Countable random $p$-groups with prescribed Ulm-invariants
HTML articles powered by AMS MathViewer
- by Manfred Droste and Rüdiger Göbel PDF
- Proc. Amer. Math. Soc. 139 (2011), 3203-3216 Request permission
Abstract:
In this paper we present a probabilistic construction of countable abelian $p$-groups with prescribed Ulm-sequence. This result provides a different proof for the existence theorem of abelian $p$-groups with any given countable Ulm-sequence due to Ulm, which is sometimes called Zippin’s theorem. The basic idea, applying probabilistic arguments, comes from a result by Erdős and Rényi. They gave an amazing probabilistic construction of countable graphs which, with probability $1$, produces the universal homogeneous graph, therefore also called the random graph. P. J. Cameron says about this in his book Oligomorphic Permutation Groups [Cambridge University Press, 1990]: In 1963, Erdős and Rényi proved the following paradoxical result. … It is my contention that mathematics is unique among academic pursuits in that such an apparently outrageous claim can be made completely convincing by a short argument. The algebraic tool in the present paper needs methods developed in the 1970s, the theory of valuated abelian $p$-groups. Valuated abelian $p$-groups are natural generalizations of abelian $p$-groups with the height valuation, investigated in detail by F. Richman and E. Walker, and others. We have to establish extensions of finite valuated abelian $p$-groups dominated by a given Ulm-sequence. Probabilistic results of a similar nature have been established by A. Blass and G. Braun, and by M. Droste and D. Kuske.References
- Samy Abbes and Klaus Keimel, Projective topology on bifinite domains and applications, Theoret. Comput. Sci. 365 (2006), no. 3, 171–183. MR 2267074, DOI 10.1016/j.tcs.2006.07.047
- Andreas Blass and Gábor Braun, Random orders and gambler’s ruin, Electron. J. Combin. 12 (2005), Research Paper 23, 27. MR 2134186
- Peter J. Cameron, Oligomorphic permutation groups, London Mathematical Society Lecture Note Series, vol. 152, Cambridge University Press, Cambridge, 1990. MR 1066691, DOI 10.1017/CBO9780511549809
- Peter Crawley, The cancellation of torsion abelian groups in direct sums, J. Algebra 2 (1965), 432–442. MR 188293, DOI 10.1016/0021-8693(65)90004-9
- Manfred Droste, Universal homogeneous causal sets, J. Math. Phys. 46 (2005), no. 12, 122503, 10. MR 2194024, DOI 10.1063/1.2147607
- M. Droste and G.-Q. Zhang, Random event structures, International Journal of Software and Informatics 2 (2008), 77 – 88.
- Manfred Droste and Rüdiger Göbel, A categorical theorem on universal objects and its application in abelian group theory and computer science, Proceedings of the International Conference on Algebra, Part 3 (Novosibirsk, 1989) Contemp. Math., vol. 131, Amer. Math. Soc., Providence, RI, 1992, pp. 49–74. MR 1175872
- Manfred Droste and Dietrich Kuske, On random relational structures, J. Combin. Theory Ser. A 102 (2003), no. 2, 241–254. MR 1979531, DOI 10.1016/S0097-3165(03)00004-9
- M. Droste and D. Kuske, Almost every domain is universal, 23rd Conf. on the Mathematical Foundations of Programming Semantics (MFPS), Electronic Notes in Theoretical Computer Science 173 (2007) 104 – 119.
- P. Erdős and A. Rényi, Asymmetric graphs, Acta Math. Acad. Sci. Hungar. 14 (1963), 295–315. MR 156334, DOI 10.1007/BF01895716
- Paul Erdős and Joel Spencer, Probabilistic methods in combinatorics, Probability and Mathematical Statistics, Vol. 17, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR 0382007
- Roland Fraïssé, Sur l’extension aux relations de quelques propriétés des ordres, Ann. Sci. Ecole Norm. Sup. (3) 71 (1954), 363–388 (French). MR 0069239, DOI 10.24033/asens.1027
- László Fuchs, Infinite abelian groups. Vol. I, Pure and Applied Mathematics, Vol. 36, Academic Press, New York-London, 1970. MR 0255673
- R. Göbel, Helmut Ulm: his work and its impact on recent mathematics, Abelian group theory (Perth, 1987) Contemp. Math., vol. 87, Amer. Math. Soc., Providence, RI, 1989, pp. 1–10. MR 995259, DOI 10.1090/conm/087/995259
- Martin Goldstern, The typical countable algebra, Contributions to general algebra. 18, Heyn, Klagenfurt, 2008, pp. 57–71. MR 2407575
- Wilfrid Hodges, Model theory, Encyclopedia of Mathematics and its Applications, vol. 42, Cambridge University Press, Cambridge, 1993. MR 1221741, DOI 10.1017/CBO9780511551574
- Roger Hunter, Fred Richman, and Elbert Walker, Existence theorems for Warfield groups, Trans. Amer. Math. Soc. 235 (1978), 345–362. MR 473044, DOI 10.1090/S0002-9947-1978-0473044-4
- B. Jónsson, Homogeneous universal relational systems, Math. Scand. 8 (1960), 137–142. MR 125021, DOI 10.7146/math.scand.a-10601
- Ray Mines, Fred Richman, and Wim Ruitenburg, A course in constructive algebra, Universitext, Springer-Verlag, New York, 1988. MR 919949, DOI 10.1007/978-1-4419-8640-5
- Fred Richman, The constructive theory of countable abelian $p$-groups, Pacific J. Math. 45 (1973), 621–637. MR 344088, DOI 10.2140/pjm.1973.45.621
- Fred Richman, A guide to valuated groups, Abelian group theory (Proc. Second New Mexico State Univ. Conf., Las Cruces, N.M., 1976) Lecture Notes in Math., Vol. 616, Springer, Berlin, 1977, pp. 73–86. MR 0480779
- Fred Richman and Elbert A. Walker, Valuated groups, J. Algebra 56 (1979), no. 1, 145–167. MR 527162, DOI 10.1016/0021-8693(79)90330-2
- Helmut Ulm, Zur Theorie der abzählbar-unendlichen Abelschen Gruppen, Math. Ann. 107 (1933), no. 1, 774–803 (German). MR 1512826, DOI 10.1007/BF01448919
- Leo Zippin, Countable torsion groups, Ann. of Math. (2) 36 (1935), no. 1, 86–99. MR 1503210, DOI 10.2307/1968666
Additional Information
- Manfred Droste
- Affiliation: Institute of Computer Science, Universität Leipzig, PF 100920, 04009 Leipzig, Germany
- Email: droste@informatik.uni-leipzig.de
- Rüdiger Göbel
- Affiliation: Fakultät für Mathematik, Universität Duisburg-Essen, Campus Essen, 45117 Essen, Germany
- Email: ruediger.goebel@uni-due.de
- Received by editor(s): March 13, 2010
- Received by editor(s) in revised form: July 9, 2010, and August 22, 2010
- Published electronically: February 25, 2011
- Additional Notes: The authors are supported by the project No. 963-98.6/2007 of the German-Israeli Foundation for Scientific Research & Development and by a project AOBJ 548025 of the Deutsche Forschungsgemeinschaft.
- Communicated by: Birge Huisgen-Zimmermann
- © Copyright 2011 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 139 (2011), 3203-3216
- MSC (2000): Primary 20K10, 20K30; Secondary 60F20, 16W20
- DOI: https://doi.org/10.1090/S0002-9939-2011-10756-1
- MathSciNet review: 2811276