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Abstract. The MAPLE package “Janet ” ⋆ comes in two parts, the first dealing with polynomials and commutative algebra, the second with linear PDEs. Here the first part, called “Involutive”, is described. Amongst others it contains a MAPLE and a C++ implementation of the involutive technique for polynomial modules as an alternative for conventional Gröbner basis techniques. 1

Algebra" in 1967 [Lee70]. Its proceedings contain a survey of what had been tried until then [Neu70] but also some papers that lead into the Decade of discoveries (1967--1977). At the Oxford conference some of those computational methods were presented for the first time that are now, in some cases varied and improved, work horses of CGT systems: Sims' methods for handling big permutation groups [Sim70], the Knuth-Bendix method for attempting to construct a rewrite system from a presentation [KB70], variations of the Todd-Coxeter method for the determination of presentations of subgroups [Men70]. Others, like J. D. Dixon's method for the determination of the character table [Dix67], the p-Nilpotent-Quotient method of I. D. Macdonald [Mac74] and the Reidemeister-Schreier method of G. Havas [Hav74] for subgroup presentations were published within a few years from that conference. However at least equally important for making group theorists aware of CGT were a number of applications of...

this paper. Our special interest is in groups which are presented to us as being generated by a small set of elements, be these permutations of vertices of a graph, matrices describing automorphisms of linear codes, or classes of homotopies of a knot described only by the relations that they satisfy (see [Hac 87]). Although the axioms that define the notion of a group are rather simple, and in spite of the abundance of knowledge about large classes of groups, one is frequently frustrated by the paucity of methods for dealing with groups described by a small set of generators and relations that hold between them. What is lacking in the standard texts of classical algebra and group theory is a counterpart of numerical methods in differential equations. Yet, such computational methods in group theory have been developed along the years under the influence of external problems as well as from within, especially by the needs of the classification of finite simple groups and the Burnside Groups problem. It should be emphasized that the present computational methods build on careful analysis of algorithmic aspects of known theories. Also, that the practical use of these algorithms became possible in a meaningful way only with the advent of computer technology. We will present in this article some of the known computational methods for investigating groups given by generators and relations, comment

We present practical algorithms to compute subgroups such as Hall systems, system normalizers, F-normalizers and F-covering subgroups in finite solvable groups. An application is an algorithm to calculate head complements in finite solvable groups; that is, complements which are closely related to maximal subgroups. Our algorithms use the technique of exhibiting subgroups. Subgroups such as Hall systems, system normalizers, F-normalizers and F-covering subgroups arise naturally in the study of finite solvable groups. Here we present practical algorithms to compute such subgroups. Besides being of interest in the theory of formations, the algorithms also yield fast methods for constructing head complements. These complements are closely related to the maximal subgroups of the underlying group and they also arise in the determination of special polycyclic generating sequences for solvable groups. Such sequences have proved to be of central importance for fast computations with finite solvable groups, and thus a practical method to obtain them has important applications. Our approach is to start with a special type of generating set for a given solvable

Mal’cev showed in the 1950s that there is a correspondence between radicable torsion-free nilpotent groups and rational nilpotent Lie algebras. In this paper we show how to establish the connection between the radicable hull of a finitely generated torsion-free nilpotent group and its corresponding Lie algebra algorithmically. We apply it to fast multiplication of elements of polycyclically presented groups. 1

. Isomorphism problems often can be solved by determining orbits of a group acting on the set of all objects to be classified. The paper centers around algorithms for this topic and shows how to base them on the same idea, the homomorphism principle. Especially it is shown that forming Sims chains, using an algorithmic version of Burnside's table of marks, computing double coset representatives, and computing Sylow subgroups of automorphism groups can be explained in this way. The exposition is based on graph theoretic concepts to give an easy explanation of data structures for group actions. 1. A General Point of View A natural goal in mathematical theories is a full description of the objects that are investigated. This goal has been successfully achieved in some cases, for example all finite abelian groups and with much more effort all finite simple groups. More often one restricted the research activity firstly to more modest problems like the pure existence of any object with som...