NILPOTENT-BY-FINITE COFINITARY GROUPS

Abstract

Throughout this paper, D denotes a division ring (possibly commutative) and V a left vector space over D, usually, but not exclusively, infinite-dimensional. We consider irreducible subgroups G of GL(V) and are particularly interested in such G that contain an element g the fixed-point set C_V(g) of which is non-zero but finite-dimensional (over D). We then use this to derive conclusions about cofinitary groups, an element g of GL(V) being cofinitary if dim_DC_V(g) is finite, and a subgroup of GL(V) being cofinitary if all its non-identity elements are cofinitary.

Suppose that G is a cofinitary subgroup of GL(V). There are two extreme cases. If dim_DV is finite the cofinitary condition is vacuous. At the other extreme, if G acts fixed-point freely on V then the fixed-point sets C_V(g) for g∈G[setmn ]〈1〉 are as small as possible, namely {0}. Work of Blichfeldt and his successors shows that certain irreducible linear groups G of dimension at least 2 over, for example, the complexes are always imprimitive. This is the case if G is nilpotent, or supersoluble, or metabelian. Apart from the two extreme cases, the same is frequently true for irreducible cofinitary subgroups G of GL(V). For example, this is the case if G is finitely generated nilpotent [9, 1.2] or more generally if G is supersoluble [10, 1.1], but not in general if G is metabelian [10, 7.1] or parasoluble (a group G is parasoluble if it has a normal series of finite length such that every subgroup of each of its factors is Abelian and normalised by G) (see [10, 7.2]). Further, it is also the case if G is Abelian-by-finite [10, 3.4], and every supersoluble group is finitely generated and nilpotent-by-finite. Collectively, these results suggest that one should consider nilpotent-by-finite groups.