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 2ℵ02ℵ0. 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.