  We study the Hirsch-Plotkin radical of stability groups of (general) subspace
series of infinite dimensional vector spaces. We show that in countable
dimension and some other cases, the HP-radical of the stability group coincides
with the set of all space automorphisms that fix a finite sub-series; this
implies that the HP radical is a Fitting group. Conversely, we prove that every
countable Fitting group, which is either torsion-free or a p-group may be
represented as a subgroup of the HP radical of a series stabilizer.
