GENERATING MAXIMAL SUBGROUPS OF SPORADIC SIMPLE GROUPS

We obtain short programs for calculating representatives of every conjugacy class of maximal subgroups of each of the sporadic simple groups of order less than 10¹⁴.     

ON THE SPREAD OF THE 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 defined a finite group G to have spread r. 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 [11] Ganief, S. 1996. 2-Generations of the Sporadic Simple Groups, Ph.D Thesis University of Natal.  [Google Scholar] in his PhD thesis proved that if G is a sporadic simple group, then s(G) ≥ 2. In this paper, by using probabilistic methods, for each sporadic simple group G we find a reasonable lower bound for s(G).

     

UNIPOTENT ALMOST CHARACTERS OF SIMPLE p-ADIC GROUPS,II

This paper contains some conjectures about the unipotent almost characters of a simple p-adic group in terms of a matrix which generalizes the nonabelian Fourier transform matrix introduced by the author in 1979.     

SOME SIMPLE GROUPS OF LIE TYPE CONSTRUCTED BY THE INNER-AUTOMORPHISM

当\[L \cong {C_l}\]，l为偶数，且l≥4,域\[\mathcal{H}\]=\[{\mathcal{H}_0}(\sqrt { - 1} )\]，其中\[{\mathcal{H}_0}\]为一有序域（或\[{\mathcal{H}_0}\]满足： а)\[\sqrt { - 1} \notin {\mathcal{H}_0},{(\sqrt { - 1} )^2} = - 1\]; b)\[ch{\mathcal{H}_0} > 3\]; c)若\[a,b \in {\mathcal{H}_0}\]，则 \[{a^2} + {b^2} \ne - 1\]，设Ф和 \[\prod :\{ {\alpha _1},{\alpha _2},...,{\alpha _l}\} \]，\[{\alpha _l}\]为长根分别为L的一组根系和素根系.令\[\{ {h_r},r \in \prod ,{e_r}r \in \Phi \} \]为L的一组Chevalley基;\[G = L({\cal H})\]为对于这一组Chevalley基在域\[{\cal H}\]上的L型 Chevalley群,令\[{w_0} = {w_{{\alpha _1}}}{w_{{\alpha _2}}}...{w_{{\alpha _{l - 1}}}}\]，其中\[{\alpha _i} \in \prod \]且为对于垂直于\[{\alpha _i}\]的平面的反射，显然\[{w_0}\]为L的Weyl群中的元素.设N为G的单项子群，\[{n_0} \in N\]，\[{n_0}\]的自然同态 像为 \[{w_0}\]，且\[{n_0}^2{\rm{ = }}I\]，存在域\[{\cal H}\]的自同构f:f(a)=a,\[a \in {{\cal H}_0}\] , \[{\rm{f(}}\sqrt { - 1} {\rm{) = }} - \sqrt { - 1} \],f在G中的扩充为G的一个域自同构(仍记为f)，且令U(V)为G对于正(负)根生 成的么幂子群，令\[{U^1}\{ u \in U|{n_0}f(u){n_0}^{ - 1} = u\} \];\[{V^1}\{ v \in V|{n_0}f(v){n_0}^{ - 1} = v\} \], 本文证明了 \[{}^2{C_l}({\cal H}) = < {U^1},{V^1} > \]为一单群.     

ON THE SUBGROUP LATTICE CHARACTERIZATION OF FINITE SIMPLE GROUPS OF LIE TYPE

Any finite simple group of Lie type is proved to be determined by its subgroup lattice up to isomorphism.     

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.     

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.     

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.     

严有效点与Henig真有效点

文章证明了严有效点等价于Henig真有效点．利用这个等价关系，得到了局部凸空间中Henig真有效点的存在性条件。纯量化特征和稠密性定理．并且改进了已知的有关结果．     

初等收敛群和拟初等收敛群

作为－／R^n上的离散收敛群中的初等群的推广,本文定义了－／R^n上一般收敛群中的初等群和拟初等群,并得到了初等收敛群、拟初等收敛群和非拟初等敛群各自的一些性质和特征。     

关于有效点与真有效点的几个定理

本文推广了Ｂｅｎｓｏｎ真有效点的定义，给出了凸锥相对内部的一个刻划，并在无尖性假设情况下，利用凸锥相对内部的刻划，证明了Ｈａｒｔｌｅｙ真有效点与改进的Ｂｅｎｓｏｎ真有效点是等价的；同时给出了一些有效点与真有效点标量化的一些结果．     

量子群上同调群的合成因子

A＝Z[v]Ω，Ω是Z[v]的由v-1和奇素数p生成的理想，U是A上的量子群，设k是特征为零的代数闭域，A→k(v|→ξ）是代数同态，ζ是p次本原根，命Uk=U与Ak的集,W是weyl群，X^ 是支配权集，本文的主要结果是如下的定理。     

同伦群和上同调群的上乘积

<正> 在[1]中,作者研究了一个(N—1)维连通空间的同伦群和同调群的密切关系,那里所考虑的同伦群的维数是在(2N—2)以内,本文可以看成[1]的继续,我偿将考虑维数在(2N—1)以上而又在(3N—3)以下的同伦群,Betti 数;和上乘积的密切关系.我们     

ON COLEMAN OUTER AUTOMORPHISM GROUPS OF FINITE GROUPS

Let G be a finite group and OutCol(G) the Coleman outer automorphism group of G(for the definition, see below). The question whether OutCol(G) is a p′-group naturally arises from the study of the normalizer problem for integral group rings, where p is a prime.In this article, some sufficient conditions for OutCol(G) to be a p′-group are obtained. Our results generalize some well-known theorems.     

POWER GROUPS OF FREE ABELIAN GROUPS OF FINITE RANK

ABSTRACT

The power set of a group G has an induced semigroup structure, some subsets of which will form groups in their own right. We are especially interested in such subsets that are maximal. We demonstrate that even when G is a free abelian group of finite rank, the groups which arise in this way can be diverse profinite abelian groups.     

泛欧氏空间的Clifford群、扭群、旋群及它们的李代数

本文研究了泛欧氏空间的Clifford群、扭群、旋群,它们为Clifford代数中选出极好的一类子群.利用Clifford代数理论方法,获得了泛欧氏空间中Clifford群、扭群、旋群及其李代数的结构及它们之间的关系,并且得到了它们的李代数.