On the Characterizability of the Automorphism Groups of Sporadic Simple Groups by Their Element Orders

For G a finite group, π _e (G) denotes the set of orders of elements in G. If Ω is a subset of the set of natural numbers, h(Ω) stands for the number of isomorphism classes of finite groups with the same set Ω of element orders. We say that G is k-distinguishable if h(π _e (G)) = k < ∞, otherwise G is called non-distinguishable. Usually, a 1-distinguishable group is called a characterizable group. It is shown that if M is a sporadic simple group different from M ₁₂, M ₂₂, J ₂, He, Suz, M ^c L and O′N, then Aut(M) is characterizable by its element orders. It is also proved that if M is isomorphic to M ₁₂, M ₂₂, He, Suz or O′N, then h(π _e (Aut(M))) ∈¸ {1,∞}.     

On the Characterizability of the Automorphism Groups of Sporadic Simple Groups by Their Element Orders

For G a finite group,π_e(G) denotes the set of orders of elements in G.If Ω is a subsetof the set of natural numbers,h(Ω) stands for the number of isomorphism classes of finite groups withthe stone set Ω of element orders.We say that G is k-distinguishable if h(π_e(G))=k<∞,otherwiseG is called non-distinguishable.Usually,a 1-distinguishable group is called a characterizable group.Itis shown that if M is a sporadic simple group different from M_(12),M_(22),J_2,He,Suz,M~cL and O'N,then Aut(M) is characterizable by its element orders.It is also proved that if M is isomorphic toM_(12),M_(22),He,Suz or O'N,then h(π_e(Aut(M)))∈{1,∞}.     

An Adjacency Criterion for the Prime Graph of a Finite Simple Group

For every finite non-Abelian simple group, we give an exhaustive arithmetic criterion for adjacency of vertices in a prime graph of the group. For the prime graph of every finite simple group, this criterion is used to determine an independent set with a maximal number of vertices and an independent set with a maximal number of vertices containing 2, and to define orders on these sets; the information obtained is collected in tables. We consider several applications of these results to various problems in finite group theory, in particular, to the recognition-by-spectra problem for finite groups. Supported by RFBR grant No. 05-01-00797; by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-2069.2003.1; by the RF Ministry of Education Developmental Program for Scientific Potential of the Higher School of Learning, project No. 8294; by FP “Universities of Russia,” grant No. UR.04.01.202; and by Presidium SB RAS grant No. 86-197. __________ Translated from Algebra i Logika, Vol. 44, No. 6, pp. 682–725, November–December, 2005.     

On Generators and Representations of the Sporadic Simple Groups

In this paper we determine the irreducible projective representations of sporadic simple groups over an arbitrary algebraically closed field F, whose image contains an almost cyclic matrix of prime-power order. A matrix M is called cyclic if its characteristic and minimum polynomials coincide, and we call M almost cyclic if, for a suitable α ∈F, M is similar to diag(α·Id _h , M ₁), where M ₁ is cyclic and 0 ≤ h ≤ n. The paper also contains results on the generation of sporadic simple groups by minimal sets of conjugate elements.     

散在单群的一个新刻画

设G是一个有限群,K_1(G)表示G的最高阶元的阶.证明了每一个散在单群G均可被|G|和K_1(G)唯一刻画.     

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.     

特征标次数图是一种连通图的单群

对有限单群G,假设其不可约特征标次数图Δ(G)连通,且图顶点集ρ(G)=π_1∪π_2∪{p},其中|π_1|,|π_2|≥1,π_1∩π_2=θ,且π_1与π_2中顶点不相邻.证明了Δ(G)满足上面的假设的有限单群G只有4种:M_(11),J_1,PSL_3(4)或²B_2(q2B_2(q²),其中q2),其中q²一1是Mersenne素数.     

