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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Periodic Groups Covered by Transitive Subgroups of Finitary Permutations or by Irreducible Subgroups of Finitary Transformations


Authors: Felix Leinen and Orazio Puglisi
Journal: Trans. Amer. Math. Soc. 352 (2000), 1913-1934
MSC (1991): Primary 20B07, 20E25, 20F50, 20H20; Secondary 03C20, 20E22
Published electronically: December 10, 1999
MathSciNet review: 1603922
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Abstract: Let $\mathfrak{X}$ be either the class of all transitive groups of finitary permutations, or the class of all periodic irreducible finitary linear groups. We show that almost primitive $\mathfrak{X}$-groups are countably recognizable, while totally imprimitive $\mathfrak{X}$-groups are in general not countably recognizable. In addition we derive a structure theorem for groups all of whose countable subsets are contained in totally imprimitive $\mathfrak{X}$-subgroups. It turns out that totally imprimitive $p$-groups in the class $\mathfrak{X}$ are countably recognizable.


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Additional Information

Felix Leinen
Affiliation: Fachbereich 17 – Mathematik, Johannes Gutenberg–Universität Mainz, D–55099 Mainz, Germany
Address at time of publication: Department of Mathematics, University of Newcastle, Newcastle upon Tyne, NE1 7RU, England
Email: f.a.leinen@ncl.ac.uk

Orazio Puglisi
Affiliation: Dipartimento di Matematica, Università degli Studi di Trento, I–38050 Povo (Trento), Italy
Email: puglisi@alpha.science.unitn.it

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02309-0
PII: S 0002-9947(99)02309-0
Keywords: Countable recognizability, finitary linear groups, finitary permutation groups, locally finite groups, wreath products, ultraproducts
Received by editor(s): February 10, 1997
Received by editor(s) in revised form: October 22, 1997
Published electronically: December 10, 1999
Additional Notes: Each of the two authors would like to thank the university of his coauthor for inviting him to a visit, during which essential parts of the work on this paper could be carried out.
Article copyright: © Copyright 2000 American Mathematical Society