# gr.group theory – Finite simple groups of automorphisms of finite simple Lie algebras

I begin by briefly recalling some basic facts in order to pose my question in context.

According to the classification, the finite simple groups are cyclic of prime order, are alternating on $$n geq 5$$ letters, are of so-called Lie-type, or belong to a family of $$26$$ sporadic groups that cannot be squeezed into any of the previous families.

According to Chevalley and co., the finite simple groups of Lie-type can (roughly) be constructed as follows. We first consider the classification of the finite-dimensional simple Lie algebras over the complex numbers. We can then define (in a natural way) analogues $$L$$ of these Lie algebras in prime characteristic $$p$$. Finally, we consider some well-chosen subgroups of the automorphism group of $$L$$.

However, the above Lie algebras $$L$$ do not exhaust all simple Lie algebras in characteristic $$p$$. According to the classification of the finite-dimensional simple Lie algebras over an algebraically-closed field of characteristic $$p>5$$, the remaining simple Lie algebras belong to one of four additional families (of so-called Cartan-type): the algebras of Witt-type, of Special-type, of Hamiltonian-type, and of Contact-type. In characteristic $$p = 5$$, there is exactly one more family, namely that of Melikyan-type.

Roughly speaking, I want to ask whether Chevalley’s programme has been extended to the simple Lie algebras that are not of classical type. More precisely:

Question: Have there been any systematic efforts to realize/construct finite
simple groups as subgroups of the automorphism groups of non-classical simple Lie
algebras (of finite dimension over a suitable field of prime
characteristic)?

I understand that, in view of the CFSG and Chevalley’s programme, there is no strict need to perform such work: we already know of at least one way to obtain the simple groups of Lie-type. I also understand that the simple groups of Lie-type can even be obtained in a second way, using the theory of algebraic groups with Frobenius automorphisms. But I would argue that it would still be interesting to know exactly which finite simple groups naturally appear as automorphism groups of simple Lie algebras that are not considered classical.

I also acknowledge that I have used the word “natural” in a rather imprecise way.

Caveat: In the modular case, the simple Lie algebras have only been classified over algebraically-closed fields. And their forms over finite subfields may have different automorphism groups. So, unlike in the classification over the complex numbers, it may not immediately be clear which forms to choose when trying to extend Chevalley’s programme as outlined above.