Skip to main content
SpringerLink
Log in
Book cover

Generators and Relations in Groups and Geometries

  • Book
  • © 1991

Overview

Editors:
  1. A. Barlotti

    1. Dipartimento di Matematica “Ulisse Dini”, Università degli Studi di Firenze, Florence, Italy

    View editor publications

    You can also search for this editor in PubMed   Google Scholar

  2. E. W. Ellers

    1. Department of Mathematics, University of Toronto, Toronto, Canada

    View editor publications

    You can also search for this editor in PubMed   Google Scholar

  3. P. Plaumann

    1. Mathematisches Institut, Universität Erlangen-Nürnberg, Erlangen, Germany

    View editor publications

    You can also search for this editor in PubMed   Google Scholar

  4. K. Strambach

    1. Mathematisches Institut, Universität Erlangen-Nürnberg, Erlangen, Germany

    View editor publications

    You can also search for this editor in PubMed   Google Scholar

Part of the book series: Nato Science Series C: (ASIC, volume 333)

This is a preview of subscription content, log in via an institution to check access.

Access this book

Log in via an institution
eBook USD 169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Other ways to access

Licence this eBook for your library

Institutional subscriptions

Table of contents (14 chapters)

  1. Front Matter

    Pages i-xv
    Download chapter PDF

  2. Optimal factorization of matrices, length problems

    1. Classical Groups

      • Erich W. Ellers
      Pages 1-45

    2. Generators of Automorphism Groups of Modules

      • H. Ishibashi
      Pages 47-67

    3. Generators of Automorphism Groups of Cayley Algebras

      • Huberta Lausch
      Pages 69-94

    4. Products of Matrices

      • Thomas J. Laffey
      Pages 95-123

  3. Reflection geometry

    1. Reflection groups

      • Frieder Knüppel
      Pages 125-164

    2. Unitary geometry

      • Martin Götzky
      Pages 165-177

    3. Lie and Algebraic Johnsen Groups

      • Peter Plaumann
      Pages 179-193

  4. Nice generators and relations, applications

    1. 2-Generation of finite simple groups and some related topics

      • L. Di Martino, M. C. Tamburini
      Pages 195-233

    2. Coxeter Groups and three Related Topics

      • Arjeh M. Cohen
      Pages 235-278

    3. Geometric structure of conjugacy classes in algebraic groups

      • T. A. Springer
      Pages 279-290

    4. Groups with Polynomial Growth and Differential Geometry

      • E. Musso, F. Tricerri
      Pages 291-320

    5. Analyticity and Growth of Pro-p-Groups

      • A. Caranti
      Pages 321-341

    6. Intersections of Local Algebraic Extensions of a Hilbertian Field

      • Moshe Jarden
      Pages 343-405

    7. Generators and Relations for Discontinuous Groups

      • Bruno Zimmermann
      Pages 407-436

  5. Back Matter

    Pages 437-447
    Download chapter PDF
Back to top

Keywords

About this book

Every group is represented in many ways as an epimorphic image of a free group. It seems therefore futile to search for methods involving generators and relations which can be used to detect the structure of a group. Nevertheless, results in the indicated direction exist. The clue is to ask the right question. Classical geometry is a typical example in which the factorization of a motion into reflections or, more generally, of a collineation into central collineations, supplies valuable information on the geometric and algebraic structure. This mode of investigation has gained momentum since the end of last century. The tradition of geometric-algebraic interplay brought forward two branches of research which are documented in Parts I and II of these Proceedings. Part II deals with the theory of reflection geometry which culminated in Bachmann's work where the geometric information is encoded in properties of the group of motions expressed by relations in the generating involutions. This approach is the backbone of the classification of motion groups for the classical unitary and orthogonal planes. The axioms in this char­ acterization are natural and plausible. They provoke the study of consequences of subsets of axioms which also yield natural geometries whose exploration is rewarding. Bachmann's central axiom is the three reflection theorem, showing that the number of reflections needed to express a motion is of great importance.

Editors and Affiliations

  • Dipartimento di Matematica “Ulisse Dini”, Università degli Studi di Firenze, Florence, Italy

    A. Barlotti

  • Department of Mathematics, University of Toronto, Toronto, Canada

    E. W. Ellers

  • Mathematisches Institut, Universität Erlangen-Nürnberg, Erlangen, Germany

    P. Plaumann, K. Strambach

Bibliographic Information

Publish with us

Policies and ethics

Back to top

Search

Navigation