Maximal unipotent subgroups of finitary linear groups (Q1913973)

Let \(V\) be a vector space over the field \(F\). An element \(g\) of \(\text{GL}(V)\) is finitary if \(\dim_FV(g-1)\) is finite and the subset \(\text{FGL}(V)\) of \(\text{GL}(V)\) of all such elements is a subgroup, called the finitary general linear group over \(V\). The author studies the maximal unipotent subgroups of \(\text{FGL}(V)\), motivated by the fact that if \(\text{char }F=p>0\), then these are exactly the Sylow \(p\)-subgroups of \(\text{FGL}(V)\). In a series of theorems, the author proves the following. Any maximal unipotent subgroup of \(\text{FGL}(V)\) is the stabilizer of some composition series of \(V\). If \(G\) and \(H\) are the stabilizers of the composition series \(\mathcal C\) and \(\mathcal D\) of \(V\) with \(G\) and \(H\) isomorphic, and if \(\text{char }F\neq 2\), then \(\mathcal C\) and \(\mathcal D\) are order isomorphic and \(G\) and \(H\) are conjugate by an \(F\)-semi-linear automorphism of \(V\). Often every isomorphism of \(G\) to \(H\) is induced by some such \(F\)-semi-linear map. The characteristic-2 case presents extra technical difficulties.