THE WIDTH OF VERBAL SUBGROUPS IN THE GROUP OF UNITRIANGULAR MATRICES OVER A FIELD Agnieszka Bier e-mail: agnieszka.bier@polsl.pl Abstract Let K be a field and let UT n (K) and T n (K) denote the groups of all unitriangular and triangular matrices over field K, respectively. In the paper, the lattices of verbal subgroups of these 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 char- acterization and results for the groups of infinitely dimensional triangular and unitriangular matrices are established in the last part of the paper. 1 INTRODUCTION & STATEMENT OF RESULTS. Let K be a field. We consider the subgroups UT n (K), T n (K) and D n (K) of the general linear group GL n (K), n ∈ N. UT n (K) is a group consisting of all upper triangular matrices of size n × n with unity entries on the main diagonal: UT n (K)= {A ∈ GL n (K) | A = 1 n +  1≤i<j ≤n a i,j e i,j , a i,j ∈ K}, where 1 n denotes the unity matrix of size n × n and e i,j denotes the matrix with unity in the place (i, j ) and zeros elsewhere. UT n (K) is a normal subgroup in the group T n (K) of all invertible upper triangular matrices of size n × n. Moreover, T n (K) can be represented as a semidirect product: T n (K)= D n (K) ⋌ UT n (K), where D n (K) is the group of all invertible diagonal matrices of size n × n. In [9] A. Weir determined the characteristic subgroups in UT n (K) over fields of odd prime characteristic. A more general description of characteristic subgroups of UT n (K) for an arbitrary field K, |K| > 2, was given later by Levchuk in [5] MSC 20E15, 20F14, 20F18, 20G15 1