Invariant theory: Mostly in the modular case

The intent of this thesis is to study the essential problems of invariant theory of finite groups: find relationships between the structure of a group and of its invariant ring; find the generators for the invariant ring or, failing to do so, find an upper bound on the degrees of a set of generators. We study the polynomial and complete intersection properties of invariant rings with a focus on permutation representations, and the Cohen-Macaulay property of the invariant rings. The study of properties of the two-dimensional vector invariant rings of Abelian p-groups is a major topic of the thesis. We also study the degree bounds of the invariant rings. We give a new degree bound from which a result of Fleischman and a result of Goebel for regular representations follow. We also prove a result generalizing a theorem of Richman.