On lattices of closed subgroups in the group of infinite triangular matrices over a field

We investigate a special type of closed subgroups of the topological group UT(∞,K) of infinite-dimensional unitriangular matrices over a field K (|K|>2), considered with the natural inverse limit topology. Namely, we generalize the concept of partition subgroups introduced in [23] and define partition subgroups in UT(∞,K). We show that they are all closed and discuss the problem of their invariance to various group homomorphisms. We prove that a characteristic subgroup of UT(∞,K) is necessarily a partition subgroup and characterize the lattices of characteristic and fully characteristic subgroups in UT(∞,K). We conclude with some implications of the given characterization on verbal structure of UT(∞,K) and T(∞,K) and use some topological properties to discuss the problem of the width of verbal subgroups in groups defined over a finite field K.