LOCALLY SOLUBLE COFINITARY GROUPS WITH FEW FIXED-POINTS

Let V be an infinite-dimensional vector space over some division ring D. In earlier work ([7], [8], [10]) we proved that if G is a cofinitar y primitive irreducible subgroup of GL(V) with a normal subgroup N of finite index that is hypercyclic and eremitic, then N acts fixed-point freely on V. We speculated that G itself should act fixed-point freel y on V. With a view to eventually resolving this point we study here g roups G with a normal subgroup N of finite index acting fixed-point fr eely on V and with an element g whose fixed-point space is non-zero bu t of finite dimension. Thus the elements of G have some, but few, fixe d points; hence the title of this note. Our investigations revolve aro und three concerns. Firstly, determine when G generates a crossed prod uct in End(D)V over a small normal subgroup, for example over the FC-c entre Delta(G) of G. Secondly, does the potentially larger subgroup N. Delta(G) act fixed-point freely on V? Thirdly, if M is a normal subgr oup of G containing g, determine conditions under which M is completel y reducible. This third investigation also involves the FC-centre of G . Note that 'Clifford's Theorem' does not hold in general in this cont ext, e.g. see [9].