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?



Is any $G$-set a coset geometry (in the sense of Tits-Buekenhout)?
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](http://magma.maths.usyd.edu.au/magma/handbook/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](http://www.gap-system.org/) is free, and you can make coset geometries by [following the documentation](http://cage.ugent.be/geometry/fining/chap13.html).


