Computing with infinite polycyclic groups (Q2759624)

\textit{G. Baumslag, F. B. Cannonito, D. J. S. Robinson} and \textit{D. Segal} [J. Algebra 142, No. 1, 118-149 (1991; Zbl 0774.20019)] have shown a large number of group theoretic problems to be decidable in polycyclic groups. The paper under review is devoted to describe practical algorithms for some group theoretic questions in such groups. Each polycyclic group can be represented by a consistent power-conjugate presentation (pc presentation) [see \textit{C. C. Sims}, Computation with finitely presented groups, Encyclopedia of Mathematics and Its Applications. 48. Cambridge: Cambridge University Press (1994; Zbl 0828.20001) for details]. A pc presentation can be used to design practical algorithms for polycyclic groups. A group defined by a pc presentation or a subgroup of such a group is said to be a pc group. In finite pc groups practical methods for several group-theoretic concepts are known. The aim of the paper is to describe some practical algorithms for arbitrary, possibly infinite pc groups. The basis of the algorithms in this paper is the determination of complement classes to a normal subgroup in a pc group which is described in Section 4 of the paper.NEWLINENEWLINENEWLINEIn Section 5 the computation of certain finite subgroups in a pc group is considered. A polycyclic group has only finitely many conjugacy classes of finite subgroups [see \textit{A. I. Mal'cev}, Mat. Sb., Nov. Ser. 28(70), 567-588 (1951; Zbl 0043.02301)]. The author introduced a method to compute a set of representatives for these classes and their normalizers in a pc group. An algorithm to test in a pc group whether the set of elements with finite order is a subgroup is described. A method to compute the largest finite normal subgroup in a pc group is outlined.NEWLINENEWLINENEWLINEIn Section 6 subgroups of finite index in a pc group are considered. A ``low index subgroups'' algorithm for pc groups is described; that is, a method to compute all subgroups or all normal subgroups of a given finite index. Every maximal subgroup of a polycyclic group has prime power index. A first approach to determine all maximal subgroups of a pc group with \(p\)-power index for a given prime \(p\) is outlined. Each polycyclic group has a normal subgroup of finite index which is poly-\(\mathbb{Z}\); that is, which has a polycyclic series with infinite factors only. It is shown how to determine a characteristic subgroup of this type together with a free Abelian normal series through the subgroup.NEWLINENEWLINENEWLINEThe algorithms introduced in this paper have been implemented in the computer algebra system GAP 4 [see The GAP group, GAP-Groups, Algorithms, and Programming, Version 4.2; Aachen, St. Andrews, 2000 (\url{http://www.gap-system.org/gap/})], and they are available as part of the ``Polycyc'' package [see \textit{B. Eick} and \textit{W. Nickel}, Algorithms for polycyclic groups, GAP package (see the latter reference)]. Finally for showing the practicability of the introduced methods, example runtimes of the main methods and comparisons to other methods are included in Section 7.NEWLINENEWLINEFor the entire collection see [Zbl 0959.00030].