The Thomson problem on the $S^2$ sphere asks what configuration(s) of $N$ points minimize a particular function which is symmetric under all permutations of its arguments, $$F(x_1,\ldots,x_N) = \sum_{i<j} f(x_i-x_j)$$ where $f(x)=f(-x)$ is the electrostatic potential but in principle could be something else. For given $N$ one typically gets some more or less equidistant points as much as this is possible. Clearly, if all points at the minimum are rotated by $SO(3)$ that will be a minimum as well, at least in the electrostatic case, so one of the points can be chosen as a prescribed point, for example the North pole. There is still a remaining $SO(2)$ symmetry which rotates around the North pole - South pole axis.
Now I was wondering if instead of $S^2$ we have a compact, connected, semi-simple Lie group $G$, is it possible to define a relatively simple function $f$ such that the minima of $F$ will be a finite subgroup? This of course can only possibly work if $N$ is such that a finite subgroup exists with that order. More precisely, $$F(g_1,\ldots,g_N) = \sum_{i<j} f(g_i g_j^{-1})$$ with $f(g) = f(g^{-1})$, and minimizing it seems to make sense for any $N$ so we'll always have some configuration of points on $G$, but is it possible to choose $f$ such that the minima consists of a finite subgroup (up to over-all multiplication by $G$) if a subgroup of order $N$ exists? Again if a minimum is found at $g_1,g_2,\ldots,g_N$, the right multiplication by any element of all points will be a minimum as well. So we can assume $g_1 = 1$. This is what I mean by "up to over-all multiplication by $G$".
Of course $f$ should not depend on $N$.
Even $G = SU(2) = S^3$ would be useful to look at which bears some similarity to the original Thomson problem.
