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Black Box Classical Groups
 
William M. Kantor University of Oregon, Eugene, OR
Ákos Seress Ohio State University, Columbus, OH
Black Box Classical Groups
eBook ISBN:  978-1-4704-0299-0
Product Code:  MEMO/149/708.E
List Price: $66.00
MAA Member Price: $59.40
AMS Member Price: $39.60
Black Box Classical Groups
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Black Box Classical Groups
William M. Kantor University of Oregon, Eugene, OR
Ákos Seress Ohio State University, Columbus, OH
eBook ISBN:  978-1-4704-0299-0
Product Code:  MEMO/149/708.E
List Price: $66.00
MAA Member Price: $59.40
AMS Member Price: $39.60
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 1492001; 168 pp
    MSC: Primary 20; Secondary 68

    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
     
     
    • Chapters
    • 1. Introduction
    • 2. Preliminaries
    • 3. Special linear groups: PSL($d, q$)
    • 4. Orthogonal groups: $P\Omega ^\epsilon (d, q)$
    • 5. Symplectic groups: $\mathrm {PSp}(2m, q)$
    • 6. Unitary groups: $\mathrm {PSU}(d, q)$
    • 7. Proofs of Theorems 1.1 and 1.1′, and of Corollaries 1.2–1.4
    • 8. Permutation group algorithms
    • 9. Concluding remarks
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 1492001; 168 pp
MSC: Primary 20; Secondary 68

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.

  • Chapters
  • 1. Introduction
  • 2. Preliminaries
  • 3. Special linear groups: PSL($d, q$)
  • 4. Orthogonal groups: $P\Omega ^\epsilon (d, q)$
  • 5. Symplectic groups: $\mathrm {PSp}(2m, q)$
  • 6. Unitary groups: $\mathrm {PSU}(d, q)$
  • 7. Proofs of Theorems 1.1 and 1.1′, and of Corollaries 1.2–1.4
  • 8. Permutation group algorithms
  • 9. Concluding remarks
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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