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Towards a soluble quotient algorithm. (English) Zbl 0635.20013
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 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 J. W. Wamsley [Lect. Notes Math. 573, 118-125 (1977; Zbl 0358.20031)], J. R. Howse and D. L. Johnson [Lond. Math. Soc. Lect. Note Ser. 71, 237-243 (1982; Zbl 0515.20018)] and 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 (§ 3), (b) for each such irreducible module M construct all extensions of M by H (§ 4), (c) for each extension \(\tilde H\) test whether the epimorphism \(\epsilon\) lifts to an epimorphism from G onto \(\tilde H\) (§ 2). If any such lifted epimorphism is found, the problem is solved. The author outlines an algorithm for deciding (in effect) whether N/N’ is infinite which is based on knowing all the irreducible rational representations for H (§ 6). He also outlines an algorithm for the case when N/N’ is finite for determining a finite set of primes which would include all the primes occurring as orders of elements in N/N’ (§ 7). The ideas outlined in §§ 2,6,7 have been used in a related context with H insoluble in a forthcoming book by D. F. Holt and the author. An implementation of these ideas is being attempted in Aachen. Note that G. Baumslag, F. B. Cannonito and C. F. Miller [Math. Z. 178, 289-295 (1981; Zbl 0455.20027)] have shown that there is an algorithm for determining for arbitrary polycyclic groups H whether G/N’ is polycyclic and, if so, for giving a polycyclic presentation for G/N’.
Reviewer: M.F.Newman

MSC:
20F05 Generators, relations, and presentations of groups
20-04 Software, source code, etc. for problems pertaining to group theory
20F16 Solvable groups, supersolvable groups
20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
20C20 Modular representations and characters
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