Constructive recognition of classical groups in their natural representation. (Q1426140)

The author gives a new algorithm to recognize constructively classical groups (SP, SU, \(\Omega^\varepsilon\)) in their natural representation. (See \textit{W. M. Kantor} and \textit{Á. Seress} [Groups, combinatorics and geometry. Proceedings of the L.M.S. Durham symposium, Durham, UK, July 16-26, 2001. River Edge, NJ: World Scientific. 123-137 (2003; Zbl 1052.20001)] for a description of the context.) The algorithm is ``Las Vegas'' in that it uses random elements, but verifies the result to be correct. The heart of the construction is to find subgroups of type \(\Omega^+(4,q)\), respectively \(\text{Sp}(4,q)\) or \(\text{SU}(4,q)\), these then are used to span the whole group. A complete complexity analysis is given that shows the algorithm to perform notably better than the black-box algorithm of \textit{W. M. Kantor} and \textit{Á. Seress} [Mem. Am. Math. Soc. 708, 168 p. (2001; Zbl 1053.20045)] and comparable with a similar algorithm for SL due to \textit{M. Conder} and \textit{C. R. Leedham-Green} [Groups and computation III. Proceedings of the international conference at the Ohio State University, Columbus, OH, USA, June 15-19, 1999. Berlin: Walter de Gruyter. Ohio State Univ. Math. Res. Inst. Publ. 8, 113-121 (2001; Zbl 1012.20043)]. Also concrete runtimes from the author's implementation are given.