A generalization of the first Malcev theorem on nilpotent semigroups and nilpotency of the wreath product of semigroups

We describe all [0-]simple semigroups that are nilpotent in the sense of Malcev. This generalizes the first Malcev theorem on nilpotent (in the sense of Malcev) semigroups. It is proved that if the extended standard wreath product of semigroups is nilpotent in the sense of Malcev and the passive semigroup is not nilpotent, then the active semigroup of the wreath product is a finite nilpotent group. In addition to that, the passive semigroup is uniform periodic. Necessary and sufficient conditions are found under which the extended standard wreath product of semigroups is nilpotent in the sense of Malcev in the case where each of the semigroups of the wreath product generates a variety of finite step.     

An Analog of the Baer-Suzuki Theorem for Infinite Groups

We consider a group without infinite ascending chains of nilpotent subgroups and prove that if every two elements of some conjugacy class generates a nilpotent subgroup then the whole class also generates a nilpotent subgroup.     

Locally nilpotent groups satisfying the weak minimum or maximum condition for subgroups of bounded nilpotency class

We study locally nilpotent groups containing subgroups of classc, c>1, and satisfying the weak maximum condition or the weak minimum condition on c-nilpotent subgroups. It is proved that nilpotent groups of this type are minimax and periodic locally nilpotent groups of this type are Chernikov groups. It is also proved that if a group G is either nilpotent or periodic locally nilpotent and if all of its c-nilpotent subgroups are of finite rank, then G is of finite rank. If G is a non-periodic locally nilpotent group, these results, in general, are not valid.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 3, pp. 384–389, March, 1992.     

On nilpotent subsemigroups of the matrix semigroup over an antiring

In this paper, nilpotent subsemigroups in the matrix semigroup over a commutative antiring are discussed. Some basic properties and characterizations for the nilpotent subsemigroups are given, and some equivalent conditions for the matrix semigroup over a commutative antiring to have a maximal nilpotent subsemigroup are obtained. Also, the maximal nilpotent subsemigroups in the matrix semigroup are described.     

Essentially nilpotent rings

In this note we introduce a class of nil rings (called essentially nilpotent) which properly contains the class of nilpotent rings. A nil ring is said to be essentially nilpotent if it contains an essential right ideal which is nilpotent. Various properties of essentially nilpotent rings are investigated. A nil ring is essentially nilpotent if and only if it contains an essential right ideal which is leftT-nilpotent.     

两类幂零的n-Lie代数

本文提出并构造了两类幂零的n-Lie代数:特征幂零的n-Lie代数与最大秩的幂零的n-Lie代数.证明了n-Lie代数是特征幂零的n-Lie代数的充分必要条件,以及最大秩的幂零的n-Lie代数的结构特征.     

Fuzzy nilpotent algebras

We define the concept of fuzzy nilpotent algebra, prove that the homomorphic inverse image of a fuzzy nilpotent algebra is also nilpotent and study the intersection and union of fuzzy nilpotent algebras.     

关于有限群两个幂零子群积的问题

众所周知,有限群的两个幂零子群的积不一定是幂零的．本文研究了Ｅｎｇｅｌ条件对两个幂零子群的影响,得到两个幂零子群的积为幂零群的几个充分条件。     

Infinite periodic words and almost nilpotent varieties

An almost nilpotent variety of linear growth is constructed in the paper for any infinite periodic word in an alphabet of two letters. A discrete series of different almost nilpotent varieties is also constructed. Only a few almost nilpotent varieties were studied previously and their existence was proved often under some additional assumptions. The existence of almost nilpotent varieties of arbitrary integer exponential growth with a fractional exponent is proved as well as the existence of a continual family of almost nilpotent varieties with not more than quadratic growth.     

The algebraic and geometric classification of nilpotent terminal algebras

We give algebraic and geometric classifications of 4-dimensional complex nilpotent terminal algebras. Specifically, we find that, up to isomorphism, there are 41 one-parameter families of 4-dimensional nilpotent terminal (non-Leibniz) algebras, 18 two-parameter families of 4-dimensional nilpotent terminal (non-Leibniz) algebras, 2 three-parameter families of 4-dimensional nilpotent terminal (non-Leibniz) algebras, complemented by 21 additional isomorphism classes (see Theorem 13). The corresponding geometric variety has dimension 17 and decomposes into 3 irreducible components determined by the Zariski closures of a one-parameter family of algebras, a two-parameter family of algebras and a three-parameter family of algebras (see Theorem 15). In particular, there are no rigid 4-dimensional complex nilpotent terminal algebras.     

The commutator width of some relatively free lie algebras and nilpotent groups

We determine the exact values of the commutator width of absolutely free and free solvable Lie rings of finite rank, as well as free and free solvable Lie algebras of finite rank over an arbitrary field. We calculate the values of the commutator width of free nilpotent and free metabelian nilpotent Lie algebras of rank 2 or of nilpotency class 2 over an arbitrary field. We also find the values of the commutator width for free nilpotent and free metabelian nilpotent Lie algebras of finite rank at least 3 over an arbitrary field in the case that the nilpotency class exceeds the rank at least by 2. In the case of free nilpotent and free metabelian nilpotent Lie rings of arbitrary finite rank, as well as free nilpotent and free metabelian nilpotent Lie algebras of arbitrary finite rank over the field of rationals, we calculate the values of commutator width without any restrictions. It follows in particular that the free or nonabelian free solvable Lie rings of distinct finite ranks, as well as the free or nonabelian free solvable Lie algebras of distinct finite ranks over an arbitrary field are not elementarily equivalent to each other. We also calculate the exact values of the commutator width of free ?-power nilpotent, free nilpotent, free metabelian, and free metabelian nilpotent groups of finite rank.     

