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L. A. Kurdachenko

Publications and source records attributed to L. A. Kurdachenko.

13 recordsLinked to original sources
arXiv2026

Analogues of Gr\"un's lemma and Baer's theorem for skew left braces

We prove in this paper some analogues of the well-known group-theoretical Gr\"un's lemma, stating that in a perfect group the first and the second centre coincide, and Baer's theorem, stating that if the quotient by the nth centre of a group is finite, then so is the $(n + 1)$th term of the lower central series, in the scope of nfinite slew left braces. These results represent significant improvements over previous work. The trifactorised group associated with a skew left brace will be crucial for our proofs.

math.GR↗
arXiv2026

On left braces in which every subbrace is an ideal II

The aim of this paper is to take the study of Dedekind braces, that is, left braces for which every subbrace is an ideal, started in a previous paper, further. Dedekind braces $A$ whose additive group is non-periodic are analysed. We prove sufficient conditions for $A$ to be abelian: it is enough that every element is $2$-nilpotent for the star operation; and, if $A$ is hypermultipermutational, it suffices that the additive group of the socle is torsion-free. Both conditions can be translated in terms of set-theoretical solutions of the Yang-Baxter equation. In addition, we prove a structural theorem for the case of $A$ to be a multipermutational brace of level $2$.

math.GR↗
arXiv2025

Leibniz rings: some basic and structural results

In this paper, we study the fundamental properties of Leibniz rings. Special attention is given to the structure of Leibniz rings whose additive group is "small". The results obtained illustrate a significant difference between the classes of Leibniz rings and Lie rings.

math.RA↗
arXiv2025

On finitely generated left nilpotent braces

A description of finitely generated left nilpotent braces of class at most two is presented in this paper. The description heavily depends on the fact that if $B$ is left nilpotent of class at most $2$, that is $B^3 = 0$, then $B$ is right nilpotent of class at most $3$, that is $B^{(4)} = 0$. In addition, we construct a free object in the category of finitely generated left nilpotent braces of class at most $2$.

math.GR↗
arXiv2025

On left nilpotent skew braces of class 2

The main objective of this article is to initiate a detailed structure theory of left nilpotent skew braces $B$ of class $2$, i.e. skew braces with $B^3 = 0$. We prove that if $B$ is of nilpotent type, then $B$ is centrally nilpotent. In fact, we show that $B$ is right nilpotent of class at most $2+mr$, i.e. $B^{(2+mr+1)} = 0$, where $m$ and $r$ are the nilpotency classes of the additive group of $B$ and $B^2$, respectively. If $B$ is of abelian type, then $B$ is actually right nilpotent of class $3$, i.e. $B^{(4)} = 0$, and this bound is best possible.

math.GR↗
arXiv2024

On left braces in which every subbrace is an ideal

The aim of this paper is to introduce and study the class of all left braces in which every subbrace is an ideal. We call them Dedekind left braces. It is proved that every finite Dedekind left brace is centrally nilpotent. Structural results about Dedekind left braces and a complete description of those ones whose additive group is elementary abelian are also shown. As a consequence, every hypermultipermutational Dedekind left brace whose additive group is elementary abelian is multipermutational of level $2$. A new class of left braces, the extraspecial left braces, is introduced and plays a prominent role in our approach.

math.GR↗
arXiv2023

On the automorphism groups of some nilpotent 3-dimensional Leibniz algebras

Let $L$ be an algebra over a field $F$ with the binary operations $+$ and $[,]$. Then $L$ is called a left Leibniz algebra if it satisfies the left Leibniz identity: $[[a,b],c]=[a,[b,c]]-[b,[a,c]]$ for all elements $a,b,c\in L$. The structure of the automorphism group of $3$-dimensional Leibniz algebras, which have nilpotency class $2$ and a one-dimensional center, is studied.

math.RA↗
arXiv2023

Automorphism groups of some 3-dimensional Leibniz algebras

Let $L$ be an algebra over a field $F$ with the binary operations $+$ and $[,]$. Then $L$ is called a left Leibniz algebra if it satisfies the left Leibniz identity: $[[a,b],c]=[a,[b,c]]-[b,[a,c]]$ for all elements $a,b,c\in L$. A linear transformation $f$ of $L$ is called an endomorphism of $L$, if $f([a,b])=[f(a),f(b)]$ for all elements $a,b\in L$. A bijective endomorphism of $L$ is called an automorphism of $L$. It is easy to show that the set of all automorphisms of the Leibniz algebra is a group with respect to the operation of multiplication of automorphisms. The description of the structure of the automorphism groups of Leibniz algebras is one of the natural and important problems of the general Leibniz algebra theory. The main goal of this article is to describe the structure of the automorphism group of a certain type of nilpotent three-dimensional Leibniz algebras.

math.RA↗
arXiv2021

Modules over some group rings having d-generator property

For modules over group rings we introduce the following numerical parameter. We say that a module A over a ring R has finite r-generator property if each f.g. (finitely generated) R-submodule of A can be generated exactly by r elements and there exists a f.g. R-submodule D of A, which has a minimal generating subset, consisting exactly of r elements. Let FG be the group algebra of a finite group G over a field F. In the present paper modules over the algebra FG having finite generator property are described.

math.AC↗
arXiv2021

Some ranks of modules over group rings

A commutative ring R has finite rank r, if each ideal of R is generated at most by r elements. A commutative ring R has the r-generator property, if each finitely generated ideal of R can be generated by r elements. Such rings are closely related to Prüfer domains. In the present paper we investigate some analogs of these concepts for modules over group rings.

math.AC↗
arXiv2011

On a Generalization of Baer Theorem

R. Baer has proved that if the factor-group G/ζ_{n}(G) of a group G by the member ζ_{n}(G) of its upper central series is finite (here n is a positive integer) then the member γ_{n+1}(G) of the lower central series of G is also finite. In particular, in this case, the nilpotent residual of G is finite. This theorem admits the following simple generalization that has been published very recently by M. de Falco, F. de Giovanni, C. Musella and Ya. P. Sysak: "If the factor-group G/Z of a group G modulo its upper hypercenter Z is finite then G has a finite normal subgroup L such that G/L is hypercentral". In the current article we offer a new simpler very short proof of this theorem and specify it substantially. In fact, we prove that if |G/Z|=t then |L|\leqt^{k}, where k=(1/2)(log_{p}t+1), and p is the least prime divisor of t.

math.GR↗
arXiv2004

Abnormal subgroups and Carter subgroups in some infinite groups

Some properties of abnormal subgroups in generalized soluble groups will be considered. In particular, the transitivity of abnormality in metahypercentral groups is proven. Also it will be proven that a subgroup H of a radical group G is abnormal in G if and only if every intermediate subgroup for H coincides with its normalizer in G. This result will extend on radical groups the well-known criterion of abnormality for finite soluble groups obtained by D. Taunt. For some infinite groups (not only periodic) the existence of Carter subgroups and their conjugations will be also proven.

math.GR↗