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Simple groups generated by involutions interchanging residue classes modulo lattices in \(\mathbb Z^d\). (English) Zbl 1287.20041

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.

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

Citations:

Zbl 1210.20026
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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)
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