On Residual Properties of Pure Braid Groups of Closed Surfaces

We prove that pure braid groups of closed surfaces are almost-direct products of residually torsion free nilpotent groups and hence residually torsion free nilpotent. As a corollary, we prove also that braid groups on 2 strands of closed surfaces are residually nilpotent.     

Determination of 7-Dimensional Indecomposable Nilpotent Complex Lie Algebras by Adjoining a Derivation to 6-Dimensional Lie Algebras

For any complex 6-dimensional nilpotent Lie algebra \mathfrakg,\mathfrak{g}, we compute the strain of all indecomposable 7-dimensional nilpotent Lie algebras which contain \mathfrakg\mathfrak{g} by the adjoining a derivation method. We get a new determination of all 7-dimensional complex nilpotent Lie algebras, allowing to check earlier results (some contain errors), along with a cross table intertwining nilpotent 6- and 7-dimensional Lie algebras.     

由环的幂零性所确定的根

结合环中的环的幂零性不是根性质。为此，本文将结合环中的幂零理想概念扩展为次拟幂零理想和拟幂零理想，定义次拟幂零根ＳＮ和拟幂零根ＱＮ，证明它们均为Ａｍｉｔｓｕｒ－Ｋｕｒｏｓｈ根，且二者相等，进一步，我们给出了ＱＮ－半单环的构造命题和ＱＮ－根的模刻划。     

Dominions in finitely generated nilpotent groups

In the first part, we prove that the dominion (in the sense of Isbell) of a subgroup of a finitely generated nilpotent group is trivial in the category of all nilpotent groups. In the second part, we show that the dominion of a subgroup of a finitely generated nilpotent group of class two is trivial in the category of all metabelian nilpotent groups.     

An uncountable family of almost nilpotent varieties of polynomial growth

A non-nilpotent variety of algebras is almost nilpotent if any proper subvariety is nilpotent. Let the base field be of characteristic zero. It has been shown that for associative or Lie algebras only one such variety exists. Here we present infinite families of such varieties. More precisely we shall prove the existence of1) a countable family of almost nilpotent varieties of at most linear growth and2) an uncountable family of almost nilpotent varieties of at most quadratic growth.     

Powerfully nilpotent groups of maximal powerful class

In this paper we continue the study of powerfully nilpotent groups started in Traustason and Williams (J Algebra 522:80–100, 2019). These are powerful p-groups possessing a central series of a special kind. To each such group one can attach a powerful class that leads naturally to the notion of a powerful coclass and classification in terms of an ancestry tree. The focus here is on powerfully nilpotent groups of maximal powerful class but these can be seen as the analogs of groups of maximal class in the class of all finite p-groups. We show that for any given positive integer r and prime $$p>r$$, there exists a powerfully nilpotent group of maximal powerful class and we analyse the structure of these groups. The construction uses the Lazard correspondence and thus we construct first a powerfully nilpotent Lie ring of maximal powerful class and then lift this to a corresponding group of maximal powerful class. We also develop the theory of powerfully nilpotent Lie rings that is analogous to the theory of powerfully nilpotent groups.     

The Problem of the Classification of the Nilpotent Class 2 Torsion Free Groups up to Geometric Equivalence

In this article, we consider the problem of classification of the nilpotent class 2 finitely generated torsion free groups up to geometric equivalence. By a very easy technique it is proved that this problem is equivalent to the problem of classification of the complete in the Maltsev sense nilpotent torsion free finite rank groups up to isomorphism. This result leads to better understanding of the complexity of the problem of the classification of the quasi-varieties of the nilpotent class 2 groups. It is well known that the variety of the nilpotent class s groups is Noetherian for every s ∈ ?. So the problem of the classification of the quasi-varieties generated even by a single nilpotent class 2 finitely generated torsion free group is equivalent to the problem of classification of the complete in the Maltsev sense nilpotent torsion free finite rank groups up to isomorphism.     

The classification of two step nilpotent complex Lie algebras of dimension 8

A Lie algebra g is called two step nilpotent if g is not abelian and [g, g] lies in the center of g. Two step nilpotent Lie algebras are useful in the study of some geometric problems, such as commutative Riemannian manifolds, weakly symmetric Riemannian manifolds, homogeneous Einstein manifolds, etc. Moreover, the classification of two-step nilpotent Lie algebras has been an important problem in Lie theory. In this paper, we study two step nilpotent indecomposable Lie algebras of dimension 8 over the field of complex numbers. Based on the study of minimal systems of generators, we choose an appropriate basis and give a complete classification of two step nilpotent Lie algebras of dimension 8.     

Nilpotent Blocks of Quasisimple Groups for the Prime Two

We examine nilpotency amongst blocks of positive defect of the quasisimple groups for the prime 2. We show that every nilpotent block of a quasisimple group has abelian defect groups, and prove a conjecture of Puig concerning the recognition of nilpotent blocks in the case of quasisimple groups. Explicit characterisations of nilpotent blocks are given for the classical, alternating and sporadic simple groups.