ABSTRACT
In Chapter 1 we collect together concepts and results that arise when linear groups are first studied. As usual, the F-algebra of all endomor-phisms from F to A and the F-algebra of suitably sized matrices are isomorphic. Various aspects of G-invariant subspaces such as the G-center, ζ G (A), are discussed and such notions as the invariator of a subspace introduced. These ideas are then extended and the notions of G-hypercentrality and G-nilpotence introduced together with basic properties are obtained. General series of G-invariant subspaces are introduced as is the notion of a Kurosh-Chernikov system. We also illustrate some of the differences from the finite dimensional case. For example, for finitely generated groups G, if A is a G-hypercentral space, then G is residually nilpotent. On the other hand, for each prime p infinite finitely generated residually finite p-groups can be realized as linear groups acting hypercentrally on a vector space of countable dimension over Fp. We also see that UTN(Fp) contains periodic subgroups which are not locally finite. Properties of unipotent and algebraic elements are obtained. We show that certain ideas important in the finite dimensional theory play a less significant role in the study of infinite dimensional linear groups; the situations arising in the theory of infinite dimensional linear groups are more diverse and have a different level of complexity. Approaches that are effective in the study of finite dimensional groups turn out to be far from being so effective in the study of infinite dimensional linear groups.
