On blocks with K(B)-L(B) = 1

We use the method of local representation and original method of Brauer to study the block with K(B) - L(B) = 1, and get some properties on the defect group and the structure of this kind of blocks. Then, we show that K(B) conjecture holds for this kind of blocks.     

The structure of saturated critical blocks

A 2‐connected graph G is a critical block if G ? v is not 2‐connected for every vertex v ∈ V(G). A critical block G is a saturated critical block if G + e is not a critical block for any new edge e. The structure of all saturated critical blocks and a procedure for constructing every saturated critical block are determined. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 223–237, 2003     

A note on blocks with dihedral defect groups

We prove that the 2-regular subspace of the center of a block with a dihedral defect group is multiplicatively closed. Received: 25 April 2008     

On Brauer’s problem 21

For ap-blockB of a finite groupG, we give a bound of the order of its defect groupD in terms ofk(B), the number of the irreducible ordinary characters inB.     

The extension of the first main theorem for π-blocks

Let π be a set of primes and G a π-separable group. Isaacs defines the Bπ characters, which can be viewed as the "π-modular" characters in G, such that the Bp′ characters form a set of canonical lifts for the p-modular characters. By using Isaacs' work, Slattery has developed some Brauer's ideals of p-blocks to the π-blocks of a finite π-separable group, generalizing Brauer's three main theorems to the π-blocks. In this paper, depending on Isaacs' and Slattery's work, we will extend the first main theorem for π-blocks.     

Brauer第24问题的推广

本文对Ｂｒａｕｅｒ第２４问题［１］作了推广．利用Ｄａｄｅ关于循环块的理论得到如下结果：设Ｇ是有限群，Ｐ是Ｇ的循环Ｓｙｌｏｗｐ子群．设｜Ｐ｜＝ｐａ，ａ为正整数．令Ｐｉ为Ｐ中唯一的ｐｉ阶子群，１ｉａ．则｜Ｃｌ（Ｇ）｜＝｜Ｃｌ（ＮＧ（Ｐｉ））｜的充分必要条件为ＰｉＧ．     

Some Results on the Problems of Block Theory

In this paper, we obtain two results on Brauer‘s k (B) - problem about finite groups under some conditions. Furthermore, we obtain that Olsson‘s conjecture holds under the same conditions on the finite groups.     

On Blocks with Nilpotent Coefficient Extensions

For modular group algebras over an arbitrary field we define new type of blocks: blocks with nilpotent extensions, and describe their source algebras. To do it, a general pattern is proposed for relations between the source algebra of a block and the source algebra of a block appearing in its decomposition in a suitable extension of the field of coefficients.     

The strong Frobenius numbers for cyclic defect blocks are equal to one

Abstract

The purpose of this note is to provide a reference for the fact that the strong Frobenius number, in the sense of Eaton and Livesey, of a block of a finite group with a cyclic defect group is equal to 1. This answers a question of Farrell and Kessar.     

The structure of geodetic blocks with diameter two

In this paper we prove that all geodetic blocks of diameter two can be divided in four types, i.e., Moore graphs with diameter two, regular pyramids with altitude 2, type AP and type PP. We also give the answers to the questions posed by J. G. Stemple in 1974.     

正则半群上的矩形群同余

文[1]中PetrichM定义了同余的核与迹，用它们描述了逆半群上的同余，Gomes在文[2]中定义了同余的核与超迹并描述了正则半群上的R-幂单(R-unipo-tent)同余，本文利用同余的核与超迹描述正则半群上的另一类重要同余，即矩形群同余．     

The structure of abelian pro-Lie groups

A pro-Lie group is a projective limit of a projective system of finite dimensional Lie groups. A prodiscrete group is a complete abelian topological group in which the open normal subgroups form a basis of the filter of identity neighborhoods. It is shown here that an abelian pro-Lie group is a product of (in general infinitely many) copies of the additive topological group of reals and of an abelian pro-Lie group of a special type; this last factor has a compact connected component, and a characteristic closed subgroup which is a union of all compact subgroups; the factor group modulo this subgroup is pro-discrete and free of nonsingleton compact subgroups. Accordingly, a connected abelian pro-Lie group is a product of a family of copies of the reals and a compact connected abelian group. A topological group is called compactly generated if it is algebraically generated by a compact subset, and a group is called almost connected if the factor group modulo its identity component is compact. It is further shown that a compactly generated abelian pro-Lie group has a characteristic almost connected locally compact subgroup which is a product of a finite number of copies of the reals and a compact abelian group such that the factor group modulo this characteristic subgroup is a compactly generated prodiscrete group without nontrivial compact subgroups.Mathematics Subject Classification (1991): 22B, 22E     

Riordan群的反演链及在组合和中的应用

利用函数复合关系,本文在Ｒｉｏｒｄａｎ群中引入Ｒｉｏｒｄａｎ反演链的概念及其Ｒｉｏｒ－ｄａｎ反演链存在的充要条件,给出计算组合和式的递推方法．进一步讨论了二项式系数所对应的Ｒｉｏｒｄａｎ反演链问题,建立了一个Ｒｉｏｒｄａｎ求和公式,该式蕴含了某些与Ｆｉｂｏｎａｃｃｉ数相关的恒等式在内的一系列组合恒等式     

Morita equivalences between some blocks for -solvable groups

We prove that any Morita equivalence between some blocks with Abelian defect groups and cyclic inertia quotients for -solvable groups is basic.

     

On blocks with trivial source simple modules

Motivated by an observation in Danz and Külshammer (2009) [3], we determine the source algebra, and therefore all the structure, of the blocks without essential Brauer pairs where the simple modules of all the Brauer corespondents have trivial sources.     

The Hyperoctahedral Group,Symmetric Group Representations and the Moments of the Real Wishart Distribution

In this paper, we compute all the moments of the real Wishart distribution. To do so, we use the Gelfand pair (S_2k,H), where H is the hyperoctahedral group, the representation theory of H and some techniques based on graphs.     

半正规n-极大子群对有限群结构的影响

设△↓n(G）为有限群G的n次极大子群的全体。1．若△↓4（G）中的子群均在G中半正规,则下述结论之一成立：（1）G是可解群;（2）G／φ（G）＝A5,（3）G/φ(G)=PSL(2,13);(4)G／φ（G）＝PSL（2,p),满足p=4p1 1=6p2-1,这里p1≥43,p2≥29;(5)G/φ(G)=PSL(2,p),满足p=6p1 1=4p2-1,这里p1≥7,p2≥11.2。2．设3不属于π（G）,若△↓（G）中的子群均在G中半正规,则G是可解群,或G／φ(G)=Sz(2^3).     

The cohomology group of weak entwining structure

In this paper, we reveal that a weak entwining structure admits a rich cohomology theory. As an application we compute the cohomology of a weak entwining structure associated to a weak coalgebra-Galois extension.