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**1 - 3**of**3**### The Classification of the Finite Simple Groups: An Overview

- MONOGRAFÍAS DE LA REAL ACADEMIA DE CIENCIAS DE ZARAGOZA. 26: 89–104, (2004)
, 2004

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### ON THE ORDER OF FINITE SEMISIMPLE GROUPS

, 2004

"... 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 ( A3(2), A2(4) ) and ( Bn(q), Cn(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 fi ..."

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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 ( A3(2), A2(4) ) and ( Bn(q), Cn(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(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. 1.

### CLASSIFICATION OF SOLVABLE 3-DIMENSIONAL LIE TRIPLE SYSTEMS

, 2003

"... Abstract. We give the classification of solvable and splitting Lie triple systems and it turn that, up to isomorphism there exist 7 non isomorphic canonical Lie triple systems and, 6 non isomorphic splitting canonical Lie triple systems and find the solvable Lie algebras associated. 1. ..."

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Abstract. We give the classification of solvable and splitting Lie triple systems and it turn that, up to isomorphism there exist 7 non isomorphic canonical Lie triple systems and, 6 non isomorphic splitting canonical Lie triple systems and find the solvable Lie algebras associated. 1.