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Cardinal invariants of infinite groups

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Abstract

LetG be a group. CallG akC-group if every element ofG has less thank conjugates. Denote byP(G) the least cardinalk such that any subset ofG of sizek contains two elements which commute.

It is shown that the existence of groupsG such thatP(G) is a singular cardinal is consistent withZFC. So is the existence of groupsG which are notkC but haveP(G)<k wherek is a limit cardinal. On the other hand, ifk is a singular strong limit cardinal, andG is akC-group, thenP(G)≠k. This partially answers questions, and improves results, of Faber, Laver and McKenzie.

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References

  1. Ehrenfeucht, A., Faber, V.: Do infinite abelian groups always have equipotent subgroups? Nederl. Akad. Wetensch. Indag. Math.34, 202–209 (1972)

    Google Scholar 

  2. Erdös, P., Hajnal, A., Máté, A., Rado, R.: Combinatorial set theory: partition relations for cardinals. Amsterdam: North-Holland 1984

    Google Scholar 

  3. Faber, V., Laver, R., McKenzie, R.: Coverings of groups by abelian subgroups. Can. J. Math.30, 933–945 (1978)

    Google Scholar 

  4. Felgner, U.: On %0-categorical extraspecialp-groups. In: Henrard, P. (ed.) Six days of Model Theory. (Proceedings of a conference held at Louvain-la Neuve in March 1975, pp. 175–176) Albeuve Switzerland: Editions Castella 1977

    Google Scholar 

  5. Gorenstein, D.: Finite groups (2nd ed.). New York: Chelsea Publishing Company 1980

    Google Scholar 

  6. Hall, J.I.: The number of trace-valued forms and extraspecialp-groups. J. Lond. Math. Soc., II. Ser.37, 1–13 (1988)

    Google Scholar 

  7. Kunen, K.: Set theory, an introduction to independence proofs. Amsterdam: North-Holland 1980

    Google Scholar 

  8. Levy, A.: Basic set theory. Berlin Heidelberg New York: Springer 1979

    Google Scholar 

  9. Scott, W.R.: Group theory. Englewood Cliffs, NJ: Prentice-Hall 1964

    Google Scholar 

  10. Shelah, S.: Classification theory and the number of non-isomorphic Models. Amsterdam: North-Holland 1978

    Google Scholar 

  11. Shelah, S., Steprāns, J.: Extraspecialp-groups. Ann. Pure Appl. Logic34, 87–97 (1987)

    Google Scholar 

  12. Todorević, S.: Partitioning pairs of countable ordinals. Acta Math.159, 261–294 (1987)

    Google Scholar 

  13. Tomkinson, M.J.:FC-groups. London: Pitman 1984

    Google Scholar 

  14. Tomkinson, M.J.: Groups covered by abelian subgroups. In: Robertson, E.F., Campbell, C.M. (eds.) Proceedings of Groups — St. Andrews 1985. (Lond. Math. Soc. Lect. Note Ser., vol. 121, pp. 332–334) Cambridge: Cambridge University Press 1986

    Google Scholar 

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Authors and Affiliations

  1. Mathematisches Institut der Universität Tübingen, Auf der Morgenstelle 10, W-7400, Tübingen, Federal Republic of Germany

    Jörg Brendle

  2. Department of Mathematics and Computer Science, Dartmouth College, Bradley Hall, 03755, Hanover, NH, USA

    Jörg Brendle

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  1. Jörg Brendle
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The present paper has non-trivial intersection with the author's Diplomarbeit written under the direction of Prof. Ulrich Felgner at the University of Tübingen, W. Germany, 1988

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Brendle, J. Cardinal invariants of infinite groups. Arch Math Logic 30, 155–170 (1990). https://doi.org/10.1007/BF01621468

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  • DOI: https://doi.org/10.1007/BF01621468

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