OD-characterization of Almost Simple Groups Related to U3（5）

Let G be a finite group with order ｜G｜=p1^α1p2^α2……pk^αk, where p1 〈 p2 〈……〈 Pk are prime numbers. One of the well-known simple graphs associated with G is the prime graph （or Gruenberg- Kegel graph） denoted .by г（G） （or GK（G））. This graph is constructed as follows： The vertex set of it is π（G） = {p1,p2,…,pk} and two vertices pi, pj with i≠j are adjacent by an edge （and we write pi - pj） if and only if G contains an element of order pipj. The degree deg（pi） of a vertex pj ∈π（G） is the number of edges incident on pi. We define D（G） ：= （deg（p1）, deg（p2）,..., deg（pk））, which is called the degree pattern of G. A group G is called k-fold OD-characterizable if there exist exactly k non- isomorphic groups H such that ｜H｜ = ｜G｜ and D（H） = D（G）. Moreover, a 1-fold OD-characterizable group is simply called OD-characterizable. Let L ：= U3（5） be the projective special unitary group. In this paper, we classify groups with the same order and degree pattern as an almost simple group related to L. In fact, we obtain that L and L.2 are OD-characterizable; L.3 is 3-fold OD-characterizable; L.S3 is 6-fold OD-characterizable.     

A RECOGNITION OF SIMPLE GROUPS PSL（3, q） BY THEIR ELEMENT ORDERS

For any group G, denote byπe(G) the set of orders of elements in G. Given a finite group G, let h(πe (G)) be the number of isomorphism classes of finite groups with the same set πe(G) of element orders. A group G is called k-recognizable if h(πe(G)) = k <∞, otherwise G is called non-recognizable. Also a 1-recognizable group is called a recognizable (or characterizable) group. In this paper the authors show that the simple groups PSL(3,q), where 3 < q≡±2 (mod 5) and (6, (q-1)/2) = 1, are recognizable.     

The Characterization of Finite Simple Groups, L3（3^2m-1）（m≥2）, by Their Element Orders

In this paper the following theorem is proved： Every group L3（q） for q = 3^（2m-1）（m≥2） is characterized by its set of element orders.     

关于阶为2~（α1）3~（α2）5~（α3）7~（α4）p~（α5）的单群

本文证明了以下结果：设α１，α２，α３，α４，α５均为正整数，ｐ为素数且．如果Ｇ是阶为的单群，则Ｇ同构于下列单群之一：Ａ１１，Ａ１２；Ｍ２２，Ｈｉ－Ｓ２ＭｃＬ，Ｈｅ；Ａ１（ｑ）（ｑ＝２６，５３，７４，２９，４１，７１，２５１，４４９，４８０１），Ａ２（３２），Ａ３（２２），Ａ３（７），Ａ４（２），Ａ５（２），Ｂ２（２３），Ｂ２（７２），Ｂ３（３），Ｂ４（２），Ｃ３（３），Ｄ４（３），Ｇ２（２），Ｇ２（５），２Ａ２（１９），２Ａ３（５），２Ａ３（７），２Ａ４（３），２Ａ５（２），２Ｄ４（２）．     

SL（2，q）的一个特征性质

本文证明了非单群系列ＳＬ（２，ｑ）（ｑ＝ｐ￣ｎ＞３）可以仅用其极大子群阶之集来刻划，从而得到了ＳＬ（２，ｑ）的一个特征性质．     

SUZUKI GROUPS Sz（q） AND 2-（v,k, 1） BLOCK DESIGNS

It is proved that if D be a 2-(v,k,1) design with G≤Aut D block primitive then G does not have a Suzuki group Sz(q) as the socle.     

EXPLICIT CONSTRUCTION OF HARMONIC MAPS FROM R^2 TO U（N）

ＥＸＰＬＩＣＩＴＣＯＮＳＴＲＵＣＴＩＯＮＯＦＨＡＲＭＯＮＩＣＭＡＰＳＦＲＯＭＲ~２ＴＯＵ（Ｎ）￥ＧｕＣＨＡＯＨＡＯＨｕＨＥＳＨＥＮＧＡｂｓｔｒａｃｔ：Ｄａｒｂｏｕｘｔｒａｎｓｆｏｒｍａｔｉｏｎｍｅｔｈｏｄｉｓｕｓｅｄｆｏｒｃｏｎｓｔｒｕｃｔｉｎｇｈａ?..     

ON THE CLASSIFICATION OF AF-ALGEBRAS AND THEIR DIMENSION GROUPS （Ⅱ）

ＯＮＴＨＥＣＬＡＳＳＩＦＩＣＡＴＩＯＮＯＦＡＦ－ＡＬＧＥＢＲＡＳＡＮＤＴＨＥＩＲＤＩＭＥＮＳＩＯＮＧＲＯＵＰＳ（Ⅱ）￥ＨＵＡＮＧＺＨＡＯＢＯＡｂｓｔｒａｃｔ：Ｔｈｉｓｐａｐｅｒｉｓａｃｏｎｔｉｎｕａｔｉｏｎｏｆ［１］．Ｉｔｇｉｖｅｓｓｏｍｅａｐｐｌｉ...     

