Abstract
Let K be a field and let UTn(K) and Tn(K) denote the groups of all unitriangular and triangular matrices over field K, respectively. In the paper, the lattices of verbal subgroups of the above groups are characterized. Consequently, the equalities between certain verbal subgroups and their verbal width are determined. The considerations bring a series of verbal subgroups with exactly known finite width equal to 2. An analogous characterization and results for the groups of infinitely dimensional triangular and unitriangular matrices are established in the last part of the paper.
Highlights
INTRODUCTION & STATEMENT OF RESULTSLet K be a field
U Tn(K) is a group consisting of all upper triangular matrices of size n × n with unity entries on the main diagonal: U Tn(K) = {A ∈ GLn(K) | A = 1n +
We introduce two endomorphisms β, δ : U Tn(K) −→ U Tn(K): β(
Summary
The characterization of verbal subgroups in U Tn(K) enables further considerations on verbal structure in the group of triangular matrices Tn(K). Since the group Tn(K) is a semidirect product of the subgroup Dn(K) and the normal subgroup U Tn(K), every verbal subgroup of Tn(K) must be of the form VW (Tn(K)) = VW (Dn(K))H for some subgroup H of U Tn(K), normal in Tn(K) and including the verbal subgroup VW (U Tn(K)). We consider the verbal subgroups generated by various types of words separately
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