On the Hirsch-Plotkin radical of stability groups

We study the stability group of subspace series of infinite dimensional vector spaces. In [1], the authors proved that when the vector space has countable dimension then the Hirsch-Plotkin radical of the stability group coincides with the set of all space automorphisms that fix a finite subseries and they conjectured that this would hold in all dimensions. We give a counter example of dimension 2ℵ0. We however show that in general the result remains true if the Hirsch-Plotkin radical is replaced by the Fitting group, the product of all the normal nilpotent subgroups of the stability group. We also show that the Hirsch-Plotkin radical has a certain strong local nilpotence property.