Quasivariety Generated by Free Metabelian and 2-Nilpotent Groups

Let qG be a quasivariety generated by a group G and be a non-Abelian quasivariety of groups with a finite lattice of subquasivarieties. Suppose is contained in a quasivariety generated by the following two groups: a free 2-nilpotent group F₂( ₂) of rank 2 and a free metabelian (i.e., with an Abelian commutant) group F₂( ₂) of rank 2. It is proved that either = qF₂( ₂) or = qF₂( ₂) in this instance.__________Translated from Algebra i Logika, Vol. 44, No. 4, pp. 389–398, July–August, 2005.     

On K 2-group of a formal matrix ring

Let S be a formal matrix ring, T the subring consisting of all diagonal elements, I the set consisting of all off-diagonal elements. Then I is a split radical ideal under certain conditions. In this paper, we show that K2 (S)■K2 (T) ? K2 (S,I), and a presentation of K2 (S,I) is given.     

Group Quasivarieties with Infinitely Many Maximal Subquasivarieties

We construct an example of a 2-generated group G and describe a set of proper maximal subquasivarieties in the quasivariety qG. This set proves to be infinite, which gives an affirmative answer to Question 14.25 posed by A. I. Budkin in the Kourovka Notebook. Moreover, it is shown that every proper subquasivariety in qG is contained in a proper maximal one.     

Rationality and strong reality for Sylow 2-subgroups of Weyl and alternating groups

It is proved that values of all complex characters for a Sylow 2-subgroup P of any Weyl group are rational, and every element of P is a product of two involutions in P. Similar results hold also for Sylow 2-subgroups of alternating groups.Supported by RFBR grant No. 03-01-00905.Translated from Algebra i Logika, Vol. 44, No. 1, pp. 44–53, January–February, 2005.     

n-极大子群为C-正规的有限群

首先,利用Hall子群的C－正规性,得到了有限群成为。可解群的一个充分条付,推广了Schur-Zassenhaus定理;其次,考查了 n－极大子群或其 2阶及 4阶循环子群为 C-正规对有限群结构的影响．     

On cyclic semi-regular subgroups of certain 2-transitive permutation groups

We determine the cyclic semi-regular subgroups of the 2-transitive permutation groups and with n a suitable power of a prime number p.     

Notes on 2-parameter quantum groups I

A simpler definition for a class of 2-parameter quantum groups associated to semisimple Lie algebras is given in terms of Euler form. Their positive parts turn out to be 2-cocycle deformations of each other under some conditions. An operator realization of the positive part is given. This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 10431040, 10728102), the TRAPOYT, the FUDP and the Priority Academic Discipline from the MOE of China, the SRSTP from the STCSM, the Shanghai Priority Academic Discipline from the SMEC     

Notes on 2-parameter quantum groupsⅠ

A simpler definition for a class of 2-parameter quantum groups associated to semisimple Lie algebras is given in terms of Euler form.Their positive parts turn out to be 2-cocycle deformations of each other under some conditions.An operator realization of the positive part is given.     

二环链系统的稳定性和能控性

提出了二环链系统,并利用二环链特征向量,给出了一个系统可实现二环链分解的充分必要条件,给出了可实现二环链线性系统的分解算法,研究了二环链系统的稳定性和能控性.     

The classification of flag-transitive Steiner 4-designs

Among the properties of homogeneity of incidence structures flag-transitivity obviously is a particularly important and natural one. Consequently, in the last decades flag-transitive Steinert-designs (i.e. flag-transitive t-(v,k,1) designs) have been investigated, whereas only by the use of the classification of the finite simple groups has it been possible in recent years to essentially characterize all flag-transitive Steiner 2-designs. However, despite the finite simple group classification, for Steiner t-designs with parameters t > 2 such characterizations have remained challenging open problems for about 40 years (cf. [11, p. 147] and [12 p. 273], but presumably dating back to around 1965). The object of the present paper is to give a complete classification of all flag-transitive Steiner 4-designs. Our result relies on the classification of the finite doubly transitive permutation groups and is a continuation of the author's work [20, 21] on the classification of all flag-transitive Steiner 3-designs. 2000 Mathematics Subject Classification. Primary 51E10 . Secondary 05B05 . 20B25     

On the Artin exponent of some rational groups

A finite group G is called rational if all its irreducible complex characters are rational valued. In this paper, we show that if G is a direct product of finitely many rational Frobenius groups then every rationally represented character of G is a generalized permutation character. Also we show that the same assertion holds when G is a solvable rational group with a Sylow 2-subgroup isomorphic to the dihedral group of order 8 and an abelian normal Sylow 3-subgroup.     

On sharply 2-transitive groups with point stabilizer of exponent 2n⋅3

We describe sharply 2-transitive groups whose point stabilizer is a nilpotent {2,3}-group without elements of order 9 and, more generally, in which the third power of each element belongs to the FC-center. In particular, we will prove that these groups are finite.     

闭2-极大子群指数为素数幂的有限群

设G为有限群, 如果G的每个非2-闭极大子群的指数均为素数幂, 那么 G-S(G)≌PSL2(7)或1, 其中S(G)为G的最大可解正规子群.     

有限群的c-半置换子群

研究了有限群的结构问题,利用子群c-半置换和完全c-半置换的定义和性质,通过对有限群sylow子群的2-极大子群的研究,获得了有限群幂零、p-幂零的充分条件和另外两个决定群结构的充要条件.     

Endomorphism rings and automorphism groups of almost completely decomposable groups

To every non-cuspidal K-rational point on the modular curve X_l(n) a non-commutative Noetherian domain of global dimension 3 can be associated : the Sklyanin algebra. In this paper we give the defining equations of the Sklyanin algebras and their centers when X_l(n) is rational, i.e. n <= 10 or n = 12.