L7（3）与GL7（3）的OD-刻画

对于任意一个有限群G,令π（G）表示由它的阶的所有素因子构成的集合.构建一种与之相关的简单图,称之为素图,记作Γ（G）.该图的顶点集合是π（G）,图中两顶点p,g相连（记作p～q）的充要条件是群G恰有pq阶元.设π（G）={P₁,p2,…,p_x}.对于任意给定的p∈π（G）,令deg（p）:=|{q∈π（G）|在素图Γ（G）中,p～q}|,并称之为顶点p的度数.同时,定义D（G）:=（deg（p₁）,deg（p₂）,…,deg（p_s））,其中p₁2<…相似文献   

L_2（q）的一个特征性质

设Ｇ是有限群，πｓ（Ｇ）为Ｇ的极大子群阶之集．本文证明了若ｑ＝ｐｎ＞２，ｐ素，则Ｇ≌Ｌ２（ｑ）当且仅当πｓ（Ｇ）＝πｓ（Ｌ２（ｑ））．对一些其它的单群也证明了同样的结论．     

维数分别为51和52的对称群的OD-刻画

利用有限群的阶和它的次数型分别对对称群S51和S_（52）进行了刻画,得到：对称群S_（51）和S_（52）均可3-重OD-刻画.     

OD-characterization of all simple groups whose orders are less than 108

Let M be a simple group whose order is less than 10⁸. In this paper, we prove that if G is a finite group with the same order and degree pattern as M, then the following statements hold: (a) If M ≠ A ₁₀, U ₄(2), then G ≅ M; (b) If M = A ₁₀, then G ≅ A ₁₀ or J ₂ × ℤ₃; (c) If M = U ₄(2), then G is isomorphic to a 2-Frobenius group or U ₄(2). In particular, all simple groups whose orders are less than 10⁸ but A ₁₀ and U ₄(2) are OD-characterizable. As a consequence of this result, we can give a positive answer to a conjecture put forward by W. J. Shi and J. X. Bi in 1990 [Lecture Notes in Mathematics, Vol. 1456, 171–180].      

作用在有限线性空间上基柱为2F4(q)的几乎单群

本文研究了基柱同构于2F4(q)的线传递的几乎单群.利用了有限线性空间上线传递的自同构群的经典结论,以及2F4(q)的阶和极大子群等性质,获得了点稳定子群和线稳定子群的分类.     

New upper bounds for the chromatic number of a graph*

We show that for any graph G, the chromatic number χ(G) ≤ Δ₂(G) + 1, where Δ₂(G) is the largest degree that a vertex ν can have subject to the condition that ν is adjacent to a vertex whose degree is at least as big as its own. Moreover, we show that the upper bound is best possible in the the following sense: If Δ₂(G) ≥ 3, then to determine whether χ(G) ≤ Δ₂(G) is an NP‐complete problem. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 117–120, 2001     

Quasirecognition by the set of element orders of the groups E 6( q ) and 2 E 6( q )

We prove that if L is one of the simple groups E ₆(q) and ² E ₆(q) and G is some finite group with the same spectrum as L, then the commutant of G/F(G) is isomorphic to L and the quotient G/G′ is a cyclic {2,3}-group. Original Russian Text Copyright ? 2007 Kondrat’ev A. S. The author was supported by the Russian Foundation for Basic Research (Grant 04-01-00463) and the RFBR-NSFC (Grant 05-01-39000). __________ Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 48, No. 6, pp. 1250–1271, November–December, 2007.     

例外型Chevalley群G_2(q)的h函数值

设G是有限群,πe(G)表示G中元素的阶的集合,h(πe(G))表示满足πe(H)=πe(G)条件的有限群H的同构类类数,本文证明了例外型Chevalley群G2(q)的h函数值为1或∞.     

OD-Characterization of alternating and symmetric groups of degrees 16 and 22

Let G be a finite group and π(G) be the set of all prime divisors of its order. The prime graph GK(G) of G is a simple graph with vertex set π(G), and two distinct primes p, q ∈ π(G) are adjacent by an edge if and only if G has an element of order pq. For a vertex p ∈ π(G), the degree of p is denoted by deg(p) and as usual is the number of distinct vertices joined to p. If π(G) = {p ₁, p ₂,...,p _k }, where p ₁ < p ₂ < ... < p _k , then the degree pattern of G is defined by D(G) = (deg(p ₁), deg(p ₂),...,deg(p _k )). The group G is called k-fold OD-characterizable if there exist exactly k non-isomorphic groups H satisfying conditions |H| = |G| and D(H) = D(G). In addition, a 1-fold OD-characterizable group is simply called OD-characterizable. In the present article, we show that the alternating group A ₂₂ is OD-characterizable. We also show that the automorphism groups of the alternating groups A ₁₆ and A ₂₂, i.e., the symmetric groups S ₁₆ and S ₂₂ are 3-fold OD-characterizable. It is worth mentioning that the prime graph associated to all these groups are connected.