The concept of algebraic simplification is of great importance for the field of symbolic computation in computer algebra. In this paper we review some fundamental concepts concerning reduction rings in the spirit of Buchberger. The most important properties of reduction rings are presented. The techniques for presenting monoids or groups by string rewriting systems are used to define several types of reduction in monoid and group rings. Grobner bases in this setting arise naturally as generalizations of the corresponding known notions in the commutative and some noncommutative cases. Several results on the connection of the word problem and the congruence problem are proven. The concepts of saturation and completion are introduced for monoid rings having a finite convergent presentation by a semi-Thue system. For certain presentations, including free groups and context-free groups, the existence of finite Grobner bases for finitely generated right ideals is shown and a procedure to com...

this paper into a survey of the state of the art in that wider field. In this paper a reminiscence of that overview will be confined to some very sketchy historical remarks in this introduction, but the bibliography should still be sufficient to obtain a reasonable coverage of the literature on the field by tracing back. For investigations that involve calculation with the elements of a finite group the elements must be represented in a form that allows efficient multiplication, inversion and, last but not least, comparison. The first programs of that kind, for the determination of the subgroup lattice, already used both permutations for arbitrary groups and, for p-groups, ordered words in a generating set corresponding to a central series with cyclic factors [Neu 61]. The systematic use in such programs of basic notions from permutation group theory, in particular the stabilizer chain, was initiated by C. Sims in three fundamental papers in the late sixties and made generally available with many refinements and extensions in the precursors and early versions of the CAYLEY system (cf. [Can 84] for a comprehensive overview). On the other hand a systematic use of the properties of composition and chief series of finite soluble groups for the investigation of their structure only started with an algorithm for the determination of conjugacy classes of p-groups [FeN 79] and the subsequent development of the SOGOS system [LNS 84], preceded by the invention [Mac 74], further development, and very successful application [New 76] of the Nilpotent Quotient Algorithm, which constructs central series of p-factor groups of a finitely presented group. Computing methods for soluble groups have recently received much interest: collection methods for the multiplication of the ordered ...

Suppose G is a group given by a finite presentation. Let G (n) denote the n-th term in the derived series of G. In 1981, Baumslag, Cannonito and Miller described an algorithm to determine whether the quotient G=G (n) is polycyclic and to find G=G (n) when it is polycyclic. However, in its original form, the algorithm is not practical for computer implementation. Here, a practical algorithm to determine whether G=G (n) is polycyclic and to find it when it is polycyclic is described. The algorithm involves a generalization of the Grobner basis method of commutative ring theory to the integral group ring of a polycyclic group. An implementation of this algorithm in the language C has been developed and its efficiency is encouraging.

There are two main results contained in this dissertation. The first result is a description of an algorithm for the computation of polycyclic presentations for nilpotent factor groups of a given finitely presented group. This algorithm is a generalization of the methods employed in the p-quotient algorithm (Havas & Newman, 1980) to possibly in nite nilpotent groups. The second is a method for the computation of the Schur multiplicator of a group given by a polycyclic presentation and a method for the classification of the isomorphism types of Schur covering groups for finite soluble groups. Both algorithms can be treated in a similar context, namely forming central downward extensions of polycyclic groups.

übertragen. Die meisten dieser Ansätze setzen eine zulässige Ordnung auf den