# Looking for root system in finite simple groups

When looking into sizes of finite simple group of "Lie type", I observed that power of $q$ 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 $q$. Number $q=p^l$ is power of certain prime $p$. Perpendicular axes correspond to commuting elements. Not perpendicular axes correspond to not commuting elements.

I admit that maybe elements of order $p$ can be used. I tested groups $L_3(2)$ and $U_4(2)$ and I found respectively $3$ and $6$ involutions having described property. Three involutions generated whole group $L_3(2)$. Six involutions generated subgroup of size $2^6$ in $U_4(2)$. Product of not commuting involutions have order 4.

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

Motivation

In Robert Wilson's book about finite simple groups I can find phrase "maximal torus" used when discussing subgroups of $F_4(q)$ and $E_6(q)$. This means that we can use terms from Lie groups in finite groups of Lie type. My next goal would be defining multiplication in the group by using the root system and number $q=p^l$.

Yet another goal is to see how we can understand sporadic group by looking at its p-Sylow group for certain prime p, whose power at least 3 divide order of the group.

End of motivation

• The question is natural but difficult, and versions of it have (I believe) been raised at times in connection with the classification of finite simple groups. But a big problem at the outset is to assign a "natural" characteristic $p$ (or a power $q$ of it) to an arbitrary simple group. You recognize this problem but try to avoid it at first even though it's unavoidable. – Jim Humphreys Aug 16 '17 at 13:01
• The groups of "Lie type" have natural characteristic. Can you explain your last sentence, what do you mean by "try to avoid it at first" ? It is kind of paradox you use. I encountered word "maximal torus" used for finite groups of Lie type. This means that people try to use notions from compact Lie groups to finite ones. – Marek Mitros Aug 16 '17 at 13:35
• I just meant that in your question you use $q$ for an arbitrary finite group without specifying it. Of course, groups of Lie type do have a natural characteristic, but others such as alternating groups get tricky. (Also, the notion of "maximal torus" doesn't immediately make sense for an arbitrary finite group.) – Jim Humphreys Aug 16 '17 at 15:32
• You mention "We can use terms from Lie groups in finite groups of Lie type." The reason for this is that many such terms have been generalised from (linear) Lie groups to the setting of linear algebraic groups over arbitrary fields, and that, at least according to most definitions (but see @JimHumphreys's post discussing the fact that there is no one correct such), a finite group of Lie type is the group of rational points of a linear algebraic group over a finite field. – LSpice Jan 1 at 16:49