Infinite Linear Groups

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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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.

• Abelian group
• Finite
• Group theory
• Groups
• Groups of Matrices
• Morphism
• Unendliche lineare Gruppe
• matrix theory

• Queen Mary College, London University, London, England

  Bertram A. F. Wehrfritz

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