On the Structure of the Units of Group Algebra of Dihedral Group

In this paper, we completely determine the structure of the unit group of the group algebra of some dihedral groups D2 n over the finite field Fpk, where p is a prime.     

The Engel Length of the Product of Engel Elements

Considering the relation between the Engel lengths of elements and their product, we give a counterexample to Question 11.88 in the “Kourovka Notebook” [1].     

On Complemented Subgroups of Finite Groups

51. IntroductionIt is quite clear that the ekistence of complements for some families of subgroups of agroup gives a lot ofinfor~ion about its structure. FOr instance, Hall[6] proved that a groupG is supersoluble with elementary abelian Sylow subgroups if and only if G is complemellted,that is, every subgroup of G is comPlemeded in G. The same anchor also proved that agroup is soluble if and only if every Sylow subgroup is complemellted (see [3;I,3.5]). Morerecelltly, Arad and Wardll] pro…     

Spectra of finite linear and unitary groups

The spectrum of a finite group is the set of its element orders. An arithmetic criterion determining whether a given natural number belongs to a spectrum of a given group is furnished for all finite special, projective general, and projective special linear and unitary groups. Supported by RFBR (grant Nos. 08-01-00322 and 06-01-39001) and by the Council for Grants (under RF President) and State Aid of Leading Scientific Schools (project NSh-344.2008.1). __________ Translated from Algebra i Logika, Vol. 47, No. 2, pp. 157–173, March–April, 2008.     

Classes of abelian groups related to sequential convergence

We study classes of abelian groups related to sequential com¬pactness and its generalizations (completeness, coarseness and sequential pre-compactness) in convergence groups. In particular, we describe the algebraic structure of the abelian groups on which every coarse convergence is complete and we prove that: i) every abelian group admits a sequentially precompact convergence; ii) every algebraically compact abelian group admits a sequen¬tially compact convergence.     

Notions of Discrimination

As an outgrowth of the study of algebraic geometry over groups, discriminating groups were introduced. Many important universal type groups such as Higman's universal group and Thompson's group F were shown to be discriminating. Squarelike groups were then introduced to better capture axiomatic properties of discrimination. In the present article squarelike groups are reinterpreted in terms of discrimination of quasi varieties, and the relationship with an older version of discrimination, termed varietal discrimination here, is studied.     

双曲群,自动机群与群的Dehn函数

本文第一部分对组台群论的发展史作了一个简要的回顾．然后重点介绍近20年来组合群论领域中的三个热点课题：即双曲群，自动机群和群的Dehn函数的有关概念，并对相关的重要研究成果给予概述，其中包括作者本人在群的二阶Dehn函数的研究工作中的若干成果．最后提出9个公开问题。     

几类非Abel群之增广商群的结构

记ZG为有限群G的整群环,△n(G)为增广理想△(G)的n次幂,Qn(G)=△"(G)/△n 1(G)为G的增广商群.本文考虑了二面体群D2tk(k 奇)和m次对称群Sm,证明了Qn(D2tk)为秩不超过2t 1的基本2-群以及Qn(Sm)≌Z2.     

Lie group structures on symmetry groups of principal bundles

In this paper we describe how one can obtain Lie group structures on the group of (vertical) bundle automorphisms for a locally convex principal bundle P over the compact manifold M. This is done by first considering Lie group structures on the group of vertical bundle automorphisms Gau(P). Then the full automorphism group Aut(P) is considered as an extension of the open subgroup DiffP(M) of diffeomorphisms of M preserving the equivalence class of P under pull-backs, by the gauge group Gau(P). We derive explicit conditions for the extensions of these Lie group structures, show the smoothness of some natural actions and relate our results to affine Kac-Moody algebras and groups.     

Quasi-isometrically embedded subgroups of braid and diffeomorphism groups

We show that a large class of right-angled Artin groups (in particular, those with planar complementary defining graph) can be embedded quasi-isometrically in pure braid groups and in the group of area preserving diffeomorphisms of the disk fixing the boundary (with respect to the -norm metric); this extends results of Benaim and Gambaudo who gave quasi-isometric embeddings of and for all . As a consequence we are also able to embed a variety of Gromov hyperbolic groups quasi-isometrically in pure braid groups and in the group . Examples include hyperbolic surface groups, some HNN-extensions of these along cyclic subgroups and the fundamental group of a certain closed hyperbolic 3-manifold.

     

Conjugacy Separability of Descending HNN-Extensions of Finitely Generated Abelian Groups

It is proved that an arbitrary descending HNN-extension of a finitely generated Abelian group is conjugacy separable.     

On mixed Abelian groups with periodic groups of automorphisms

In the paper, sufficient conditions for the splittability of mixed Abelian groups with periodic automorphism groups are established. Classes of mixed splittable Abelian groups with perfect holomorphs are distinguished. Translated fromMaternaticheskie Zametki, Vol. 61, No. 4, pp. 483–493, April, 1997. Translated by A. I. Shtern     

The abelianization of hypercyclic groups

Let G be a hypercyclic group. The most substantial results of this paper are the following. a) If G/G′ is 2-divisible, then G is 2-divisible. b) If G/G′ is a 2′-group, then G is a 2′-group. c) If G/G′ is divisible by finite-of-odd-order, then G/V is divisible by finite-of-odd-order, where V is the intersection of the lower central series (continued transfinitely) of O _2′ (G).      

Connecting monomiality questions with the structure of rational group algebras

The purpose of this paper is to show the vastness of the class of generalized strongly monomial groups and the implication of its study on monomiality questions studied by group theorists. The so called generalized strongly monomial groups, which is a generalization of strongly monomial groups introduced by Olivieri, del Río and Simón, arose in a recent work of authors while understanding the algebraic structure of rational group algebras.     

The unit group of FS 4

A complete characterization is given for the unit group U(FS ₄) of the group algebra FS ₄ of the symmetric group S ₄ of degree 4 over a finite field F.      

Finite groups with 2-nilpotent subgroups of even index

Finite groups with 2-nilpotent subgroups of even index are described. Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 700–705, May, 1997. Translated by A. I. Shtern     

On the group of polynomial functions in a group

Let G be a group and let n be a positive integer. A polynomial function in G is a function from G ⁿ to G of the form , where f(x ₁, . . . , x _n ) is an element of the free product of G and the free group of rank n freely generated by x ₁, . . . , x _n . There is a natural definition for the product of two polynomial functions; equipped with this operation, the set of polynomial functions is a group. We prove that this group is polycyclic if and only if G is finitely generated, soluble, and nilpotent-by-finite. In particular, if the group of polynomial functions is polycyclic, then necessarily it is nilpotent-by-finite. Furthermore, we prove that G itself is polycyclic if and only if the subgroup of polynomial functions which send (1, . . . , 1) to 1 is finitely generated and soluble.