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Black Box Classical Groups
William M. Kantor, University of Oregon, Eugene, OR, and Ákos Seress, Ohio State University, Columbus, OH
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Memoirs of the American Mathematical Society
2001; 168 pp; softcover
Volume: 149
ISBN-10: 0-8218-2619-0
ISBN-13: 978-0-8218-2619-5
List Price: US$66
Individual Members: US$39.60
Institutional Members: US$52.80
Order Code: MEMO/149/708
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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 group-theoretic 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 3-dimensional 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 1.2-1.4
  • Permutation group algorithms
  • Concluding remarks
  • References
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