On Connection Between the Structure of a Finite Group and the Properties of Its Prime Graph

It is shown that the condition of nonadjacency of 2 and at least one odd prime in the Gruenberg-Kegel graph of a finite group G under some natural additional conditions suffices to describe the structure of G; in particular, to prove that G has a unique nonabelian composition factor. Applications of this result to the problem of recognition of finite groups by spectrum are also considered.Original Russian Text Copyright © 2005 Vasilev A. V.The author was supported by the Russian Foundation for Basic Research (Grant 05-01-00797), the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grant NSh-2069.2003.1), the Program Development of the Scientific Potential of Higher School of the Ministry for Education of the Russian Federation (Grant 8294), the Program Universities of Russia (Grant UR.04.01.202), and a grant of the Presidium of the Siberian Branch of the Russian Academy of Sciences (No. 86-197).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 511–522, May–June, 2005.     

Recognition of the Finite Simple Groups F 4(2m) by Spectrum

The spectrum of a finite group is the set of its element orders. A finite group G is said to be recognizable by spectrum, if every finite group with the same spectrum as G is isomorphic to G. The purpose of the paper is to prove that for every natural m the finite simple Chevalley group F ₄(2 ^m ) is recognizable by spectrum.     

On the Soluble Graph of a Finite Simple Group

The maximal independent sets of the soluble graph of a finite simple group G are studied and their independence number is determined. In particular, it is shown that this graph in many cases has an independent set with three vertices.     

On Recognition by Spectrum of Finite Simple Linear Groups over Fields of Characteristic 2

A finite group G is said to be recognizable by spectrum, i.e., by the set of element orders, if every finite group H having the same spectrum as G is isomorphic to G. We prove that the simple linear groups L _n (2^k) are recognizable by spectrum for n = 2^m ≥ 32.Original Russian Text Copyright © 2005 Vasil’ev A. V. and Grechkoseeva M. A.The authors were supported by the Russian Foundation for Basic Research (Grant 05-01-00797), the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grant NSh-2069.2003.1), the Program “ Development of the Scientific Potential of Higher School” of the Ministry for Education of the Russian Federation (Grant 8294), the Program “Universities of Russia” (Grant UR.04.01.202), and a grant of the Presidium of the Siberian Branch of the Russian Academy of Sciences (No. 86-197).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 749–758, July–August, 2005.     

The Prime Graph of a Sporadic Simple Group

Abstract

Let Gbe a finite group and Sa sporadic simple group. We denote by π(G) the set of all primes dividing the order of G. The prime graph Γ(G) of Gis defined in the usual way connecting pand qin π(G) when there is an element of order pqin G. The main purpose of this paper is to determine finite group Gsatisfying Γ(G) = Γ(S) (See Theorem 3) and to give applications which generalize Abe (Abe, S. Two ways to characterize 26 sporadic finite simple groups. Preprint) and Chen (Chen, G. (1996). A new characterization of sporadic simple groups. Algebra Colloq.3:49–58). The results are elementary but quite useful.     

2-Signalizers of Finite Simple Groups

Maximal 2-signalizers and centralizers of Sylow 2-subgroups in all finite simple groups are described. Also normalizers are computed for Sylow 2-subgroups in the finite simple groups of exceptional Lie type over a field of odd characteristic.     

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.     

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)     

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.     

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).     

On Recognition of the Finite Simple Orthogonal Groups of Dimension 2m, 2m+1, and 2m+2 over a Field of Characteristic 2

The spectrum (G) of a finite group G is the set of element orders of G. A finite group G is said to be recognizable by spectrum (briefly, recognizable) if HG for every finite group H such that (H)=(G). We give two series, infinite by dimension, of finite simple classical groups recognizable by spectrum.     

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).     

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.      

The Number of Isomorphism Classes of Finite Groups with Given Element Orders

Let G be a finite group and _e(G) the set of element orders of G. Denote by h( _e(G)) the number of isomorphism classes of finite groups H satisfying _e(H) = _e(G). We prove that if G has at least three prime graph components, then h( _e (G)){1, }.     

Non-Nilpotent Graph of a Group

We associate a graph 𝒩 _G with a group G (called the non-nilpotent graph of G) as follows: take G as the vertex set and two vertices are adjacent if they generate a non-nilpotent subgroup. In this article, we study the graph theoretical properties of 𝒩 _G and its induced subgraph on G \ nil(G), where nil(G) = {x ∈ G | ? x, y ? is nilpotent for all y ∈ G}. For any finite group G, we prove that 𝒩 _G has either |Z*(G)| or |Z*(G)| +1 connected components, where Z*(G) is the hypercenter of G. We give a new characterization for finite nilpotent groups in terms of the non-nilpotent graph. In fact, we prove that a finite group G is nilpotent if and only if the set of vertex degrees of 𝒩 _G has at most two elements.     

Quasirecognition of One Class of Finite Simple Groups by the Set of Element Orders

We prove that a finite group, having the same set of element orders as a finite simple group L and the prime graph with at least three connected components, possesses a (unique) nonabelian composition factor which is isomorphic to L, unless L is isomorphic to the alternating group of degree 6.     

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.     

有限域上单变量代数函数域中的素数定理

设q是素数的幂次，Fq为一有限域；F为Fq上的单变量代数函数域．在这篇文章中我们证明了下面的素数定理，πF（x）=1/（q-1）.x/logqx＋O（x/log^2qx）.x=q^n→∞其中logqx以q为底的对数，这一结果改进了M．Kruse，H．Stichtenoth的结果．