Volume 58, Nº 5 (2017)

Under study are the dual automorphism-invariant modules and pseudoprojective modules. Some conditions were found under which the dual automorphism-invariant module over a perfect ring is quasiprojective. We also show that if R is a right perfect ring then a pseudoprojective right R-module M is finitely generated if and only if M is a Hopf module.

The authors have previously constructed two representations of the virtual braid group into the automorphism group of the free product of a free group and a free abelian group. Using them, we construct the two groups, each of which is a virtual link invariant. By the example of the virtual trefoil knot we show that the constructed groups are not isomorphic, and establish a connection between these groups as well as their connection with the group of the virtual trefoil knot which was defined by Carter, Silver, and Williams.

Given a finite-dimensional associative commutative algebra A over a field F, we define the structure of a Lie algebra using a nonzero derivation D of A. If A is a field and charF > 3; then the corresponding algebra is simple, presenting a nonisomorphic analog of the Zassenhaus algebra W₁(m).

This article is a part of our effort to explain the foundations of algebraic geometry over arbitrary algebraic structures [1–8]. We introduce the concept of universal geometrical equivalence of two algebraic structures A and B of a common language L which strengthens the available concept of geometrical equivalence and expresses the maximal affinity between A and B from the viewpoint of their algebraic geometries. We establish a connection between universal geometrical equivalence and universal equivalence in the sense of equality of universal theories.

We prove area formulas for classes of the mappings that are Hölder continuous in the sub-Riemannian sense and defined on nilpotent graded groups. Moreover, in one of the model cases, we establish an area formula for calculating the initial measure and a measure close to it.

We consider a general system of functional equations of the second kind in L₂ with a continuous linear operator T satisfying the condition that zero lies in the limit spectrum of the adjoint operator T*. We show that this condition holds for the operators of a wide class containing, in particular, all integral operators. The system under study is reduced by means of a unitary transformation to an equivalent system of linear integral equations of the second kind in L₂ with Carleman matrix kernel of a special kind. By a linear continuous invertible change, this system is reduced to an equivalent integral equation of the second kind in L₂ with quasidegenerate Carleman kernel. It is possible to apply various approximate methods of solution for such an equation.

We establish the existence and uniqueness of a weak solution to an initial boundary value problem for the system of the motion equations of a fluid that is a fractional analog of the Voigt viscoelasticity model. The rheological equation of the model contains fractional derivatives.

We prove Jordan’s Theorem for infinite sharply 2-transitive groups satisfying the finiteness (a, b)-condition, with |a| · |b| even.

A normal subgroup N of a finite group G is called n-decomposable in G if N is the union of n distinct G-conjugacy classes. We study the structure of nonperfect groups in which every proper nontrivial normal subgroup is m-decomposable, m+1-decomposable, or m+2-decomposable for some positive integer m. Furthermore, we give classification for the soluble case.

Let G be a finite group and let σ = {σ_i|i ∈ I} be some partition of the set ℙ of all primes. Then G is called σ-nilpotent if G = A₁ × ⋯ × A_r, where A_i is a \({\sigma _{{i_j}}}\)-group for some i_j = i_j(A_i). A collection ℋ of subgroups of G is a complete Hall σ-set of G if each member ≠ 1 of ℋ is a Hall σ_i-subgroup of G for some i ∈ I and ℋ has exactly one Hall σ_i-subgroup of G for every i such that σ_i ∩ π(G) ≠ ø. A subgroup A of G is called σ-quasinormal or σ-permutable [1] in G if G possesses a complete Hall σ-set ℋ such that AH^x = H^xA for all H ∈ ℋ and x ∈ G. The symbol r(G) (r_p(G)) denotes the rank (p-rank) of G.

Assume that ℋ is a complete Hall σ-set of G. We prove that (i) if G is soluble, r(H) ≤ r ∈ ℕ for all H ∈ ℋ, and every n-maximal subgroup of G (n > 1) is σ-quasinormal in G, then r(G) ≤ n+r − 2; (ii) if every member in ℋ is soluble and every n-minimal subgroup of G is σ-quasinormal, then G is soluble and r_p(G) ≤ n + r_p(H) − 1 for all H ∈ ℋ and odd p ∈ π(H).