On the orders of finite semisimple groups

Abstract

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 up to isomorphism, but it determines the field F q under some mild conditions. We then put a group structure on 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 1 (F q ) and H 2 (F q ) 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.