×

Modular permutations on \(\mathbb{Z}\). (English) Zbl 1166.20002

Summary: The group \(\mathcal M\) of permutations \(\sigma\) of \(\mathbb{Z}\) for which an integer \(n=n(\sigma)>0\) exists such that \((z+n)\sigma=z\sigma+n\) for every \(z\in\mathbb{Z}\) is studied. \(\mathcal M\) is countably infinite locally (Abelian-by-finite) and contains all finitely generated (Abelian-by-finite) groups as subgroups. The commutator subgroup \(\mathcal M'\) is an infinite simple group and the quotient group \(\mathcal{M/M}'\) is isomorphic to \(\mathbb{Z}\). Finally, all Abelian groups that can be represented as modular permutation groups are determined: these are countable Abelian groups whose quotient over the torsion subgroup is free.

MSC:

20B35 Subgroups of symmetric groups
20B07 General theory for infinite permutation groups
20E32 Simple groups
20E07 Subgroup theorems; subgroup growth
20E25 Local properties of groups
PDFBibTeX XMLCite
Full Text: EuDML

References:

[1] M. R. DIXON - M. J. EVANS - H. SMITH, Embedding groups in locally (solubleby-finite) simple groups, Journal of Group Theory 9 (2006), pp. 383-395. Zbl1120.20030 MR2226620 · Zbl 1120.20030 · doi:10.1515/JGT.2006.026
[2] M. I. KARGAPOLOV - JU. I. MERZLJAKOV, Fundamentals of the Theory of Groups, Graduate Texts in Mathematics, vol. 62, Springer (1979). Zbl0549.20001 MR551207 · Zbl 0549.20001
[3] N. POUYANNE, On the number of permutations admitting an m-th root, The Electronic Journal of Combinatorics 9 (2002). Zbl0990.05003 MR1887084 · Zbl 0990.05003
[4] H. WIELANDT, Permutationsgruppe, Math. Inst. Univ. Tübingen (1955). [Translated in english: Finite Permutation Groups, Academic Press, New York (1964). Reprinted in Mathematische Werke, Walter de Gruyter, Berlin, Vol. 1 (1994), pp. 119-198.]
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.