ABSTRACT
In Chapter 5 we discuss groups that are close to finite dimensional. If a group G has finite central dimension, then G acts like a finite dimensional linear group. If G has finite central dimension, A has a large subspace fixed by G and the case when all proper subspaces have finite central dimension is considered. More generally, G satisfies min-icd, if the family of subgroups, ordered by inclusion, satisfies the minimal condition and in similar fashion G satisfies max-icd if the family of subgroups ordered by inclusion satisfies the maximal condition. In this chapter groups satisfying min-icd or max-icd are studied in detail. It is shown for example that if G is a subgroup of GL(F, A) with infinite central dimension and satisfies the condition min-icd, then G either has the minimal condition on all subgroups or G is finitary. For locally generalized radical such groups G it turns out that G is soluble-by-finite. The condition max-icd is not quite so amenable, but still it is shown, for example, that infinite dimensional groups with max-icd are finitely generated or finitary. A crucial role is played by the subgroup Fin(G), consisting of all g G for which g is finitary. For example, when G is locally generalized radical and has max-icd, then G/ Fin(G) is polycyclic-by-finite. The chapter ends with a discussion of antifinitary groups, those groups G of infinite central dimension, but every proper subgroup of G which is not finitely generated has finite central dimension, so opposite to being finitary.
