Finite groups in which the normalizers of Sylow 3-subgroups are of odd or primary index

We determine the composition factors of the finite groups in which the normalizers of Sylow 3-subgroups are of odd or primary index.     

Finite groups acting 2-transitively on their sylow subgroups

In this paper we have completely determined: (1) all almost simple groups which act 2-transitively on one of their sets of Sylow p-subgroups. (2) all non-abelian simple groups T whose automorphism group acts 2-transitively on one of the sets of Sylow p-subgroups of T. (3) all finite groups which are 2-transitive on all their sets of Sylow subgroups. The first author acknowledges the support of OPR Scholarship of Australia The second author is supported by the National Natural Science Foundation of China. Thanks are also due to the Department of Mathematics, the University of Western Australia, where he did his part of this work for its hospitality     

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.     

The p-local ranks of finite simple groups with abelian Sylow p-subgroups

As a continuation of previous work, in this paper, we mainly study the finite simple groups which have abelian Sylow subgroups in term of p-local rank, especially a group theoretic characterization will be given.     

Rationality and strong reality for Sylow 2-subgroups of Weyl and alternating groups

It is proved that values of all complex characters for a Sylow 2-subgroup P of any Weyl group are rational, and every element of P is a product of two involutions in P. Similar results hold also for Sylow 2-subgroups of alternating groups.Supported by RFBR grant No. 03-01-00905.Translated from Algebra i Logika, Vol. 44, No. 1, pp. 44–53, January–February, 2005.     

Mod 7 Cohomology Algebra of Held Simple Group

We calculate the mod7 cohomology algebra of the Held simple group, whose Sylow 7-subgroups are extraspecial of order 7³ and exponent 7. Our method depends on the analysis of the Carlson module of a regular class belonging to a system of parameters.     

On Weakly Factorizable Groups

Groups with complemented subgroups, which are also called completely factorizable groups, were studied by P. Hall, S. N. Chernikov, and N. V. Chernikova (Baeva). For complete factorizability, it is sufficient (Theorem 1) that each proper subgroup have a normal complement in some larger subgroup. A group is said to be weakly factorizable if each of its proper subgroups is complemented in some larger subgroup; the problem of describing finite groups with this property is posed (Question 8.31) in the Kourovka Notebook. Some properties of these groups are considered. The question is studied for Sylow p-subgroups of Chevalley-type groups of characteristic p. The main theorem, Theorem 2, establishes the weak factorizability of the Sylow p-subgroups in the symmetric and alternative groups and in the classical linear groups over fields of characteristic p> 0, excluding the unitary groups of odd dimension > p.     

在其Sylow子群上作用双传递的有限群

本文完全确定了下述类型的有限群：（１）对于某个素数ｐ，在其Ｓｙｌｏｗｐ－子群的集合上作用双传递的所有几乎单群；（２）所有的非交换单群Ｔ，其自同构群在Ｔ的某个Ｓｙｌｏｗｐ－子群的集合上作用双传递；（３）在其所有的Ｓｙｌｏｗ子群上双传递的全部有限群．     

On Some Groups with Finite Involution Saturated with Finite Simple Groups

In the paper, on the basis of the notion of saturation introduced in [1], we give a characterization of locally finite simple groups with an Abelian Sylow 2-subgroup in the classes of mixed and periodic groups. A part of the main results was announced in [2]. Theorem 2 of the paper generalizes results of [3, 4].     

Recognizability of finite simple groups <Emphasis Type="Italic">L</Emphasis><Subscript>4</Subscript>(2<Superscript>m</Superscript>) and <Emphasis Type="Italic">U</Emphasis><Subscript>4</Subscript>(2<Superscript>m</Superscript>) by spectrum

It is proved that finite simple groups L₄(2^m), m ⩾ 2, and U₄(2^m), m ⩾ 2, are, up to isomorphism, recognized by spectra, i.e., sets of their element orders, in the class of finite groups. As a consequence the question on recognizability by spectrum is settled for all finite simple groups without elements of order 8. Supported by RFBR (grant Nos. 05-01-00797 and 06-01-39001), by SB RAS (Complex Integration project No. 1.2), and by the Ministry of Education of China (Project for Retaining Foreign Expert). Supported by NSF of Chongqing (CSTC: 2005BB8096). __________ Translated from Algebra i Logika, Vol. 47, No. 1, pp. 83–93, January–February, 2008.     

具有阿贝尔Sylow 2-子群的有限群的整群环的正规化子性质

Mazur猜想:具有阿贝尔Sylow 2-子群的有限群有正规化子性质.设G是一个有限群,N是G的一个正规子群且Z(G/N)仅有平凡单位,本文建立了由Z(G/N)中单位诱导的G的自同构与N的Coleman自同构之间的联系,在此基础上证明了若G是一个具有阿贝尔Sylow 2-子群的有限群且Z(G/F*(G))仅有平凡单位,则Mazur猜想对G成立.     

Characters and Sylow 2-subgroups of maximal class revisited

We give two ways to distinguish from the character table of a finite group G if a Sylow 2-subgroup of G has maximal class. We also characterize finite groups with Sylow 3-subgroups of order 3 in terms of their principal 3-block.     

On 2-sylow intersections

LetG be a finite group. We say thatG satisfiesI(k) if anyk+1 distinct Sylow 2-subgroups ofG interesect trivially. In this paper we classify all finite groups satisfyingI (3), and all finite simple groups satisfyingI (5).     

Relative Projectivity of Modules and Cohomology Theory of Finite Groups

In the modular representation theory of finite groups the theory of projectivity relative to subgroups is of fundamental importance. To generalize this notion the theory of projectivity relative to modules was introduced by the first author. Our aim is to show some aspects of cohomology theory of finite groups concerning projectivity of modules relative to both subgroups and modules. We shall give some applications to the cohomology theory; especially we shall calculate the mod 2 cohomology algebras of finite groups with wreathed Sylow 2-subgroups.     

Recognizing Abelian Sylow Subgroups in Finite Groups

Let p be a prime. We prove that if a finite group G has non-abelian Sylow p-subgroups, and the class size of every p-element in G is coprime to p, then G contains a simple group as a subquotient which exhibits the same property. In addition, we provide a list of all the simple groups and primes such that the Sylow p-subgroups are non-abelian and all p-elements have class size coprime to p.     

The Cohomologies of the Sylow 2-Subgroups of a Symplectic Group of Degree Six and of the Third Conway Group

The third Conway group Co ₃ is one of the 26 sporadic finite simple groups. The cohomology of its Sylow 2-subgroup S is computed, an important step in calculating the mod 2 cohomology of Co ₃. The spectral sequence for the central extension of S is described; it collapses at the sixth page. Generators are described in terms of the Evens norm or transfers from subgroups. The central quotient S′ = S/2 is the Sylow 2-subgroup of the symplectic group Sp ₆(𝔽₂) of six-by-six matrices over the field of two elements. The cohomology of S′ is computed and is detected by restriction to elementary abelian 2-subgroups.