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
CV(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
dimDCV(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 dimDV is finite the cofinitary
condition is vacuous. At the other extreme, if G acts
fixed-point freely on V then the fixed-point sets
CV(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.