Symmetric Presentations for the Fischer Groups II: The Sporadic Groups

We give symmetric presentations for groups closely related to the three sporadic Fischer groups Fi₂₂, Fi₂₃ and Fi₂₄.Mathematics Subject Classiffications (2000). 20D08, 20D06, 20F05This research was supported by an EPSRC grant ref. GR/N27491/01     

On the Exact Spread of Sporadic Simple Groups

A group is 2-generated if it can be generated by two elements x and y. In this case y is called a mate for x. Brenner and Wiegold (1975a Brenner , J. L. , Wiegold , J. ( 1975a ). Two-generator groups. I . Michigan Math. J. 22 : 53 – 64 .[Crossref], [Web of Science ®] , [Google Scholar]) defined a finite group G to have spread r if for every set {x ₁, x ₂,…, x _r } of distinct nontrivial elements of G, there exists an element y ? G such that G = 〈 x _i , y〉 for all i. A group is said to have exact spread r if it has spread r but not r + 1. The exact spread of a group G is denoted by s(G). Ganief (1996 Ganief , M. S. ( 1996 ). 2-Generations of the Sporadic Simple Groups , Ph.D thesis , University of Natal . [Google Scholar]) in his Ph.D. thesis proved that if G is a sporadic simple group, then s(G) ≥ 2. In Ganief and Moori (2001 Ganief , M. S. , Moori , J. ( 2001 ). On the spread of the sporadic simple groups . Comm. Algebra 29 : 3239 – 3255 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) the second author and Ganief used probabilistic methods and established a reasonable lower bound for the exact spread s(G) of each sporadic simple group G. The present article deals with the search for reasonable upper bounds for the exact spread of the sporadic simple groups.     

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.     

Recognition of Finite Simple Groups S 4(q) by Their Element Orders

It is proved that among simple groups $S_4(q)$ in the class of finite groups, only the groups $S_4(3^n)$, where $n$ is an odd number greater than unity, are recognizable by a set of their element orders. It is also shown that simple groups $U_3(9)$, ${^3D}_4(2)$, $G_2(4)$, $S_6(3)$, $F_4(2)$, and ${^2E}_6(2)$ are recognizable, but $L_3(3)$ is not.     

Classifying Finite Simple Groups with Respect to the Number of Orbits Under the Action of the Automorphism Group

Abstract

Let ω(G) denote the number of orbits on the finite group G under the action of Aut(G). Using the classification of finite simple groups, we prove that for any positive integer n, there is only a finite number of (non-abelian) finite simple groups G satisfying ω(G) ≤ n. Then we classify all finite simple groups G such that ω(G) ≤ 17. The latter result was obtained by computational means, using the computer algebra system GAP.     

On Generation of Sporadic Simple Groups by Three Involutions Two of Which Commute

We prove the following result: Let G be one of the 26 sporadic simple groups. The group G cannot be generated by three involutions two of which commute if and only if G is isomorphic to M ₁₁, M ₂₂, M ₂₃, or M ^cL .     

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.     

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).     

On the Validity of Thompson's Conjecture for Finite Simple Groups

In this article, we prove a conjecture of J. G. Thompson for the finite simple group ² D _n (q). More precisely, we show that every finite group G with the property Z(G) = 1 and N(G) = N(² D _n (q)) is necessarily isomorphic to ² D _n (q). Note that N(G) is the set of lengths of conjugacy classes of G.     

Prime graphs and exponential composition of species

In this paper, we enumerate prime graphs with respect to the Cartesian multiplication of graphs. We use the unique factorization of a connected graph into the product of prime graphs given by Sabidussi to find explicit formulas for labeled and unlabeled prime graphs. In the case of species, we construct the exponential composition of species based on the arithmetic product of species of Maia and Méndez, and express the species of connected graphs as the exponential composition of the species of prime graphs.     

带作用的群系的临界结构

在本文中，我们讨论了带作用群的群系的临界结构，给出了处理允许某些作用群的群的问题的一般方法，作为应用，我们特别给出了关于幂零群及ｐ－幂零群的一些结果。