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Infinite Linear Groups

An Account of the Group-theoretic Properties of Infinite Groups of Matrices

  • Book
  • © 1973

Overview

Authors:
  1. Bertram A. F. Wehrfritz

    1. Queen Mary College, London University, London, England

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Part of the book series: Ergebnisse der Mathematik und ihrer Grenzgebiete. 2. Folge (MATHE2, volume 76)

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Table of contents (14 chapters)

  1. Front Matter

    Pages I-XIV
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  2. Basic Concepts

    • Bertram A. F. Wehrfritz
    Pages 1-16

  3. Some Examples of Linear Groups

    • Bertram A. F. Wehrfritz
    Pages 17-40

  4. Soluble Linear Groups

    • Bertram A. F. Wehrfritz
    Pages 41-49

  5. Finitely Generated Linear Groups

    • Bertram A. F. Wehrfritz
    Pages 50-71

  6. CZ-Groups and the Zariski Topology

    • Bertram A. F. Wehrfritz
    Pages 72-81

  7. The Homomorphism Theorems

    • Bertram A. F. Wehrfritz
    Pages 82-89

  8. The Jordan Decomposition and Splittable Linear Groups

    • Bertram A. F. Wehrfritz
    Pages 90-100

  9. The Upper Central Series in Linear Groups

    • Bertram A. F. Wehrfritz
    Pages 101-111

  10. Periodic Linear Groups

    • Bertram A. F. Wehrfritz
    Pages 112-133

  11. Rank Restrictions, Varietal Properties and Wreath Products

    • Bertram A. F. Wehrfritz
    Pages 134-154

  12. Supersoluble and Locally Supersoluble Linear Groups

    • Bertram A. F. Wehrfritz
    Pages 155-173

  13. A Localizing Technique and Applications

    • Bertram A. F. Wehrfritz
    Pages 174-185

  14. Module Automorphism Groups over Commutative Rings

    • Bertram A. F. Wehrfritz
    Pages 186-201

  15. Appendix on Algebraic Groups

    • Bertram A. F. Wehrfritz
    Pages 202-218

  16. Back Matter

    Pages 219-232
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Keywords

About this book

By a linear group we mean essentially a group of invertible matrices with entries in some commutative field. A phenomenon of the last twenty years or so has been the increasing use of properties of infinite linear groups in the theory of (abstract) groups, although the story of infinite linear groups as such goes back to the early years of this century with the work of Burnside and Schur particularly. Infinite linear groups arise in group theory in a number of contexts. One of the most common is via the automorphism groups of certain types of abelian groups, such as free abelian groups of finite rank, torsion-free abelian groups of finite rank and divisible abelian p-groups of finite rank. Following pioneering work of Mal'cev many authors have studied soluble groups satisfying various rank restrictions and their automor­ phism groups in this way, and properties of infinite linear groups now play the central role in the theory of these groups. It has recently been realized that the automorphism groups of certain finitely generated soluble (in particular finitely generated metabelian) groups contain significant factors isomorphic to groups of automorphisms of finitely generated modules over certain commutative Noetherian rings. The results of our Chapter 13, which studies such groups of automorphisms, can be used to give much information here.

Authors and Affiliations

  • Queen Mary College, London University, London, England

    Bertram A. F. Wehrfritz

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