Abstract

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 ...

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