Towards a soluble quotient algorithm.

*(English)*Zbl 0635.20013The 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 |

##### Keywords:

algorithm; soluble quotients of finitely presented groups; finite soluble group; polycyclic presentation; AG-presentation; coset enumeration; Schreier transversal; Reidemeister-Schreier algorithm; irreducible representation; irreducible modules; polycyclic groups
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##### References:

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