Kohl, Stefan Simple groups generated by involutions interchanging residue classes modulo lattices in \(\mathbb Z^d\). (English) Zbl 1287.20041 J. Group Theory 16, No. 1, 81-86 (2013). From the introduction: Definition 1.1. Let \(d\in\mathbb N\), and let \(L_1,L_2\in\mathbb Z^{d\times d}\) be matrices of full rank which are in Hermite normal form. Further let \(r_1+\mathbb Z^dL_1\) and \(r_2+\mathbb Z^dL_2\) be disjoint residue classes, and assume that the representatives \(r_1\) and \(r_2\) are reduced modulo \(\mathbb Z^dL_1\) and \(\mathbb Z^dL_2\), respectively. Then we define the class transposition \[ \tau_{r_1+\mathbb Z^dL_1,r_2+\mathbb Z^dL_2}\in\text{Sym}(\mathbb Z^d) \] as the involution which interchanges \(r_1+kL_1\) and \(r_2+kL_2\) for all \(k\in\mathbb Z^d\) and which fixes everything else. Definition 1.2. Let \(\text{CT}(\mathbb Z^d)\) denote the group which is generated by the set of all class transpositions of \(\mathbb Z^d\). The purpose of this article is to prove the following generalization of [S. Kohl, Math. Z. 264, No. 4, 927-938 (2010; Zbl 1210.20026), Theorem 3.4]: Theorem 1.3. The groups \(\text{CT}(\mathbb Z^d)\) are simple. The work which led to the discovery of the simple group \(\text{CT}(\mathbb Z)\) was originally motivated by Lothar Collatz’ \(3n+1\) conjecture, which dates back to the 1930’s. MathOverflow Questions: Fractal-like structures arising from the action of a group on \(\mathbb{Z}^2\) MSC: 20E32 Simple groups 20F05 Generators, relations, and presentations of groups 20B40 Computational methods (permutation groups) (MSC2010) 20B22 Multiply transitive infinite groups 20-04 Software, source code, etc. for problems pertaining to group theory Keywords:countable simple groups; groups generated by involutions; residue-class-wise affine permutations; highly transitive groups; residue classes modulo lattices Citations:Zbl 1210.20026 PDFBibTeX XMLCite \textit{S. Kohl}, J. Group Theory 16, No. 1, 81--86 (2013; Zbl 1287.20041) Full Text: DOI References: [1] Springer, print http arxiv org abs math NT part http arxiv org abs math NT part II Wirsching The Dynamical System Generated by the Function Lecture in Kohl Received revised, Notes Mathematics 13 pp 1– (2011) 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.