Kohl, Stefan Algorithms for a class of infinite permutation groups. (English) Zbl 1155.20003 J. Symb. Comput. 43, No. 8, 545-581 (2008). Motivated by the \(3n+1\) “Collatz” problem [see the survey by G. J. Wirsching, Elem. Math. 55, No. 4, 142-155 (2000; Zbl 0999.11013)], the author studies groups of permutations defined on the integers such that for a chosen \(n\) they are affine functions on the residue classes modulo \(n\). (The author calls these ‘residue-class wise affine’ groups, short RCWA.) The underlying mathematics has been studied by the author before [in Adv. Appl. Math. 39, No. 3, 322-328 (2007; Zbl 1146.11015)].) The paper under review describes basic algorithms (such as: element arithmetic, direct product, wreath product, transitivity tests) for working with such groups, as implemented in his GAP package RCWA. Methods for element order or group membership only are provided for particular well-behaved cases. (The general problem is known to be not algorithmically solvable.) Reviewer: Alexander Hulpke (Fort Collins) Cited in 2 Documents MSC: 20B40 Computational methods (permutation groups) (MSC2010) 68W30 Symbolic computation and algebraic computation 11B83 Special sequences and polynomials Keywords:computational group theory; infinite permutation groups; residue-class-wise affine groups; \(3n+1\) conjecture; Collatz conjecture; GAP; \(3n+1\) problem; Collatz problem; residue-class-wide affine mappings Citations:Zbl 0999.11013; Zbl 1146.11015 Software:RCWA; Polycyclic; GAP PDFBibTeX XMLCite \textit{S. Kohl}, J. Symb. Comput. 43, No. 8, 545--581 (2008; Zbl 1155.20003) Full Text: DOI References: [1] de la Harpe, P., (Topics in geometric group theory. Topics in geometric group theory, Chicago Lectures in Mathematics (2000)), MR 1786869 (2001i:20081) · Zbl 0965.20025 [2] Eick, B., Nickel, W., 2007. Polycyclic — Computation with polycyclic groups; Version 2.2, refereed GAP package. Published at: http://www.gap-system.org/Packages/polycyclic.html; Eick, B., Nickel, W., 2007. Polycyclic — Computation with polycyclic groups; Version 2.2, refereed GAP package. Published at: http://www.gap-system.org/Packages/polycyclic.html [3] The GAP Group, 2007. GAP — Groups, Algorithms, and Programming; Version 4.4.10. http://www.gap-system.org; The GAP Group, 2007. GAP — Groups, Algorithms, and Programming; Version 4.4.10. http://www.gap-system.org [4] Holt, D. F.; Eick, B.; O’Brien, E. A., Handbook of computational group theory, (Discrete Mathematics and its Applications (Boca Raton) (2005), Chapman & Hall / CRC: Chapman & Hall / CRC Boca Raton, FL), MR 2129747 (2006f:20001) · Zbl 1091.20001 [5] Keller, T. P., Finite cycles of certain periodically linear permutations, Missouri J. Math. Sci., 11, 3, 152-157 (1999) · Zbl 1097.05500 [6] Kohl, S., 2005. Restklassenweise affine Gruppen, Dissertation, Universität Stuttgart. Published at: http://deposit.d-nb.de/cgi-bin/dokserv?idn=977164071; Kohl, S., 2005. Restklassenweise affine Gruppen, Dissertation, Universität Stuttgart. Published at: http://deposit.d-nb.de/cgi-bin/dokserv?idn=977164071 [7] Kohl, S., 2006a. Graph theoretical criteria for the wildness of residue-class-wise affine permutations, preprint (short note). Available at: http://www.cip.mathematik.uni-stuttgart.de/ kohlsn/preprints/graphcrit.pdf; Kohl, S., 2006a. Graph theoretical criteria for the wildness of residue-class-wise affine permutations, preprint (short note). Available at: http://www.cip.mathematik.uni-stuttgart.de/ kohlsn/preprints/graphcrit.pdf [8] Kohl, S., 2006b. A simple group generated by involutions interchanging residue classes of the integers, preprint. Available at: http://www.cip.mathematik.uni-stuttgart.de/ kohlsn/preprints/simplegp.pdf; Kohl, S., 2006b. A simple group generated by involutions interchanging residue classes of the integers, preprint. Available at: http://www.cip.mathematik.uni-stuttgart.de/ kohlsn/preprints/simplegp.pdf [9] Kohl, S., 2007a. RCWA - Residue-Class-Wise Affine Groups; Version 2.5.4, refereed GAP package. Published at: http://www.gap-system.org/Packages/rcwa.html; Kohl, S., 2007a. RCWA - Residue-Class-Wise Affine Groups; Version 2.5.4, refereed GAP package. Published at: http://www.gap-system.org/Packages/rcwa.html [10] Kohl, S., Wildness of iteration of certain residue-class-wise affine mappings, Adv. in Appl. Math., 39, 3, 322-328 (2007), MR 2352043 · Zbl 1146.11015 [11] Lagarias, J.C., 2007. The \(3 x + 1\) problem: An annotated bibliography. http://arxiv.org/abs/math.NT/0309224; Lagarias, J.C., 2007. The \(3 x + 1\) problem: An annotated bibliography. http://arxiv.org/abs/math.NT/0309224 [12] Wirsching, G. J., (The Dynamical System Generated by the \(3 n + 1\) Function. The Dynamical System Generated by the \(3 n + 1\) Function, Lecture Notes in Mathematics, vol. 1681 (1998), Springer-Verlag) · Zbl 0892.11002 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.