We obtain the generalizations of the well-known Magnus theorem on isomorphism of groups with one defining relation and Greendlinger‘s theorem on isomorphism of groups with condition C ′(1/6) on the groups C(6); C(4) and T(4), C(3) and T(6).

The necessary and sufficient conditions of virtual root-class residuality for some generalized free products and HNN-extensions are obtained.

An asymptotic formula for the number of squarefree integers of the form [αn] is proved in the paper, where α is an algebraic number or a number with restricted partial quotients.

Establish the solubility in any algebraically closed group G of each equation of the form w(x1, . . . , xn) = g, where w(x1, . . . , xn) – nonempty irreducible group word unknown x1, ...., xn, and g – arbitrary element of group G.

Establish that the problem of reducibility for automorphic stencils free group F2 of rank two is a matter of solvability of equations over this group.

In this paper we consider hypergeometric functions with irrational parameters and their derivatives (including with respect to parameter). By means of specially chosen degree of zero polynomial more precise low estimates of the moduli of corresponding linear forms are obtained.

The different definitions of an artinaness are considered in the article.

It is proved that all finitely generated subgroups of generalized free product of two groups are finitely separable provided that free factors have this property and amalgamated subgroups are normal in corresponding factors and satisfy the maximum condition for subgroups.

Improved the estimation of the parameter in the group generate by three 3 × 3 matrices with elements depending on this parameter if free.

The conditional result concerning to Jacobian conjecture is obtained in the article.  

We described structure of an centralizer of elements in Artin groups with arboreal structure.

We generalize the result of A. L. Shmelkin which establishes the fact that polycyclic groups are virtually residually p–finite for any prime p.

Let K be an arbitrary root class of groups. We prove a sufficient condition of the K-residuality of the generalized free product of two K-groups with a normal amalgamated subgroup.

A complete description of the congruences of acts over right zero semigroup is obtained.