具有很多素数阶元的有限群

考察元素的阶如何影响有限群的结构是群论中的一个重要课题．本文研究存在一个正规子群N，N外的元素都是素数阶元的有阶群．主要利用熟知的Thompson的一个定理，获得了这样的有限群．     

Recognition by the Set of Element Orders of Symmetric Groups of Degree r and r+1 for Prime r

We prove that the symmetric group S _r of prime degree r17 is recognizable by its element order set. A test for recognizability is obtained for the symmetric group S _r+1 with r17 prime.     

Pure quantitative characterization of linear groups over the binary field

The purpose of this paper is to discuss the pure scalar characterization of the automorphism group Aut (L ₅(2)) and the linear group L ₆(2). It is proved that Aut(L ₅(2)) and L ₆(2) can be characterized quantitatively by the set of element orders. The main results are obtained by using William’s work on prime graph components of finite groups and Brauer characters in trivializing the possible 2-subgroups. __________ Translated from Chinese Annals of Mathematics, 2003, 24A(6): 675–682. This work was supported by the National Natural Science Foundation of China under grant number 10171074     

Recognition of simple groups U 3(Q) by element orders

An exhaustive solution is given to the recognition-by-spectrum problem for finite, simple, three-dimensional unitary groups. For every such group, the number of non-isomorphic, finite, isospectral groups is determined. In particular, a new counterexample to Problem 13.63 in the Kourovka Notebook is furnished. Supported by RFBR grant No. 05-01-00797, and by SB RAS Young Researchers Support grant No. 29 and Integration Project No. 2006.1.2. __________ Translated from Algebra i Logika, Vol. 45, No. 2, pp. 185–202, March–April, 2006.     

On Finite Simple Groups with the Set of Element Orders as in a Frobenius Group or a Double Frobenius Group

It is proved that a finite simple group with the set of element orders as in a Frobenius group (a double Frobenius group, respectively) is isomorphic to L₃(3) or U₃(3) (to U₃(3) or S₄(3), respectively).     

Matrix Groups Related to the Quaternion Group and Spherical Orbit Codes

We consider a finite matrix group with 3⁴· 2¹⁶ elements, which is a subgroup of the infinite group , where is the regular representation of the quaternion group and C is a matrix that transforms the regular representation Q to its cellwise-diagonal form. There is a number of ways to define the matrix C. Our aim is to make the group similar in a certain sense to a finite group. The eventual choice of an appropriate matrix C done heuristically. We study the structure of the group and use this group to construct spherical orbit codes on the unit Euclidean sphere in R⁸. These codes have code distance less than 1. One of them has 3²· 2⁸ = 2304 elements and its squared Euclidean code distance is 0.293. Communicated by: V. A. Zinoviev     

Recognition by spectrum for finite simple linear groups of small dimensions over fields of characteristic 2

Two groups are said to be isospectral if they share the same set of element orders. For every finite simple linear group L of dimension n over an arbitrary field of characteristic 2, we prove that any finite group G isospectral to L is isomorphic to an automorphic extension of L. An explicit formula is derived for the number of isomorphism classes of finite groups that are isospectral to L. This account is a continuation of the second author's previous paper where a similar result was established for finite simple linear groups L in a sufficiently large dimension (n > 26), and so here we confine ourselves to groups of dimension at most 26. Supported by RFBR (project Nos. 08-01-00322 and 06-01-39001), by SB RAS (Integration Project No. 2006.1.2), and by the Council for Grants (under RF President) and State Aid of Leading Scientific Schools (grant NSh-344.2008.1) and Young Doctors and Candidates of Science (grants MD-2848.2007.1 and MK-377.2008.1). Translated from Algebra i Logika, Vol. 47, No. 5, pp. 558–570, September–October, 2008.     

关于单$K_4-$群的新刻画

本文用群阶和高阶元素的阶刻画了部分单$K_4-$群. 所谓单$K_4$-群就是其阶只含4个素因子的单群.     

若干家族p-群的自同构群的阶

本文确定了Rodney James文章中P．Hall isoclinism族第一家族到第十家族所有群的自同构群的阶,并纠正了Rodney James文章里的某些错误．     

最高阶元素个数为2×5m的有限群是可解群

本文讨论了最高阶元素个数为 2× 5 m(m为正整数 )的有限群 ,证明了群G是可解群 .     

A characterization of the simple group PSL<Subscript>5</Subscript>(5) by the set of its element orders

Let G be a finite group and let ω(G) denote the set of the element orders of G. For the simple group PSL₅(5) we prove that if G is a finite group with ω(G) = ω(PSL₅(5)), then either G ? PSL₅(5) or G ? PSL₅(5): 〈θ〉 where θ is a graph automorphism of PSL₅(5) of order 2.     

A new characterization of <Emphasis Type="Italic">PSU</Emphasis>(<Emphasis Type="Italic">p,q</Emphasis>)

Summary Let G be a finite group. Order components of G were introduced in Chen [5]. Let OC(G) be the set of order components of G. Some finite groups are characterizable by their order components. This assertion was proved for the simple groups PSU(p,q), where p=3, 5, 7 and 11. In this paper, we prove that the simple groups PSU(p,q) can be uniquely determined by their order components, where p&ge;13 is a prime number. Main consequences of our results are the validity of a conjecture of J. G. Thompson and another conjecture of W. Shi and J. Bi for the groups under consideration.     

关于群的阶与共轭类数

讨论群的共轭数与群阶的关系,获得两个新的数量不等式,同时改进了一些相关的已知结果。     

A Characterization of <Emphasis Type="Italic">PSU</Emphasis><Subscript>3</Subscript>(<Emphasis Type="Italic">q</Emphasis>) for <Emphasis Type="Italic">q</Emphasis> ≥ 5

Based on the prime graph of a finite simple group, its order is the product of its order components (see [4]). We prove that the simple groups PSU₃(q) are uniquely determined by their order components. Our result immediately implies that the Thompsons conjecture and the Wujie Shis conjecture [16] are valid for these groups.AMS Subject Classification: 20D05, 20D60     

2-Recognizability by prime graph of <Emphasis Type="Italic">PSL</Emphasis>(2, <Emphasis Type="Italic">p</Emphasis><Superscript>2</Superscript>)

Let G be a finite group and let Γ(G) be the prime graph of G. Assume p prime. We determine the finite groups G such that Γ(G) = Γ(PSL(2, p ²)) and prove that if p ≠ 2, 3, 7 is a prime then k(Γ(PSL(2, p ²))) = 2. We infer that if G is a finite group satisfying |G| = |PSL(2, p ²)| and Γ(G) = Γ(PSL(2, p ²)) then G ? PSL(2, p ²). This enables us to give new proofs for some theorems; e.g., a conjecture of W. Shi and J. Bi. Some applications are also considered of this result to the problem of recognition of finite groups by element orders.     

最高阶元个数是4p的有限群

本文证明了如果有限群G恰有4p个最高阶元,p为素数,则群G为可解群,除非G同构于S5.     

三系数有限元向量尺度函数的构造

The aim of this paper is to present construction of finite element multiscaling function with three coefficients. In order to illuminate the result, two examples are given finally.