On the orders of finite semisimple groups

  • Shripad M. Garge


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 (A3(2), A2(4)) and(B n (q), C n (q)) forn ≥ 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 groupH defined over\(\mathbb{F}_{_q } \), does not determine the groupH 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 H1(\(\mathbb{F}_{_q } \)) and H2(\(\mathbb{F}_{_q } \)) are the same and the groupsH 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.


Finite semisimple groups transitive actions of compact Lie groups Artin’s theorem 


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Copyright information

© Indian Academy of Sciences 2005

Authors and Affiliations

  1. 1.School of MathematicsTata Institute of Fundamental ResearchMumbaiIndia

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