Finitely generated subgroups as von Neumann radicals of an Abelian group

Author S. S. Gabriyelyan
saak@math.bgu.ac.il
Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva, Israel

Abstract Let G be an infinite Abelian group. We give a complete characterization of those finitely generated subgroups of G which are the von Neumann radicals for some Hausdorff group topologies on G. It is proved that every infinite finitely generated Abelian group admits a complete Hausdorff minimally almost periodic group topology. The latter result resolves a particular case of Comfort’s problem.
Keywords characterized group; T-sequence; von Neumann radical; finitely generated subgroup
Reference 1. M. Ajtai, I. Havas, J. Koml.os, Every group admits a bad topology, Stud. Pure Math., Memory of P. Turan, Basel–Boston, 1983, 21–34.

 2. D.L. Armacost, The structure of locally compact Abelian groups, Monographs and Textbooks in Pure and Applied Mathematics, 68, Marcel Dekker, Inc., New York, 1981.

 3. G. Barbieri, D. Dikranjan, C. Milan, H.Weber, Answer to Raczkowski’s question on convergent sequences of integers, Topology Appl., 132 (2003), 89–101.

 4. G. Barbieri, D. Dikranjan, C. Milan, H. Weber, Topological torsion related to some sequences of integers, Math. Nachr., 281 (2008), ¹7, 930–950.

 5. W.W. Comfort, Problems on Topological Groups and Other Homogeneous Spaces, Open problems in topology, 314–347, North-Holland, 1990.

 6. W.W. Comfort, S.U. Raczkowski, F. Trigos-Arrieta, Making group topologies with, and without, convergent sequences, Applied General Topology, 7 (2006), ¹1, 109–124.

 7. D. Dikranjan, C. Milan, A. Tonolo, A characterization of the maximally almost periodic Abelian groups, J. Pure Appl. Algebra, 197 (2005), 23–41.

 8. S.S. Gabriyelyan, On T-sequences and characterized subgroups, Topology Appl., 157 (2010), 2834–2843.

 9. S.S. Gabriyelyan, Characterization of almost maximally almost-periodic groups, Topology Appl., 156 (2009), 2214–2219.

 10. E. Hewitt, K.A. Ross, Abstract Harmonic Analysis, V.I, 2nd ed. Springer-Verlag, Berlin, 1979.

 11. G. Luk.acs, Almost maximally almost-periodic group topologies determined by T-sequences, Topology Appl., 153 (2006), 2922–2932.

 12. J. von Neumann, Almost periodic functions in a group, Trans. Amer. Math. Soc., 36 (1934), 445–492.

 13. N. Noble, k-groups and duality, Trans. Amer. Math. Soc., 151 (1970), 551–561.

 14. I.V. Protasov, Review of Ajtai, Havas and J. Komlos, Zentralblatt fur Matematik, 535 (1983), 93.

 15. I.V. Protasov, E.G. Zelenyuk, Topologies on abelian groups, Math. USSR Izv., 37 (1991), 445–460. Russian original: Izv. Akad. Nauk SSSR, 54 (1990), 1090–1107.

 16. I.V. Protasov, E.G. Zelenyuk, Topologies on groups determined by sequences, Monograph Series, Math. Studies VNTL, Lviv, 1999.
Pages 124-138
Volume 38
Issue 2
Year 2012
Journal Matematychni Studii
Full text of paper PDF
Table of content of issue HTML