When looking into [sizes][1] of finite simple group of "Lie type", I observed that power of the $q$ is equal to number of axes in the root system of corresponding Lie group. It is also valid for Steinberg groups. The exception is Suzuki and Ree groups $^2B_2, ^2G_2, ^2F_4$ where this number is $2,3,12$ which is half of number of axes in corresponding root system.

**Question**

Is it possible to find representation of root system in the finite group in following way. Axis in root system is represented by element of order $q$. Perpendicular axes correspond to commuting elements. Not perpendicular axes correspond to not commuting elements.

Next question is what subgroup such "root system" of elements will generate. I guess it should be either Sylow subgroup of size $q^k$ or full group.

**End of question**

There are also alternating and sporadic groups which does not have any $q$ number assigned. When we have proper definition of "root system" in finite group then we can see if it can be extended somehow to sporadic or alternating groups.


  [1]: https://en.wikipedia.org/wiki/List_of_finite_simple_groups