We prove that every verbally closed two-generated subgroup of a free solvable group G of a finite rank is a retract of G.

Universal elements of unitriangular matrices groups The following theorems are proved for a matrix g from the group of unitriangular matrices over a commutative and associative ring K of finite dimension of greater than three with unity: 1) if the matrix g is universal then all of its elements are on the first collateral diagonal except extreme ones are nonzero; 2) if all elements of the first collateral diagonal of the matrix g, with the possible exception of the last element are reversible in K, then g is universal; 3) if the ring K is Euclidean and has no reversible elements except trivial ones, then it follows from the universality of the matrix g that all the elements of its first collateral diagonal, except the extreme ones, are reversible in K.

No abstract

A criterion for the universality of a matrix from the group U T n (R) over a commutative and associative ring R with unity, Sib.Èlektron. Mat. Izv., 2019, Volume 16, 165-174 Use of the all-Russian mathematical portal Math-Net.Ru implies that you have read and agreed to these terms of useAbstract. We find necessary and sufficient universality conditions of a matrix from the unitriangular matrix group of arbitrary finite dimension over a commutative associative ring with unity. An algorithm is used to determine the universality of the element of the unitriangular matrix group over the ring of polynomials with a finite number of variables with integer coefficients.