Solvability of Independent Systems of Equations in Finitely Generated Nilpotent Groups

Mathematical Notes - It is proved that the solvability problem for a finite independent system of equations in a finitely generated nilpotent group can effectively be reduced to a similar problem...     

Levi Classes of Quasivarieties of Groups with Commutator Subgroup of Order p

Algebra and Logic - The Levi class generated by the class ? of groups is the class of all groups in which the normal closure of each element belongs to ?. We describe Levi classes...     

On the Residual Solvability of Generalized Free Products of Finitely Generated Nilpotent Groups

In this article we study the residual solvability of the generalized free product of finitely generated nilpotent groups. We show that these groups are often residually solvable.     

Factorization of Periodic Locally Solvable Groups by Locally Nilpotent and Nilpotent Subgroups

We establish a series of new results concerning periodic locally solvable and finite solvable groups G = AB with locally nilpotent or nilpotent subgroups A and B.     

On Co-Hopfian Nilpotent Groups

Co-Hopfian finitely generated torsion-free nilpotent groupsare characterized in this paper in terms of their Lie algebraautomorphisms. Many examples of such groups are also constructed.2000 Mathematics Subject Classification 17B30, 20F18 (primary).     

Some Classes of Nilpotent Associative Algebras

We further the study of rings with no middle class by focusing on an interpretation of that property in terms of the lattice of hereditary pretorsion classes over a given ring. For non-semisimple rings, the absence of a middle class is equivalent to the requirement that the class of all semisimple right modules be a coatom in that lattice. Taking advantage of this perspective, we discover new facts and shed light on others already known with a possibly more direct interpretation without having to refer to an exhaustive analysis of the structure theorems available in the literature. Our approach also allows us to characterize rings with no middle class in terms of hereditary pretorsion classes containing the class of all singular right modules. We discuss the open problem of whether there is a ring with no right middle class which is not right Noetherian and see, in particular, that an indecomposable ring satisfying that property would have to be Morita equivalent to a certain type of subring of a full linear ring.     

Quasivarieties with Torsion-Free Nilpotent Groups

A condition is specified on which a quasivariety generated by a direct product of two groups with amalgamated central subgroups has an infinite lattice of subquasivarieties. A lattice of quasivarieties is stated finite for a quasivariety which is a union of some covers of Abelian groups in a lattice of quasivarieties of torsion-free solvable groups.     

Groups that are Pairwise Nilpotent

In this article we study groups generated by a set X with the property that every two elements in X generate a nilpotent subgroup.     

Frobenius Groups Generated by Quadratic Elements

An automorphism of a group X is said to be quadratic if there exist integers and such that for any . If is a Frobenius group then an element is said to be quadratic if induces, by conjugation in the core of , a quadratic automorphism. By definition, a group H acts on a group F freely if for and only with or . It is proved that a Frobenius group generated by two quadratic elements is finite and its core is commutative. In particular, any Frobenius group generated by two elements of order at most 4 is finite. Also we argue that a Frobenius group with finitely generated soluble core is finite. The results mentioned are used to show that a group acting freely on an Abelian group is finite if it is generated by elements of order 3, and the order of a product of every two elements of order 3 in is finite.     

On Some Classes of Nilpotent Leibniz Algebras

Siberian Mathematical Journal -     

Minimal Geodesics and Nilpotent Fundamental Groups

Hedlund [18] constructed Riemannian metrics on n-tori, n 3 for which minimal geodesics are very rare. In this paper we construct similar examples for every nilpotent fundamental group. These examples show that Bangert's existence results of minimal geodesics [4] are optimal for nilpotent fundamental groups.     

Idempotent-Conjugate Monoids with Nilpotent Unit Groups

Let M be a finite monoid with unit group G such that J-related idempotents in M are conjugate. If G is nilpotent, we prove that the complex monoid algebra CM of M is semisimple if and only if M is an inverse monoid. Conversely let G be a finite group such that for any finite idempotent-conjugate monoid M with unit group G, CM semisimple implies that M is an inverse monoid. We then show that G is a nilpotent group.     

G-Identities of Nilpotent Groups. II

The structure of a group V _n,red (G) of reduced G-identities is described subject to the condition that G is a nilpotent group of class 3. We prove the criterion for a G-variety G-var(G) to be finitely based for such G.     

G-Identities of Nilpotent Groups. I

The structure of a group V _n,red(G) of reduced G-identities of rank n is treated subject to the condition that G is a nilpotent group of class 1 or 2. The results obtained allow us to settle the question of whether a G-variety G-var(G) generated by a nilpotent group G of class 2 is finitely based. Moreover, we introduce the concepts of a d-commutator subgroup and of a main d-group, associated with G.     

Extremal Curves in Nilpotent Lie Groups

We classify extremal curves in free nilpotent Lie groups. The classification is obtained via an explicit integration of the adjoint equation in Pontryagin maximum principle. It turns out that abnormal extremals are precisely the horizontal curves contained in algebraic varieties of a specific type. We also extend the results to the nonfree case.     

Prime Ideals in Algebras Determined by Submonoids of Nilpotent Groups

The prime spectrum of the semigroup algebra K[S] of a submonoid S of a finitely generated nilpotent group is studied via the spectra of the monoid S and of the group algebra K[G] of the group G of fractions of S. It is shown that the classical Krull dimension of K[S] is equal to the Hirsch length of the group G provided that G is nilpotent of class two. This uses the fact that prime ideals of S are completely prime. An infinite family of prime ideals of a submonoid of a free nilpotent group of class three with two generators which are not completely prime is constructed. They lead to prime ideals of the corresponding algebra. Prime ideals of the monoid of all upper triangular n × n matrices with non-negative integer entries are described and it follows that they are completely prime and finite in number.