Hirsch-Plotkin radical of stability groups

TL;DR

This paper investigates the Hirsch-Plotkin radical of stability groups in infinite-dimensional vector spaces, revealing conditions under which it coincides with automorphisms fixing finite sub-series and representing countable Fitting groups.

Contribution

It characterizes the HP radical in various cases and establishes a correspondence with countable Fitting groups, expanding understanding of stability groups in infinite dimensions.

Findings

HP radical equals automorphisms fixing finite sub-series in certain cases

Every countable Fitting group can be embedded into an HP radical of a series stabilizer

The HP radical is a Fitting group under these conditions

Abstract

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.