关于亏零p-块的一些结果

本文研究了亏零p-块的存在性,得到一些亏零p-块存在的群论条件。     

有限群亏零p-块的存在性

对p-阶子群均共轭的群的亏零p-块的存在性给出了一些刻画.同时给出了偶阶群亏零p-块的存在性.     

有限群亏零p-块的存在性

本文研究了有限群亏零p-块的存在性.利用极大子群的性质,给出了有限群存在亏零p-块的充要条件.这些结论丰富了块论理论.     

关于幂零π-块中B_(π′)-特征标个数的一个注记

Broué和Puig给出了幂零p-块的概念,并指出了幂零p-块的存在性以及幂零p-块的一个性质:幂零p-块中仅含一个Brauer特征标.利用Slattery,Robinson等的一些工作,将上述思想推广到π-可分群的π-块论中,给出了幂零π-块的合理定义,并证明:幂零π-块中仅含一个B_(π′)-特征标.     

亏数零p-块的存在条件

<正> 给定有限群G的一个p-子群D,在什么条件下D是G的一个p-块的亏数群呢?这是模表示论中一个十分重要的问题。由Brauer第一基本定理,我们可以假定D是正规子群,从而进一步化为DC_G(D)/D的亏数零p-块的存在性问题.关于此间题的研究,吸引了众多的群论和模表示论工作者.但迄今进展甚微.Brauer和Fowler[1]曾经给出了存在亏数零P-块的充分条件,Tsushima[2]和wada也曾给出过类似的条件,在本文     

关于幂零π-块中Bπ'-特征标个数的一个注记

Broué和Puig给出了幂零p-块的概念, 并指出了幂零p-块的存在性以及幂零p-块的一个性质:幂零p-块中仅含一个Brauer特征标. 利用Slattery, Robinson等的一些工作, 将上述思想推广到π-可分群的π-块论中, 给出了幂零π-块的合理定义, 并证明: 幂零π-块中仅含一个B_π''-特征标.     

关于Brauer的K(B)猜想

本文证明了如下结论:设 p 是一个素数,有限 p′-群 G 忠实不可约地作用于初等交换 p-群 V.若 G的阶不能被4整除,则半直积群 GV 的共轭类个数一定不大于 V 的阶.或者等价地:设 G 是一个有限 p-可解群,且其p′-Hall 子群的阶不能被4整除,则 G 的每个 p-块中含不可约常指标的个数一定不大于这个块的亏群的阶.     

关于复杂度,亏群与不可分解模的若干说明

在本文中，我们给出了CG^ (B)≤2但不满意当p=2时CG^ (B)=CG^-(B)=2的p-可解群的块B的亏群的结构。同时亦给出关于复杂度与不可分解模的若干说明。     

关于可解群p—块的几个结果

主要结果是如下定理:设G是有限可解群使得G/F(G)是奇阶A-群,又设p是一个素数且G不含截断q~(pn):(Z_m:Z_p)。其中q~(pn):(Z_m:Z_n))是初等交换q-群q~(pn)被Z_m:Z_p的扩张,而m=(q~(pn)-1)/(q~n-1)。则G有亏数零p-块的充要条件是O_p(G)=1。     

无限正则p-群

对一类无限正则p-群进行了研究,得到了一个正则的局部幂零p-群G如果满足|G(Ω)1(G)|＜∞,那么G是幂零的且G是可除阿贝尔p-群被有限群的扩张.进而,还研究了一类无限的非正则p-群,但它的所有真商群或者真的无限子群是正则群.在假设这类群存在拟循环子群的情况下,在定理1.2和1.3给出了这类群的结构的刻画.     

关于弱半可根群

In this paper, we prove that if a torsion nilpotent group G is a weak semi-radicable group, then every Sylow p-group Gp is a central-by-finite p-group, and moreover Gp's center ζ(GP) satisfies |ζ(GP) : (ζ(GP))P| <∞, that is, ζ(GP) = D×F, where D is a divisible Abelian group, and F is a finite Abelian group.     

Tychonoff property for linear groups

A criterion for a wide class of topological groups which includes linear discrete groups and Lie groups to be Tychonoff groups is established. The main result provides a criterion for an almost polycyclic group to have the Tychonoff property. By the well-known Tits alternative, this yields the required criterion for linear discrete groups. In conclusion it is pointed out that a particular case of the presented proof yields a Tychonoff property criterion for Lie groups. In addition, an example of a polycyclic group without Tychonoff subgroups of finite index is constructed. Translated fromMatematicheskie Zametki, Vol. 63, No. 2, pp. 269–279, February, 1998. The author wishes to express his gratitude to R. I. Grigorchuk for setting the problem and his interest in the work. This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00182 and by the American Mathematical Society Fund.     

Torsion-free constructive nilpotent <Emphasis Type="Italic">R</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-groups

We consider a torsion-free nilpotent R _p -group, the p-rank of whose quotient by the commutant is equal to 1 and either the rank of the center by the commutant is infinite or the rank of the group by the commutant is finite. We prove that the group is constructivizable if and only if it is isomorphic to the central extension of some divisible torsion-free constructive abelian group by some torsion-free constructive abelian R _p -group with a computably enumerable basis and a computable system of commutators. We obtain similar criteria for groups of that type as well as divisible groups to be positively defined. We also obtain sufficient conditions for the constructivizability of positively defined groups.     

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.     

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…     

关于多克托罗夫定理

本文将多克托罗夫定理的条件减弱，得到了这样的定理：设Ｇ是有限群．如果Ｇ的每个西洛子群的正规化子有Ｈａｉｌ补，则Ｇ为σ－西洛塔解；此外，如果这些补的Ｆｉｔｔｉｎｇ子群是循环群，则Ｇ为超可解群．     

Periodic groups saturated by finite simple groups <Emphasis Type="Italic">U</Emphasis><Subscript>3</Subscript>(2<Superscript>m</Superscript>)

Let {ie166-01} be a set of finite groups. A group G is said to be saturated by the groups in {ie166-02} if every finite subgroup of G is contained in a subgroup isomorphic to a member of {ie166-03}. It is proved that a periodic group G saturated by groups in a set {U₃(2^m) | m = 1, 2, …} is isomorphic to U₃(Q) for some locally finite field Q of characteristic 2; in particular, G is locally finite. __________ Translated from Algebra i Logika, Vol. 47, No. 3, pp. 288–306, May–June, 2008.     

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].