On the order of finite semisimple groups

Abstract

It is a theorem of Artin, Tits et al. 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 > 2, 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 upto 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 give a geometric reasoning for these order coincidences.