Towards a soluble quotient algorithm
type: Item
instance of: scholarly article
title: Towards a soluble quotient algorithm
zbMATH Open document ID: 0635.20013
DOI: 10.1016/S0747-7171(87)80060-3
author: Wilhelm Plesken
published in: Journal of Symbolic Computation
publication date: 1987-01-01T00:00:00Z
review text: The aim of this paper is to outline an algorithm for computing soluble quotients of finitely presented groups. More explicitly, the problem being considered is: given a group G by a finite presentation, a finite soluble group H by a presentation derived from a composition series of H (an AG- or polycyclic presentation of a special kind), and an epimorphism \(\epsilon\) of G onto H, find an AG-presentation for a larger finite soluble quotient group of G or show that H is the largest soluble quotient of G. A simple algorithm of this kind which can use existing computer programs is: (a) use coset enumeration to find a Schreier transversal in G for the kernel N of \(\epsilon\), (b) use an abelianized form of the Reidemeister-Schreier algorithm to compute a finite presentation for N/N' (see \textit{G. Havas} [Lect. Notes Math. 372, 347-356 (1974; Zbl 0288.20047)]), (c) use integer matrix diagonalization to determine whether N/N' is trivial or not. The goal is an algorithm which will lead to programs which can perform better than this. The present paper, like earlier ones by \textit{J. W. Wamsley} [Lect. Notes Math. 573, 118-125 (1977; Zbl 0358.20031)], \textit{J. R. Howse} and \textit{D. L. Johnson} [Lond. Math. Soc. Lect. Note Ser. 71, 237-243 (1982; Zbl 0515.20018)] and \textit{C. R. Leedham-Green} [Computational group theory, Proc. Symp., Durham/Engl. 1982, 85-101 (1984; Zbl 0558.20022)], offers some theoretical ideas towards this goal. They are based on the observation that the degree of the largest irreducible representation of H is mostly significantly less than the Schreier bound for the rank of N. It proposes a three-stage algorithm: (a) compute for selected primes p all the irreducible modules for H over the field of p elements working up the given composition series of H using Clifford theory ({\S} 3), (b) for each such irreducible module M construct all extensions of M by H ({\S} 4), (c) for each extension \(\tilde H\) test whether the epimorphism \(\epsilon\)…
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20-04
Mathematics Subject Classification ID: 20F16
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 20C20
zbMATH DE Number: 4033992
zbMATH Keywords: algorithm
zbMATH Keywords: soluble quotients of finitely presented groups
zbMATH Keywords: finite soluble group
zbMATH Keywords: polycyclic presentation
zbMATH Keywords: AG-presentation
zbMATH Keywords: coset enumeration
zbMATH Keywords: Schreier transversal
zbMATH Keywords: Reidemeister-Schreier algorithm
zbMATH Keywords: irreducible representation
zbMATH Keywords: irreducible modules
zbMATH Keywords: polycyclic groups
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4033992