×

The Collatz conjecture in a group theoretic context. (English) Zbl 1400.20022

By \(r(m)\) we denote the residue class \(r+m{\mathbb Z}\), where we assume that \(0\leqslant r < m\). The Collatz conjecture asserts that iterated application of the mapping \[ C: {\mathbb Z}\rightarrow {\mathbb Z}, \;\;\;\;n\mapsto \begin{cases} \frac{n}{2} & \text{if } n\in 0(2),\\ 3n+1 & \text{if } n\in 1(2), \end{cases} \] to any positive integer yields \(1\) after a finite number of steps.
Given disjoint residue classes \(r_1(m_1)\) and \(r_2(m_2)\) of \({\mathbb Z}\), let the class transposition \(\tau_{r_1(m_1), r_2(m_2)}\) be the permutation which interchanges \(r_1+km_1\) and \(r_2+km_2\) for each integer \(k\) and which fixes all other points. We denote by \(\mathrm{CT}({\mathbb Z})\) the subgroup of all class transpositions. We also set \[ G_C:=\langle \tau_{1(2), 4(6)}, \tau_{1(3), 2(6)}, \tau_{2(3), 4(6)}\rangle \text{ and } G_T:=\langle \tau_{0(2), 1(2)}, \tau_{1(2), 2(4)}, \tau_{1(4), 2(6)}\rangle. \]
As the first result, the author proves that:
Proposition 1.2. The following hold:
(a)
The group \(G_C\) acts transitively on \({\mathbb N}\setminus 0(6)\) if and only if the Collatz conjecture holds.
(b)
The group \(G_T\) acts transitively on \({\mathbb N}_0\) if and only if the Collatz conjecture holds.

Given a set \({\mathbb P}\) of prime numbers, let \(\mathrm{CT}_{\mathbb P}(\mathbb Z)\leqslant \mathrm{CT}({\mathbb Z})\) denote the subgroup which is generated by all class transpositions \(\tau_{r_1(m_1), r_2(m_2)}\) for which all prime factors of \(m_1\) and \(m_2\) lie in \({\mathbb P}\). Clearly, both \(G_C\) and \(G_T\) are subgroups of \(\mathrm{CT}_{\{2,3\}}(\mathbb Z)\). By Theorem 2.3 in [the author, Math. Z. 264, No. 4, 927–938 (2010; Zbl 1210.20026)], the group \(\mathrm{CT}(\mathbb Z)\) is not finitely generated. By the arguments used in the proof of that theorem, it follows also that \(\mathrm{CT}_{\mathbb P}(\mathbb Z)\) is not finitely generated if \(\mathbb P\) is infinite. As the second result, the author proves that:
Proposition 3.2. Let \(\mathbb P\) be a finite set of primes. Then the group \(\mathrm{CT}_{\mathbb P}(\mathbb Z)\) is finitely generated.

MSC:

20F05 Generators, relations, and presentations of groups
11B83 Special sequences and polynomials
20B22 Multiply transitive infinite groups
20B40 Computational methods (permutation groups) (MSC2010)
20E32 Simple groups

Citations:

Zbl 1210.20026

Software:

GAP; RCWA
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] J. Cannon, W. Floyd and W. Parry, Introductory notes on Richard Thompson’s groups, Enseign. Math. (2) 42 (1996), no. 3-4, 215-256.; Cannon, J.; Floyd, W.; Parry, W., Introductory notes on Richard Thompson’s groups, Enseign. Math. (2), 42, 3-4, 215-256 (1996) · Zbl 0880.20027
[2] G. Higman, Finitely Presented Infinite Simple Groups, Notes Pure Math., Department of Pure Mathematics, Australian National University, Canberra, 1974.; Higman, G., Finitely Presented Infinite Simple Groups (1974) · Zbl 1479.20003
[3] S. Kohl, Algorithms for a class of infinite permutation groups, J. Symbolic Comput. 43 (2008), 545-581.; Kohl, S., Algorithms for a class of infinite permutation groups, J. Symbolic Comput., 43, 545-581 (2008) · Zbl 1155.20003
[4] S. Kohl, A simple group generated by involutions interchanging residue classes of the integers, Math. Z. 264 (2010), no. 4, 927-938.; Kohl, S., A simple group generated by involutions interchanging residue classes of the integers, Math. Z., 264, 4, 927-938 (2010) · Zbl 1210.20026
[5] S. Kohl, RCWA - Residue-Class-Wise Affine Groups; Version 4.4.1, GAP package (2016), .; Kohl, S., RCWA - Residue-Class-Wise Affine Groups; Version 4.4.1 (2016) · Zbl 1195.20001
[6] J. C. Lagarias, The 3x{+}1 problem and its generalizations, Amer. Math. Monthly 92 (1985), 3-23.; Lagarias, J. C., The 3x{+}1 problem and its generalizations, Amer. Math. Monthly, 92, 3-23 (1985) · Zbl 0566.10007
[7] J. C. Lagarias, The 3x{+}1 problem: An annotated bibliography, preprint (2011), (part I), <ext-link ext-link-type=”uri“ xlink.href=”>http://arxiv.org/abs/math.NT/0608208 (part II).; Lagarias, J. C., The 3x{+}1 problem: An annotated bibliography (2011)
[8] The GAP Group, GAP - Groups, Algorithms, and Programming; Version 4.8.5, 2016, .; 2020-01-09 08:49:26 Citation within text.
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.