Black box groups

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 SL2(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 SL2(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 (or "unipotent radical") O-p(G) of a black box group G is trivial or not, where G/O-p(G) is a finite simple classical group of odd characteristic p, answering a well-known question of Babai and Shalev.