OD-CHARACTERIZATION OF ALMOST SIMPLE GROUPS RELATED TO U6（2）

Let G be a finite group and π(G) = { p 1 , p 2 , ··· , p k } be the set of the primes dividing the order of G. We define its prime graph Γ(G) as follows. The vertex set of this graph is π(G), and two distinct vertices p, q are joined by an edge if and only if pq ∈π e (G). In this case, we write p ～ q. For p ∈π(G), put deg(p) := |{ q ∈π(G) | p ～ q }| , which is called the degree of p. We also define D(G) := (deg(p 1 ), deg(p 2 ), ··· , deg(p k )), where p 1 < p 2 < ··· < p k , which is called the degree pattern of G. We say a group G is k-fold OD-characterizable if there exist exactly k non-isomorphic finite groups with the same order and degree pattern as G. Specially, a 1-fold OD-characterizable group is simply called an OD-characterizable group. Let L := U 6 (2). In this article, we classify all finite groups with the same order and degree pattern as an almost simple groups related to L. In fact, we prove that L and L.2 are OD-characterizable, L.3 is 3-fold OD-characterizable, and L.S 3 is 5-fold OD-characterizable.     

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）．     

射影特殊酉群的纯数量刻画

证明了对每一个射影特殊酉群可用它的元的阶之集和群的阶加以刻画.     

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   

Recognition of the Projective Special Linear Group over GF（3）

Let P be a finite group and denote by w（P） the set of its element orders. P is called k-recognizable by the set of its element orders if for any finte group G with ω（G） =ω（P） there are, up to isomorphism, k finite groups G such that G ≌P. In this paper we will prove that the group Lp（3）, where p 〉 3 is a prime number, is at most 2-recognizable.     

The Characterizations of Weighted H^p（w）Spaces on the Certain Groups

ＴｈｅＣｈａｒａｃｔｅｒｉｚａｔｉｏｎｓｏｆＷｅｉｇｈｔｅｄ　Ｈ~ｐ（ｗ）ＳｐａｃｅｓｏｎｔｈｅＣｅｒｔａｉｎＧｒｏｕｐｓＺｏｕＪｉｎ（邹进）（Ｌｅｓｈａｎ，ＴｈｅａｃｈｅｒｓＣｏｌｌｅｇｅ，Ｃｈｉｎａ）Ａｂｓｔｒａｃｔ：ＴｈｅｗｅｉｇｈｔｅｄＨＰ（?..     

Characterizability of the Group ^2Dp（3） by its Order Components,Where p ≥ 5 is a Prime Number Not of the Form 2^m ＋ 1

The author will prove that the group ^2Dp（3） can be uniquely determined by its order components, where p ≠ 2^m ＋ 1 is a prime number, p ≥ 5. More precisely, if OC（G） denotes the set of order components of G, we will prove OC（G） = OC（^2Dp（3）） if and only if G is isomorphic to ^2Dp（3）. A main consequence of our result is the validity of Thompson＇s conjecture for the groups under consideration.     

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].      

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.     

Carter subgroups of finite almost simple groups

In the paper we work to complete the classification of Carter subgroups in finite almost simple groups. In particular, it is proved that Carter subgroups of every finite almost simple group are conjugate. Based on our previous results, together with those obtained by F. Dalla Volta, A. Lucchini, and M. C. Tamburini, as a consequence we derive that Carter subgroups of every finite group are conjugate. Supported by RFBR grant No. 05-01-00797; by the Council for Grants (under RF President) for Support of Young Russian Scientists via projects MK-1455.2005.1 and MK-3036.2007.1; by SB RAS Young Researchers Support grant No. 29; via Integration Project No. 2006.1.2. __________ Translated from Algebra i Logika, Vol. 46, No. 2, pp. 157–216, March–April, 2007.     

On recognition of finite simple groups with connected prime graph

The spectrum of a finite group is the set of its element orders. We prove a theorem on the structure of a finite group whose spectrum is equal to the spectrum of a finite nonabelian simple group. The theorem can be applied to solving the problem of recognizability of finite simple groups by spectrum.     

Almost simple groups with socle ~3D_4(q) act on finite linear spaces

After the classification of flag-transitive linear spaces, attention has now turned to line-transitive linear spaces. Such spaces are first divided into the point-imprimitive and the point-primitive, the first class is usually easy by the theorem of Delandtsheer and Doyen. The primitive ones are now subdivided, according to the O'Nan-Scotte theorem and some further work by Camina, into the socles which are an elementary abelian or non-abelian simple. In this paper, we consider the latter. Namely, T≤G≤Aut(T) and G acts line-transitively on finite linear spaces, where T is a non-abelian simple. We obtain some useful lemmas. In particular, we prove that when T is isomorphic to 3D4(q), then T is line-transitive, where q is a power of the prime p.     

Recognition by spectrum for finite simple groups of Lie type

The goal of this article is to survey new results on the recognition problem. We focus our attention on the methods recently developed in this area. In each section, we formulate related open problems. In the last two sections, we review arithmetical characterization of spectra of finite simple groups and conclude with a list of groups for which the recognition problem was solved within the last three years.      

Recognition by spectrum of the groups 2~D_(2m+1)(3)

It is proved that a finite group with the same set of element orders as the simple group 2~D_(2m+1)(3) is isomorphic to 2~D_(2m+1)(3).     

Recognition by spectrum of the groups $$ ^2 D_{2^m + 1} (3) $$

Abstract It is proved that a finite group with the same set of element orders as the simple group is isomorphic to . This work was supported by Russian Foundation for Basic Research (Grant No. 07-01-00148), RFBR-BRFBR (Grant No. 08-01-90006) and RFBR-GFEN (Grant No. 08-01-92200)