On lattices of closed subgroups in the group of infinite triangular matrices over a field.
The author investigates a special type of closed subgroups of the topological group \(G=\mathrm{UT}(\infty,K)\) of infinite-dimensional unitriangular matrices over a field \(K\) (\(|K|>2\)), considered with the natural inverse limit topology. Namely, she generalizes the concept of partition subgroups introduced by \textit{A. J. Weir} [Proc. Am. Math. Soc. 6, 454-464 (1955; Zbl 0065.01202)] and defines partition subgroups in \(G\). She shows that they are all closed and discusses the problem of their invariance under various group homomorphisms. She proves that a characteristic subgroup of \(G\) is necessarily a partition subgroup and characterizes the lattices of characteristic and fully characteristic subgroups in \(G\). She concludes with some implications of the given characterization on verbal structure of \(G\) and uses some topological properties to discuss the problem of the width of verbal subgroups in groups defined over a finite field \(K\).
groups of infinite upper triangular matrices over fields
lattices of fully characteristic subgroups
scientific article; zbMATH DE number 6493723
scientific article