BN-Pairs And Invariants Of Special Subgroups Of Classical Groups

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.