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On stable torsion-free nilpotent groups

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We show that an infinite field is interpretable in a stable torsion-free nilpotent groupG of classk, k>1. Furthermore we prove thatG/Z k-1 (G) must be divisible. By generalising methods of Belegradek we classify some stable torsion-free nilpotent groups modulo isomorphism and elementary equivalence.

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References

  1. Baudisch, A.: A new ℵ1-categorical pure group. (Preprint)

  2. Belegradek, O.V.: The Mal'cev correspondence revisited. In: Bokut, L.A. et al. (eds.) Conference on Algebra honoring A.I. Mal'cev, Novosibirsk 1989. (Cont. Math., vol. 131, pp. 37–59) Providence, Rhode Island: Am. Math. Soc. 1992

    Google Scholar 

  3. Bryant, R.M.: Groups with the minimal condition on centralizers. J. Algebra60, 371–383 (1979)

    Google Scholar 

  4. Cherlin, G.L., Reineke, J.: Categoricity and stability of commutative rings. Ann. Math. Logic9, 367–399 (1976)

    Google Scholar 

  5. Felgner, U.: ℵ0-categorical stable groups. Math. Z.160, 27–49 (1978)

    Google Scholar 

  6. Fuchs, L.: Infinite abelian groups. New York London: Academic Press 1972

    Google Scholar 

  7. Mal'cev, A.I.: A correspondence between rings and groups. In: Wells, B.F., III (ed.) The metamathematics of algebraic systems, collected papers: 1936–1967 (Studies in Logic Found. Math., vol. 66, pp. 234–137). Amsterdam: North-Holland 1971

    Google Scholar 

  8. Myasnikov, A.G., Remeslennikov, V.N.: Classification of power nilpotent groups by elementary properties. Mathematical logic and the theory of algorithms, Trudy Inst. Mat.2, 56–87 (1982) (in Russian)

    Google Scholar 

  9. Myasnikov, A.G., Remeslennikov, V.N.: Formularity of sets of Mal'cev bases and elementary theories of finite dimensional algebras. I. Sib. Mat. Z.23(5), 152–167 (1982) (Transl. in Sib. Math. J.23, 711–721 (1982))

    Google Scholar 

  10. Robinson, J.: The undecidability of algebraic rings and fields. Proc. Amer. Math. Soc.10, 950–957 (1959)

    Google Scholar 

  11. Rogers, P.: Preservation of saturation and stability in a variety of nilpotent groups. J. Symb. Logic46, 499–512 (1981)

    Google Scholar 

  12. Videla, C.R.: On the model theory of the ringNT(n, R). J. Pure Appl. Algebra55, 289–302 (1988)

    Google Scholar 

  13. Warfield, R.B.: Nilpotent groups. Lect. Notes Math. 513. Berlin Heidelberg New York: Springer 1976

    Google Scholar 

  14. Zil'ber, B.I.: Uncountable categorical nilpotent groups and Lie algebras. Soviet Math. (Iz. VUZ)26(5), 98–99 (1982)

    Google Scholar 

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  1. Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076, Tübingen, Germany

    Claus Grünenwald & Frieder Haug

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  1. Claus Grünenwald
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  2. Frieder Haug
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Supported by the Deutsche Forschungsgemeinschaft

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Grünenwald, C., Haug, F. On stable torsion-free nilpotent groups. Arch Math Logic 32, 451–462 (1993). https://doi.org/10.1007/BF01270468

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