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.

Inhalt

1. Basic Concepts.- 2. Some Examples of Linear Groups.- 3. Soluble Linear Groups.- 4. Finitely Generated Linear Groups.- 5. CZ-Groups and the Zariski Topology.- 6. The Homomorphism Theorems.- 7. The Jordan Decomposition and Splittable Linear Groups.- 8. The Upper Central Series in Linear Groups.- 9. Periodic Linear Groups.- 10. Rank Restrictions, Varietal Properties and Wreath Products.- 11. Supersoluble and Locally Supersoluble Linear Groups.- 12. A Localizing Technique and Applications.- 13. Module Automorphism Groups over Commutative Rings.- 14. Appendix on Algebraic Groups.- Suggestions for Further Reading.