Vol 60, No 3 (2019)

Let π be a finite set of primes. We prove that each soluble group of finite rank contains a finite index subgroup whose every finite homomorphic π-image is nilpotent. A similar assertion is proved for a finitely generated group of finite rank. These statements are obtained as a consequence of the following result of the article: Each soluble pro-π-group of finite rank has an open normal pronilpotent subgroup.

Let G be a finite group, and let σ = {σ_i | i ∈ I} be a partition of the set of all primes ℙ and σ(G) = {σ_i | σ_i ∩ π(G) ≠ ∅}. A set ℋ of subgroups of G is said to be a complete Hall σ-set of G if each nonidentity member of ℋ is a Hall σ_i-subgroup of G and ℋ has exactly one Hall σ_i-subgroup of G for every σ_i ∈ σ (G). A subgroup H of G is said to be σ-permutable in G if G possesses a complete Hall σ-set ℋ such that HA^x = A^xH for all A ∈ ℋ and x ∈ G. A subgroup H of G is said to be σ-n-embedded in G if there exists a normal subgroup T of G such that HT = H^G and H ∩ T ≤ H_σG, where H_σG is the subgroup of H generated by all those subgroups of H that are σ-permutable in G. A subgroup H of G is said to be σ-embedded in G if there exists a σ-permutable subgroup T of G such that HT = H^σG and H ∩ T ≤ H_σG, where H^σG is the intersection of all σ-permutable subgroups of G containing H. We study the structure of finite groups under the condition that some given subgroups of G are σ-embedded and σ-n-embedded. In particular, we give the conditions for a normal subgroup of G to be hypercyclically embedded.

The degree of a vertex or face in a 3-polytope is the number of incident edges. A k-face is one of degree k, a k⁻-face has degree at most k. The height of a face is the maximum degree of its incident vertices; and the height of a 3-polytope, h, is the minimum height of its faces. A face is pyramidal if it is either a 4-face incident with three 3-vertices or a 3-face incident with two vertices of degree at most 4. If pyramidal faces are allowed, then h can be arbitrarily large; and so we assume the absence of pyramidal faces in what follows. In 1940, Lebesgue proved that each quadrangulated 3-polytope has a face f with h(f) ≤ 11. In 1995, this bound was lowered by Avgustinovich and Borodin to 10. Recently, we improved it to the sharp bound 8. For plane triangulation without 4-vertices, Borodin (1992), confirming the Kotzig conjecture of 1979, proved that h ≤ 20, which bound is sharp. Later, Borodin proved that h ≤ 20 for all triangulated 3-polytopes. In 1996, Horňák and Jendrol’ proved for arbitrarily polytopes that h ≤ 23. Recently, we obtained the sharp bounds h ≤ 10 for triangle-free polytopes and h ≤ 20 for arbitrary polytopes. Later, Borodin, Bykov, and Ivanova refined the latter result by proving that any polytope has a 10⁻-face of height at most 20, where 10 and 20 are sharp. Also, we proved that any polytope has a 5⁻-face of height at most 30, where 30 is sharp and improves the upper bound of 39 obtained by Horňák and Jendrol’ (1996). In this paper we prove that every polytope has a 6⁻-face of height at most 22, where 6 and 22 are best possible. Since there is a construction in which every face of degree from 6 to 9 has height 22, we now know everything concerning the maximum heights of restricted-degree faces in 3-polytopes.

We consider the direct problems for poroelasticity equations. In the low-frequency approximation we prove existence and uniqueness theorems for the solution to a certain mixed problem. In the high-frequency approximation we establish the uniqueness of a weak solution to the mixed problem and its continuous dependence on the data in the cases of bounded and unbounded temporal intervals and for however many spatial variables.

We study the problem of the existence of decidable and positive \(\Pi_1^1\)- and \(\Sigma_1^1\)-numberings of the families of \(\Pi_1^1\)- and \(\Sigma_1^1\)-cones with respect to inclusion. Some laws are found that reflect the presence of decidable computable \(\Pi_1^1\)- and \(\Sigma_1^1\)-numberings of these families in dependence on the analytical complexity of the set defining a cone.

A module M is called dual automorphism invariant if whenever X₁ and X₂ are small submodules of M, then each epimorphism f : M/X₁ → M/X₂ lifts to an endomorphism g of M. A module M is said to be d-square free (dual square free) if whenever some factor module of M is isomorphic to N² for a module N then N = 0. We show that each dual automorphism invariant module over a semiperfect ring which is a small epimorphic image of a projective lifting module is a direct sum of cyclic indecomposable d-square free modules. Moreover, we prove that for each module M over a semiperfect ring which is a small epimorphic image of a projective lifting module (e.g., M is a finitely generated module), M is dual automorphism invariant iff M is pseudoprojective. Also, we give the necessary and sufficient conditions for a dual automorphism invariant module over a right perfect ring to be quasiprojective.

We introduce the general structures of Lie-admissible algebras in the spaces of Gâteaux differentiable operators and establish their connection with the symmetries of operator equations and the mechanics of infinite-dimensional systems.

We consider the lattice of subalgebras of a semifield U(X) of positive continuous functions on an arbitrary topological space X and its sublattice of subalgebras with unity. We prove that each isomorphism of the lattices of subalgebras with unity of semifields U(X) and U(Y) is induced by a unique isomorphism of the semifields. The same result holds for lattices of all subalgebras excluding the case of the double-point Tychonoff extension of spaces.