Magma contains a powerful new module for computing with invariant rings of finite groups. This module has come about thanks to a visit to the Magma group in 1996 by Gregor Kemper (IWR, Universität Heidelberg), to whom the group expresses much gratitude. The algorithms for invariant theory in Magma are based on those in the Invar package written in Maple, implemented by G. Kemper [G. Kemper, Calculating Invariant Rings of Finite Groups over Arbitrary Fields, JSC 21 (1996), 351--366], but also include many new ideas and improvements which are described in detail in a new paper [G. Kemper & A. Steel, Some Algorithms in Invariant Theory of Finite Groups (preprint---mail magma@maths.usyd.edu.au for details)]. Since a detailed understanding of this new paper is useful for better understanding of many of the functions in the chapter, it is recommended the paper be perused by anyone wishing to make serious applications of the functions. The paper will be referred to in this chapter as the "Magma Invariant Theory paper".

The primary goal of invariant theory in Magma is the computation of the invariant ring of a given finite matrix or permutation group over a ground field of arbitrary characteristic. Of particular interest is the modular case, i.e., the case where the characteristic of the ground field K divides the group order, since in that case there are still many theoretical questions unanswered. Magma also contains easy algorithms to calculate properties of modular invariant rings, such as the Hilbert series, the Cohen-Macaulay property, depth, and free resolutions.

The approach to calculating the invariant ring is broken up into two major steps: first a system of primary invariants is constructed, i.e., homogeneous invariants f_1, ..., f_n which are algebraically independent, such that the invariant ring is a finitely generated module over A = K[f_1, ..., f_n]. In the next step we calculate secondary invariants, which are generators of the invariant ring as an A-module.

Throughout this chapter, K will be a field and G is a finite matrix or permutation group acting from the right on the n-dimensional vector space V isomorphic to K^n with basis x_1, ..., x_n. Thus G also acts on the symmetric algebra K[V] = S(V), which is the multivariate polynomial ring K[x_1, ..., x_n] in the variables x_1, ..., x_n. The invariant ring R = {f in K[V] | f^(sigma) = f forall sigma in G} is denoted by K[V]^G.