On the order of finite semisimple groups

The aim of this paper is to investigate the order coincidences among the finite semisimple groups and to give a reasoning of such order coincidences through the transitive actions of compact Lie groups. It is a theorem of Artin and Tits that a finite simple group is determined by its order, with the exception of the groups (A(3)(2), A(2)(4)) and (B-n(q), C-n(q) for n >= 3, q odd. We investigate the situation for finite semisimple groups of Lie type. It turns out that the order of the finite group H(F-q) for a split semisimple algebraic group H defined over F-q, does not determine the group H LIP to isomorphism, but it determines the field F-q Under some mild conditions. We then Put a group structure oil the pairs (H-1. H-2) of split semisimple groups defined over a fixed field F-q such that the orders of the finite groups H-2 (F-q) and are the same and the groups H-i have no common simple direct factors. We obtain an explicit set of generators for this abelian, torsion-free group. We finally show that the order coincidences for some of these generators can be understood by the inclusions of transitive actions of compact Lie groups.