Black Box Groups

Authors : Şükrü YALÇINKAYA

Pages : 171-210

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Publication Date : 0000-00-00

Article Type : Research Paper

Abstract :We propose a uniform approach for recognizing all black box groups of Lie type which is based on the analysis of the structure of the centralizers of involutions. Our approach can be viewed as a computational version of the classification of the finite simple groups. We present an algorithm which constructs a long root SL2insert ignore into journalissuearticles values(q);-subgroup in a finite simple group of Lie type of odd characteristic, then we use the Aschbacher`s ``Classical Involution Theorem`` as a model in the recognition algorithm and we construct all root SL2insert ignore into journalissuearticles values(q);-subgroups corresponding to the nodes in the extended Dynkin diagram, that is, we construct the extended Curtis - Phan - Tits system of the finite simple groups of Lie type of odd characteristic. In particular, we construct all subsystem subgroups which can be read from the extended Dynkin diagram. We also present an algorithm which determines whether the p-core insert ignore into journalissuearticles values(or ``unipotent radical``); Opinsert ignore into journalissuearticles values(G); of a black box group G is trivial or not, where G/Opinsert ignore into journalissuearticles values(G); is a finite simple classical group of odd characteristic p, answering a well-known question of Babai and Shalev
Keywords : Turk J Math, 31, 2007, , 171 210 Turk J Math, vol 31, iss Suppl