Generators and relations for special linear algebras and groups (Q1080956)

Let \(\bar G\) be any simply connected algebraic group over the algebraic closure \(\bar F_ p\) of a finite field of p elements. Let \(\sigma\) be a surjective endomorphism of \(\bar G\) with finite fixed points. Assume L is a group and \(\phi_ i: G_ i\to L\) are homomorphisms. The main result in this paper is the following theorem: Let \(\bar G\) be of type \(A_ k\), so \(\bar G\cong SL(k+1,\bar F_ p)\). Suppose \(\sigma\) does not involve the graph automorphism, so \(G\cong SL(k+1,F_ q)\) and \(G_ i\cong SL(1+rank \bar G_ i,F_{q^{m_ i}})\), \(i=1,...,r\), for some p-power q. If the commutation diagram of G forms a tree, then the \(\phi_ i's\) extend to a unique homomorphism \(G\to L\).