This article is an expanded version of the material presented there. The main topic is the calculation of the invariant ring of a finite group acting on a polynomial ring by linear transformations of the indeterminates. By "calculation" I mean finding a finite system of generators for the invariant ring, and (optionally) determining structural properties of it. In this exposition particular emphasis is placed on the case that the ground field has positive characteristic dividing the group order. We call this the modular case, and it is important for several reasons. First, many theoretical questions about the structure of modular invariant rings are still open. I will address the problems which I consider the most important or fascinating in the course of the paper. Thus it is very helpful to be able to compute modular invariant rings in order to gain experience, formulate or check conjectures, and gather some insight which in fortunate cases leads to proofs. Furthermore, the computation of modular invariant ring can be very useful for the study of cohomology of finite groups (see Adem and Milgram [1]). This exposition also treats the nonmodular case (characteristic zero or coprime to the group order), where computations are much easier and the theory is for the most part settled. There are also various applications in this case, such as the solution of algebraic equations or the study of dynamical systems with symmetries (see, for example, Gatermann [11], Worfolk [26]).

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. The theorem of Hochster and Roberts says that, for every module V of a linearly reductive group G over a field K, the invariant ring K[V ] G is Cohen--Macaulay. We prove the following converse: if G is a reductive group and K[V ] G is Cohen--Macaulay for every module V , then G is linearly reductive. Introduction Linearly reductive groups play a prominent role in invariant theory. Hilbert's original proof of the finiteness theorem [3] works for linearly reductive groups. Much later, Hochster and Roberts [5] proved that if G is a linearly reductive group and V a G- module, then the invariant ring K[V ] G is Cohen--Macaulay, i.e., a free module over the subalgebra generated by a homogeneous system of parameters. The goal of this note is to prove the converse. Namely, we will show that if G is a reductive group and K[V ] G is Cohen--Macaulay for every G-module V , then G must be linearly reductive. This becomes false if the hypothesis that G be reductive is dropped (see Remar...

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Let G be a linear algebraic group over a field K of characteristic 0. An integer m is called a global degree bound for G if for every linear representation V the invariant ring K[V] G is generated by invariants of degree at most m. We prove that if G has a global degree bound, then G must be finite. The converse is well known from Noether’s degree bound.

Let G be an algebraic group acting on an irreducible variety X. We present an algorithm for computing the invariant field k(X) G. This algorithm leads to a new, constructive proof of a theorem of Rosenlicht, which says that almost all orbits can be separated by rational invariants.