Memoirs of the American Mathematical Society 2001; 168 pp; softcover Volume: 149 ISBN10: 0821826190 ISBN13: 9780821826195 List Price: US$66 Individual Members: US$39.60 Institutional Members: US$52.80 Order Code: MEMO/149/708
 If a black box simple group is known to be isomorphic to a classical group over a field of known characteristic, a Las Vegas algorithm is used to produce an explicit isomorphism. The proof relies on the geometry of the classical groups rather than on difficult grouptheoretic background. This algorithm has applications to matrix group questions and to nearly linear time algorithms for permutation groups. In particular, we upgrade all known nearly linear time Monte Carlo permutation group algorithms to nearly linear Las Vegas algorithms when the input group has no composition factor isomorphic to an exceptional group of Lie type or a 3dimensional unitary group. Readership Graduate students and research mathematicians interested in group theory and generalizations. Table of Contents  Introduction
 Preliminaries
 Special linear groups: \(\mathrm{PSL}(d,q)\)
 Orthogonal groups: \(\mathrm{P}\Omega^\varepsilon(d,q)\)
 Symplectic groups: \(\mathrm{PSp}(2m,q)\)
 Unitary groups: \(\mathrm{PSU}(d,q)\)
 Proofs of Theorems 1.1 and 1.1', and of corollaries 1.21.4
 Permutation group algorithms
 Concluding remarks
 References
