| Let G be a finite group, V n-dimensional vector space over a field F and G≤GL(V). Let F[V] be the symmetric algebra of V*(the dual of V). If we choose {x1,…,xn} as a basis for V*, then F[V] can be identified with the polynomial ring F[xi,…,xn]. Let elements of G act on F[V] as algebra automorphisms, then the subring F[F]G consisting of G-invariant polynomials, is called the ring of invariants for (G, V, F). It is well known that F[V]G is a finitely generated graded F-algebra. The purpose of invariant theory is to study the structure and properties of the ring of invariants F[V]G(for instance, Polynomiality, Cohen-Macaulayness,etc.). Here we are interested in finding the explicit generators and relations of F[V]G, especially when the characteristic of F divides the group order |G|.The first chapter contains some basic facts and motivations for this thesis. In particular,we introduce Hilbert's 14th problem and Noether's problem and present some recent important developments in the related subjects.The Chapter 2 is devoted to study the Noether's problem for certain similitude classical groups over finite fields. Following the works of L.E.Dickson, D.Carlisle and P.H.Kropholler, H.Chu et al.,we answer the Noether's problem for orthogonal (unitary and sympletic) similitude groups over a finite field by finding the explicit set of generators of fixed field.In Chapter 3, we firstly give a constructive proof of M.Kang's theorem. More precisely, we give an affirmative answer about Noether's problem by constructing an explicit set of generators of rational invariant field of finite triangular groups over any field. Secondly, for the Sylow p-subgroup of general linear groups over finite fields, we reconstruct a minimal generating set of the ring of invariants and it is different from the construction of D.Benson. Finally, we study the Dickson property of fixed fields for some finite classical groups.In Chapter 4, we generalize the classical Dickson's theorem on the ring of invariants of general linear groups over finite fields to the case of finite commutative local ring. First, we study the ring of invariants for the general linear groups over Zpm and its subgroups, and a minimal set of generators of the ring of invariants will be given. Secondly, we generalize these results to the general situation, i.e., the case of any finite commutative local rings.In Chapter 5, we study the modular vector invariant ring of finite group. More pre- cisely, we study two-dimensional and three-dimensional(characteristic of Fq is two) modular vector invariant ring of the Sylow p-subgroup of general linear groups over finite field Fq. In particular, our results partially generalize the Richman-Campell-Hughes's theorem. |
