**Memoirs of the American Mathematical Society**

2001;
168 pp;
Softcover

MSC: Primary 20;
Secondary 68

Print ISBN: 978-0-8218-2619-5

Product Code: MEMO/149/708

List Price: $66.00

Individual Member Price: $39.60

**Electronic ISBN: 978-1-4704-0299-0
Product Code: MEMO/149/708.E**

List Price: $66.00

Individual Member Price: $39.60

# Black Box Classical Groups

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*William M. Kantor; Ákos Seress*

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

# Table of Contents

## Black Box Classical Groups

- Contents vii8 free
- 1 Introduction 110 free
- 2 Preliminaries 1120 free
- 3 Special linear groups: PSL(d,q) 2029
- 4 Orthogonal groups: PΩ[sup(ε)] (d,q) 5362
- 5 Symplectic groups: PS[sub(p)](2m,q) 92101
- 6 Unitary groups: PSU(d,q) 118127
- 7 Proofs of Theorems 1.1 and 1.1', and of Corollaries 1.2–1.4 140149
- 8 Permutation group algorithms 149158
- 9 Concluding remarks 159168
- References 162171

#### Readership

Graduate students and research mathematicians interested in group theory and generalizations.