【作者】 秦玉芳；

【导师】 南基洙；

【作者基本信息】 大连理工大学 ， 基础数学， 2009， 博士

【摘要】 典型群及其特殊子群是代数学的重要研究对象。本文主要研究局部环上典型群的BN-对,二面体群的BN-对和不变式环,有限域上典型群的极大子群和根子群的有理不变式域的结构。第一章考虑任意局部环上典型群的BN-对问题,构造了局部环上一般线性群、辛群、正交群的BN-对,并且证明了局部环上一般线性群与对应的BN-对之间满足群交换图关系。第二章研究群环R上酉群的结构及其应用。首先确定群环R上Hermitian矩阵的合同标准型。其次,讨论R上酉群的BN-对和计数问题。最后,作为应用构造了一个Cartesian认证码,并计算了相应码的参数。第三章讨论一类重要的有限反射群-二面体群D_n的BN-对和Building。并且,在不变式环方面证明一个有趣的结论:当n=3m,m∈Z⁺时,D_n的BN-对中的子群B的不变式环是D_n的不变式环的一个整扩张,并且R₂作为R₁-自由模的基底的个数等于[D_n:B],从而刻划了群和对应的不变式环之间的数值关系。第四章在有限典型群的极大子群和根子群的有理不变式域方面做了一些讨论。通过构造有理不变式域的超越基,从而回答了有限典型群的G₁和G₂类极大子群及其根子群Noether问题。另外,证明有限域上一般线性群的C₁类极大子群和特殊线性群的根子群的不变式环是多项式的。更多还原

【Abstract】 It is well known that the classical groups and their special subgroups are important research objects.In this thesis,we study BN-pairs of classical groups over local rings, BN-pairs and rings of polynomial invariants of dihedral groups,the rational invariant fields of the maximal subgroups and root subgroups of classical groups over finite fields.In Chapter 1,we investigate BN-pairs of classical groups over local rings.We construct the BN-pairs arising from classical groups of linear,orthogonal and symplectic type over local rings.Furthermore,we consider the relations between the general linear groups and the corresponding BN-pairs and show that they satisfy a commutative diagram of groups.In Chapter 2,we study the structure of unitary groups over finite group rings R and its applications.First,we determine all the normal forms of Hermitian matrices over R.Secondly,we obtain BN-pairs and some Anzahl theorems of unitary groups over R. Finally,as applications,we construct a.Cartesian authentication code and compute its parameters.In Chapter 3,we discuss the BN-pairs and buildings of dihedral groups.And we also obtain an interesting conclusion for the invariant rings of dihedral groups D_n with n=3m,m∈Z⁺.Denote the invariant rings of D_n by R₂ and that of the subgroup B by R₁.Then R₁ is an integral extension over R₂.Moreover,the ring R₁,as a free R₂-module, has a basis with[D_n:B]elements.Consequently,we determine the relations between groups and invariant rings.In Chapter 4,we focus on the structure of the rational invariant fields of the maximal subgroups and root subgroups of finite classical groups.We answer Noether’s problem for those maximal subgroups of classes C₁ and C₂ and the root subgroups by constructing the explicit,transcendental bases of their invariant subfields.In addition,we prove that the invariant rings of the maximal subgroups of class C₁ of the general linear group are polvnomial,and so do the invariant rings of root subgroups of the special linear groups.更多还原