Finite Flat Group Schemes over Local Artin Rings

Let R be a local Artin ring with maximal ideal m and residue class field of characteristic p > 0. We show that every finite flat group scheme over R is annihilated by its rank, whenever m ^p = pm = 0. This implies that any finite flat group scheme over an Artin ring the square of whose maximal ideal is zero, is annihilated by its rank.     

Flat Modules over Group Rings of Finite Groups

Let k be a commutative ring of coefficients and G be a finite group. Does there exist a flat k G-module which is projective as a k-module but not as a k G-module? We relate this question to the question of existence of a k-module which is flat and periodic but not projective. For either question to have a positive answer, it is at least necessary to have |k| ≥ ?_ω. There can be no such example if k is Noetherian of finite Krull dimension, or if k is perfect.     

二面体群上的Hopf Ore扩张

In this paper, the Hopf Ore extension and corresponding module extension of the group algebra over dihedral group are studied. It turns out that the 1-dimensional and 2- dimensional simple representations can both be extended to the simple representations over a class of Hopf Ore extension.     

有限环上典型群的阶

本文通过“取模”，“取值和”，“取积”等方法，将有单位元的有限环Ｒ上典型群阶的计算转化为有限域上典型群阶的计算，并计算了Ｒ上ｎ－维自由模Ｖ_ｎ（Ｒ）中ｋ－维自由子模的个数．     

低维环上的群环

本文研究了具有较小Krull维数和同调维数的环上群环的模结构,给出了低维系数环上群环的某些外部特征.     

Finite groups of matrices over group rings

We investigate certain finite subgroups of , where is a finite nilpotent group. Such a group gives rise to a -module; we study the characters of these modules to limit the structure of . We also exhibit some exotic subgroups .

     

Association Schemes with at Most Two Nonlinear Irreducible Characters and Applications to Finite Groups

An irreducible character χ of an association scheme is called nonlinear if the multiplicity of χ is greater than 1. The main result of this paper gives a characterization of commutative association schemes with at most two nonlinear irreducible characters. This yields a characterization of finite groups with at most two nonlinear irreducible characters. A class of noncommutative association schemes with at most two nonlinear irreducible character is also given.     

特征为2的有限域上二次曲面的结合方案的注记

By using nondegenerate and degenerate quadrics in projective space over finite fields of characteristic 2, some association schemes were constructed and their parameters were computed by the authors (see Adv. in Math., 3(2000), 120-128 and Acta Math. Appl. Sinica, 1(1999), 96-103). In this note, their polynomial properties, eigenmatrices, imprimitivities, association subschemes and related quotient association schemes are studied.     

Generic Idempotent Modules for a Finite Group

Let G be a finite group and k an algebraically closed field of characteristic p. Let F _U be the Rickard idempotent k G-module corresponding to the set U of subvarieties of the cohomology variety V _G which are not irreducible components. We show that F _U is a finite sum of generic modules corresponding to the irreducible components of V _G . In this context, a generic module is an indecomposable module of infinite length over k G but finite length as a module over its endomorphism ring.     

对称群的分歧数据系统分类

含特征标的分歧数据系统能被用来分类PM箭图Hopf代数.本文给出了对称群Sn(n≠6)上含特征标的分歧数据系统的同构类个数的计算公式.     

环的约化群与群环上的投射模

本文系统地研究群环的约化群，利用约化群刻划了群环上模的结构。主要结果：（１）Ｒ为交换半遗传环且Ｋ＿０Ｒ为挠群ｉｆｆ对任何有限生成半自反Ｒ－模Ｐ，ｓ＞０，使得．（２）设Ｒ为半局部Ｄｅｄｅｋｉｎｄ环，Ｇ为有限生成Ａｂｅｌ群，则Ｋ＿０ＲＧ为挠群ｉｆｆ如果Ｇ有素数ｐ阶元，则（３）如果Ｋ＿０ＲＧ为挠群，［Ｇ∶Ｈ］＜∞，则对任何，有．这里Ｒ为整环，Ｌ为其分式域。     

Comparing First Order Theories of Modules over Group Rings

We consider R‐torsionfree modules over group rings RG, where R is a Dedekind domain and G is a finite group. We compare the (first order) theory T of al these modules and the theory T₀ of the finitely generated ones (so of RG‐lattices). It is easy to realize that they are equal iff R is a field. The obstruction is the existence of R‐divisible R‐torsionfree RG‐modules. Accordingly we consider R‐reduced R‐torsionfree RG‐modules for a local R. We show that the key conditions ensuring that their theory equals T₀ are: (1) RG‐lattices have a finite representation type; (2) each attice over the completion R̂G is isomorphic to the completion of some RG‐lattice.Some related questions are discussed.     

