A New Cover of the 3-Local Geometry of Co1

The 3-local geometry of the sporadic simple group Co₁ has been known to have a cover with a flag-transitive automorphism group which is a nonsplit extension of an elementary Abelian 2-group of rank 24 (the Leech lattice modulo 2) by Co₁. It was conjectured that was simply connected. We disprove this conjecture by constructing a double cover of . The automorphism group of is of the shape . However, it is not isomorphic to the involution centralizer of the Monster sporadic simple group.     

Semiclassical Parabolic Systems Related to M24

Semiclassical parabolic systems belong to diagram geometries whose rank two residues are generalized n-gons or a triple cover of the generalized 4-gon for Sp(4, 2), and the latter really occurs. So semiclassical parabolic systems are a generalization of Tits chamber systems of classsical type. We classify all such systems of rank four assuming in addition that there is some residue belonging to a subdiagram of type . Among others iteresting examples showing up are systems for Co₁ and M(24).     

A rank six geometry related to the McLaughlin sporadic simple group

In the literature, there are but a few incidence geometries on which the McLaughlin sporadic group acts as a flag-transitive automorphism group. Their highest rank is four. In the present paper, we construct a geometry of rank six on which acts flag-transitively and which has the following diagram.      

On the Classification of Flag-Transitive c.c*- Geometries

Let ( G) be a flag-transitive c.c*-geometry whose point-stabilizer is not an affine group. We list all known examples and show that, if (, G) is a minimal unknown example, then G is an almost simple group and the commutator subgroup G is a simple group of Lie type.     

P-Geometries of Rank 3

Under minor extra assumptions, we classify P-geometries of rank 3. Previously, a classification was known only for flag-transitive P-geometries.     

A characterization theorem for sporadic simple groups

We show that each sporadic simple group G can be uniquely determined by the set of the orders of the maximal abelian subgroups of G.     

A Criterion for Non-Simplicity of Groups with Involutions

We settle Question 10.61, posed by A. Sozutov in the Kourovka Notebook, for the case where .     

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 .     

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．     

The 13-modular character table of 2.Suz.2

We compute the 13-modular character table of the bicyclic extension 2.Suz.2 given in the Atlas. This completes the last unknown modular character table of a bicyclic extension of the sporadic simple group Suz. We explain an issue of compatibility between different modular tables and partially recompute the 7-modular table of 2.Suz.2 in order to avoid such problems.     

2-Cohomologies of the groups SL(n, q)

Let G = SL(n, q), where q is odd, V be a natural module over G, and L = S²(V) be its symmetric square. We construct a 2-cohomology group H²(G, L). The group is one-dimensional over F _q if n = 2 and q ≠ 3, and also if (n, q) = (4, 3). In all other cases H²(G, L) = 0. Previously, such groups H²(G, L) were known for the cases where n = 2 or q = p is prime. We state that H²(G, L) are trivial for n ⩾ 3 and q = p^m, m ⩾ 2. In proofs, use is made of rather elementary (noncohomological) methods. __________ Translated from Algebra i Logika, Vol. 47, No. 6, pp. 687–704, November–December, 2008.     

On the Finite Group with the Index of Maximal Subgroups of the Monster

Abstract

We study the problem concerning the influence of the index of maximal subgroup or the degree of primitive permutation representation of the finite simple groups on the structure of a group. Let G be a finite group and s be the index of maximal subgroup of the Monster M. In this paper, we prove that there exists an epimorphism from G to M or A _s if G has the primitive permutation representation of degree s, and as a consequence we prove that the Monster is determined by every s.     

A Geometric Rational Form for Artin Groups of FC Type

If A is an Artin group whose poset of finite type special subgroups is a flag complex, then A is said to be of FC type. Such groups act cocompactly on a CAT(0) cubical complex with finite type Artin groups as stabilizers. We use the geometry of this complex to obtain a rational normal form for the group.     

On Flat Flag-Transitive c.c *-Geometries

We study flat flag-transitive c.c ^*-geometries. We prove that, apart from one exception related to Sym(6), all these geometries are gluings in the meaning of [6]. They are obtained by gluing two copies of an affine space over GF(2). There are several ways of gluing two copies of the n-dimensional affine space over GF(2). In one way, which deserves to be called the canonical one, we get a geometry with automorphism group G = 2²ⁿ · L _n(2) and covered by the truncated Coxeter complex of type D ₂ ⁿ . The non-canonical ways give us geometries with smaller automorphism group (G ≤ 2²ⁿ · (2 ^n?1)n) and which seldom (never ?) can be obtained as quotients of truncated Coxeter complexes.     

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.     

A New Characterization of A 22 by Its Spectrum

It is proved that a finite group whose element order set is the same as that of an alternating group A ₂₂ of degree 22 is isomorphic to A ₂₂.     

A Geometric Characterization of the Perfect Ree-Tits Generalized Octagons

Given a generalized octagon which satisfies certain geometricconditions, we show that it must be isomorphic to the Ree–Titsoctagon arisingnaturally from a Ree group of type ²F₄ over aperfect field of characteristic 2, in which the Frobenius automorphismhas a square root. 1991 Mathematics Subject Classification:51E12.     

A Characterization of Root Classes of Groups

We prove that a class of groups is root in a sense of K. W. Gruenberg if, and only if, it is closed under subgroups and Cartesian wreath products. Using this result we obtain a condition which is sufficient for the generalized free product of two nilpotent groups to be residual solvable.     

用元素阶的集合和Sylow子群的正规化子阶的集合刻画奇数维的有限正交单群

本文研究了一类单群的数量刻画问题.运用有限单群分类定理及群的元素阶的素图表示的结构定理,获得了有限群G同构于有限正交单群PΩ2n+1(q)(n≥13),当且仅当群的元素的阶的集合一致和群的Sylow子群的正规化子的阶之集合一致.在某种意义推进施武杰教授的一个猜想.