Towards a soluble quotient algorithm
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 \(\epsilon\) lifts to an epimorphism from G onto \(\tilde H\) ({\S} 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 ({\S} 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' ({\S} 7). The ideas outlined in {\S}{\S} 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 \textit{G. Baumslag}, \textit{F. B. Cannonito} and \textit{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'.
soluble quotients of finitely presented groups
scientific article; zbMATH DE number 4033992
scientific article