Generalized Transvection Groups And Gluing Construction For Invariant Rings In The Modular Case

Let G be a finite group acting on a vector space V of dimension n over a finite field Fq via?:G? GL(n,Fq).This action arises from the left action of G on the dual space V*defined by(g · z)(v)= z(?(g)-1· v)for g ?G,z?V*and v?V,and its extensions to Fq[V]by algebra automorphisms.In this thesis,we investigate the invariant rings Fq[V]G of finite groups G and their proper-ties in the modular case.In practical terms,we consider two problems as follows.Problem(1):How shall we calculate the invariant rings of generalized transvection groups and their Cohen-Macaulay and Gorenstein properties?In Chapter 2,constructing quotient groups and tensors,we compute the invariant rings of generalized transvection groups with a given invariant subspace which involves roots of unity.Meanwhile,we prove that these invariant rings are Cohen-Macaulay and provide the necessary and sufficient conditions for their Gorenstein properties.In Chapter 3,we prove that there are totally four kinds of generalized transvection groups if their invariant subspaces are arbitrary.Combining with top Chern classes and Dickson poly-nomials,we construct the invariant rings of these groups which are all polynomial rings.Problem(2):How shall we calculate the invariant ring and the coinvariant ring of a group G from the ones of its subgroups?In Chapter 4,we develop a method of invariant gluing construction Fq[X]Gx and Fq[Y]GY through ? to obtain the invariant ring of the group G(?)(GX×GY)(?).Moreover,we also study the Cohen-Macaulay properties of these invariant rings.In Chapter 5,a method of coinvariant gluing construction Fq[X]GX and Fq[Y]GY through ?to obtain the coinvariant ring of the group G(?)(GX × GY)(?)? is provided.