• Shripad M Garge

      Articles written in Proceedings – Mathematical Sciences

    • On the orders of finite semisimple groups

      Shripad M Garge

      More Details Abstract Fulltext PDF

      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(Bn(q), Cn(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(H1,H2) 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 groupsHi 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.

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