Black box classical groups
Author(s)
Bibliographic Information
Black box classical groups
（Memoirs of the American Mathematical Society, no. 708）
American Mathematical Society, 2001
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Note
"January 2001, Volume 149, Number 708 (third of 4 numbers)"
Includes bibliographical references (p. 162-168)
Description and Table of Contents
Description
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.
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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