On the Orders of Finite Semisimple Groups

[摘要] 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 $H(mathbb{F}_q)$ for a split semisimple algebraic group 𝐻 defined over $mathbb{F}_q$, does not determine the group 𝐻 up to isomorphism, but it determines the field $mathbb{F}_q$ 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 $mathbb{F}_q$ such that the orders of the finite groups $H_1(mathbb{F}_q)$ and $H_2(mathbb{F}_q)$ 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.

[学科分类] 数学（综合）

[关键词] Finite semisimple groups;transitive actions of compact Lie groups;Artin’s theorem.