On the order of finite semisimple groups

الأصل · EN

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 (A₃(2), A₂(4)) and (Bₙ(q), Cₙ(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(Fq) for a split semisimple algebraic group H defined over Fq, does not determine the group H upto isomorphism, but it determines the field Fq under some mild conditions. We then put a group structure on the pairs (H₁, H₂) of split semisimple groups defined over a fixed field Fq such that the orders of the finite groups H₁(Fq) and H₂(Fq) are the same and the groups Hᵢ 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.