Electronic ISBN:  9781470402990 
Product Code:  MEMO/149/708.E 
List Price:  $66.00 
MAA Member Price:  $59.40 
AMS Member Price:  $39.60 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 149; 2001; 168 ppMSC: 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 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.
ReadershipGraduate 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


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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 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.
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