Vol 58, No 3 (2017)

We find necessary and sufficient conditions for the coincidence of the universal theories of partially commutative groups of metabelian varieties defined by acyclic graphs.

Let G be a finite group. It is proved that if, for every prime p, the number of nonidentity p-elements of G is divisible by the p′-part of |G|, then all element orders of G are prime powers.

We study the functions on the punctured n-dimensional sphere having zero integrals over all admissible “hemispheres.” We find a condition for the point to be a removable set for this class of functions and show that the condition cannot be dropped or substantially improved.

We prove embedding theorems for the multianisotropic Sobolev spaces generated by the completely regular Newton polyhedron. Under study is the case of the polyhedron with one anisotropic vertex. We obtain a special integral representation of functions in terms of the tuple of multi-indices of the Newton polyhedron.

A subgroup H of a group G is pronormal if the subgroups H and H^g are conjugate in 〈H,H^g〉 for every g ∈ G. It was conjectured in [1] that a subgroup of a finite simple group having odd index is always pronormal. Recently the authors [2] verified this conjecture for all finite simple groups other than PSL_n(q), PSU_n(q), E₆(q), ²E₆(q), where in all cases q is odd and n is not a power of 2, and P Sp_2n(q), where q ≡ ±3 (mod 8). However in [3] the authors proved that when q ≡ ±3 (mod 8) and n ≡ 0 (mod 3), the simple symplectic group P Sp²ⁿ(q) has a nonpronormal subgroup of odd index, thereby refuted the conjecture on pronormality of subgroups of odd index in finite simple groups.

The natural extension of this conjecture is the problem of classifying finite nonabelian simple groups in which every subgroup of odd index is pronormal. In this paper we continue to study this problem for the simple symplectic groups P Sp_2n(q) with q ≡ ±3 (mod 8) (if the last condition is not satisfied, then subgroups of odd index are pronormal). We prove that whenever n is not of the form 2^m or 2^m(2^2k+1), this group has a nonpronormal subgroup of odd index. If n = 2^m, then we show that all subgroups of P Sp_2n(q) of odd index are pronormal. The question of pronormality of subgroups of odd index in P Sp_2n(q) is still open when n = 2^m(2^2k + 1) and q ≡ ±3 (mod 8).

We obtain a criterion for the convergence of the Mellin–Barnes integral representing the solution to a general system of algebraic equations. This yields a criterion for a nonnegative matrix to have positive principal minors. The proof rests on the Nilsson–Passare–Tsikh Theorem about the convergence domain of the general Mellin–Barnes integral, as well as some theorem of a linear algebra on a subdivision of the real space into polyhedral cones.

Considering a process with independent increments under the moment Cramér condition, we establish the extended large deviation principle in the space of functions without discontinuities of the second kind which is endowed with the Borovkov metric.

We study algebraically and verbally closed subgroups and retracts of finitely generated nilpotent groups. A special attention is paid to free nilpotent groups and the groups UT_n(Z) of unitriangular (n×n)-matrices over the ring Z of integers for arbitrary n. We observe that the sets of retracts of finitely generated nilpotent groups coincides with the sets of their algebraically closed subgroups. We give an example showing that a verbally closed subgroup in a finitely generated nilpotent group may fail to be a retract (in the case under consideration, equivalently, fail to be an algebraically closed subgroup). Another example shows that the intersection of retracts (algebraically closed subgroups) in a free nilpotent group may fail to be a retract (an algebraically closed subgroup) in this group. We establish necessary conditions fulfilled on retracts of arbitrary finitely generated nilpotent groups. We obtain sufficient conditions for the property of being a retract in a finitely generated nilpotent group. An algorithm is presented determining the property of being a retract for a subgroup in free nilpotent group of finite rank (a solution of a problem of Myasnikov). We also obtain a general result on existentially closed subgroups in finitely generated torsion-free nilpotent with cyclic center, which in particular implies that for each n the group UT_n(Z) has no proper existentially closed subgroups.

We establish that the set of minimal generalized computable enumerations of every infinite family computable with respect to a high oracle is effectively infinite. We find some sufficient condition for enumerations of the infinite families computable with respect to high oracles under which there exist minimal generalized computable enumerations that are irreducible to the enumerations.