not been much studied for themselves. I don't know the answer to questions 2 and 3 (except that in the present context, no metric is involved).
References
[1] David Joyce, An Algebraic Approach to Symmetry with Applications to Knot Theory, Thesis. http://aleph0.clarku.edu/~djoyce/quandles/aaatswatkt.pdf
[2] Ottmar Loos, Symmetric Spaces. 1: General theory, Benjamin, New York, Amsterdam, 1969.
[3] Nobuo Nobusawa, On symmetric structure of a finite set, Osaka J. Math. Volume 11, Number 3 (1974), 569-575. http://projecteuclid.org/euclid.ojm/1200757525
[4] R. S. Pierce, Symmetric groupoids, Osaka J. Math. Volume 15, Number 1 (1978), 51-76. http://projecteuclid.org/euclid.ojm/1200770903
[5] M. Takasaki, Abstractions of symmetric functions, Tohoku Math. J. 49 (1943), 143-207, [Japanese]. https://www.jstage.jst.go.jp/browse/tmj1911