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
automorphism 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.