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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."
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