|Universiteit Faculteit FNWI English version||2 december 2004, e-mail|
Woensdag 15 december 2004
AbstractThis is joint work with Don Taylor of the University of Sydney, and Arjeh Cohen and Sergei Haller of the Technical University of Eindhoven.
Most of the finite simple groups are groups of Lie type. A finite group of Lie type can be described as the fixed points of a Frobenius endomorphism acting on a reductive algebraic group. Given a structure in the algebraic group, such as a conjugacy class or a maximal torus, we want to find the corresponding structures in the finite group of Lie type — I will demonstrate how this can often be achieved with Lang's Theorem. Then I will describe the first computationally efficient algorithm for Lang's Theorem in split connected reductive groups.
Glasby and Howlett (1997) have already solved this problem in a special case; our algorithm is inspired by their work and the proof of Lang's Theorem given in (Müller, 2003). Our approach also uses the Lie algebra of the group. I will discuss algorithms for computing with modular Lie algebras, which have recently been classified (except for characteristic 2 and 3) by Strade and others.
Let k be a finite field of size q and characteristic p. Let G be a k-split connected reductive linear algebraic group. Then G can be considered a subgroup of GLn and the standard Frobenius endomorphism F is the map which takes the qth power of every entry in a matrix. Lang's Theorem states that the map