Question

Hi there!

Let $X$ be a left $G$-set, and $\Delta=${$x_1,\ldots,x_n$} a fundamental domain of $G$ in $X$. In other words, $G$ acts on $X$ from the left, and {$Gx_1,\ldots,Gx_n$} is the orbit space $X/G$.

Let us call $H_i$ the stabilizer of $x_i$, for all $i=1,2,\ldots, n$. Then the set $G/{H_i}$ of left $H_i$-cosets is the orbit $Gx_i$; according, the union $$T(H_1,\ldots, H_n):=\bigcup_{i=1}^nG/{H_i}$$ is another way to write down $X$ as the union of the $G$-orbits.

Somebody calls $T(H_1,\ldots, H_n)$ a coset geometry of rank $n$, or a Tits-Buekenhout geometry; interestingly enough, $T(H_1,\ldots, H_n)$ is a natural left $G$-set, and its identification with $X$ is a $G$-set morphism.

This line of thoughts looks so obvious that I cannot spot any mistake, so I'm asking:

Is it true that any $G$-set with $n$ orbits is a rank-$n$ coset geometry (in the sense I've just explained)?

If yes, who firstly realized that?

Answer 1

Any $G$-set with $n$ orbits has a rank $n$ coset geometry structure that is unique up to relabeling. Indeed, there is computer algebra software (e.g., GAP, Magma) that will produce such a structure from a list of stabilizers of orbits.

Some Googling reveals that Tits came up with coset geometries, so it is likely that he was the first to realize your observation.

Edit: Regarding software, MAGMA is not free (although I have heard that they have some forgiving policies for students). The documentation on incidence geometry has some discussion of the construction I outlined, especially in the introduction. In particular, they point out that a coset geometry is not in general a geometry in the sense of Buekenhout, due to an extra flag-transitivity condition. They reference:

Tits, Géométries polyédriques et groupes simples Atti 2a Riunione Groupem. Math. Express. Lat. Firenze, (1962) pp66-88.

GAP is free, and you can make coset geometries by following the documentation.