Automorphisms of Association Schemes of Quadratic Forms over a Finite Field of Characteristic Two

Let X_n denote the set of quadratic forms in n variables over a finite field F_q of characteristic 2, and X_n the association scheme on X_n by defining the relations with respect to the type of quadratic forms. We prove that every automorphism of X_n is of the form X P^tXP + Y for all X X_n, where P GL_n(F_q), is an automorphism of F_q, and Y X_n.2000 Mathematics Subject Classification: 05E30Supported by the National Natural Science Foundation of China (10271039), Hobei Natural Science Foundation (102132) and Hebei Education Committee.     

差分格式的余项效应分析及其应用

本文对已有的差分格的色散关系和群速度效应的Ｆｏｕｒｉｅｒ分析提出了置疑，指出症结所在并予以纠正，并且利用差分格式的Ｍｏｄｉｆｉｅｄ ＰＤＥ思想，提出了一种新的构造性差分格式分析方法－差分格式余项效应分析。这种方法基于差分格式的耗散关系和色散关系的，人有明显的构造性和现实意义。     

自同构群是循环群被交换群扩张的有限群

设Ｃ是有限群，ＡｕｔＧ＝ＡＢ，，Ａ是交换群且每Ｓｙｌｏｗ子群的秩≤２，Ｂ是循环群，本文得出了Ｇ的结构，特别地，证明了ＡｕｔＧ是秩≤２的交换群时，Ｇ循环。     

Ramification of a finite flat group scheme over a local field

In this paper, we analyze ramification in the sense of Abbes-Saito of a finite flat group scheme over the ring of integers of a complete discrete valuation field of mixed characteristic (0,p). We deduce that its Galois representation depends only on its reduction modulo explicitly computed p-power. We also give a new proof of a theorem of Fontaine on ramification of a finite flat Galois representation, and extend it to the case where the residue field may be imperfect.     

Finite Group with Two Supersolvable Subgroups of Coprime Indices

Ｔｈｒｏｕｇｈｏｕｔｔｈｉｓｐａｐｅｒ,Ｇｉｓａｆｉｎｉｔｅｇｒｏｕｐ ,ｐｉｓａｐｒｉｍｅｉｎπ(Ｇ) ,ａｎｄＭ <·ＧｄｅｎｏｔｅｓｔｈａｔＭｉｓａｍａｘｉｍａｌｓｕｂｇｒｏｕｐｏｆＧ .Ａ/ＢｉｓｃａｌｌｅｄａｓｅｃｔｉｏｎｏｆＧｉｆＢ Ａ≤Ｇ .Ｔｈｅｒｅｓｔｏｆｎｏｔａｔｉｏｎｓｉｓｓｔａｎｄａｒｄａｎｄｔａｋｅｎｍａｉｎｌｙｆｒｏｍ [1 ]ａｎｄ [2 ] .　Ｉｎ [3] ,Ｗｅｉｌａｎｄｔｐｒｏｖｅｄｔｈｅｗｅｌｌ ｋｎｏｗｎｔｈｅｏｒｅｍ :Ｇｉｓｓｏｌｖａｂｌｅｉｆｉｔｃｏｎｔａｉｎｓｔｗｏｎｉｌｐｏｔｅ…     

Normal Bases and Their Dual-Bases over Finite Fields

In this paper, we prove the following results： 1） A normal basis N over a finite field is equivalent to its dual basis if and only if the multiplication table of N is symmetric; 2） The normal basis N is self-dual if and only if its multiplication table is symmetric and Tr（α^2） = 1, where α generates N; 3） An optimal normal basis N is self-dual if and only if N is a type-Ⅰ optimal normal basis with q = n = 2 or N is a type-Ⅱ optimal normal basis.     

具有4pq阶自同构群的有限群

本文讨论了自同构群阶为4pq(p,q为不同奇素数)的有限群,得出了它们的构造.     

关于有限群两个幂零子群积的问题

众所周知,有限群的两个幂零子群的积不一定是幂零的．本文研究了Ｅｎｇｅｌ条件对两个幂零子群的影响,得到两个幂零子群的积为幂零群的几个充分条件。