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Elementary equivalence of the automorphism groups of Abelian p-groups

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We consider Abelian p-groups (p ≥ 3) A 1 = D 1G 1 and A 2 = D 2G 2, where D 1 and D 2 are divisible and G 1 and G 2 are reduced subgroups. We prove that if the automorphism groups Aut A 1 and Aut A 2 are elementarily equivalent, then the groups D 1, D 2 and G 1, G 2 are equivalent, respectively, in the second-order logic.

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References

  1. R. Baer, “Der Kern, eine charakteristische Untergruppe,” Compositio Math., 1, 254–283 (1934).

    MathSciNet  MATH  Google Scholar 

  2. C. I. Beidar and A. V. Mikhalev, “On Malcev’s theorem on elementary equivalence of linear groups,” Contemp. Math., 131, 29–35 (1992).

    MathSciNet  Google Scholar 

  3. D. L. Boyer, “On the theory of p-basic subgroups of Abelian groups,” in: Topics in Abelian Groups, Chicago (1963), pp. 323–330.

  4. E. I. Bunina, “Elementary equivalence of unitary linear groups over fields,” Fundam. Prikl. Mat., 4, No. 4, 1265–1278 (1998).

    MathSciNet  MATH  Google Scholar 

  5. E. I. Bunina, “Elementary equivalence of unitary linear groups over rings and skewfields,” Usp. Mat. Nauk, 53, No. 2, 137–138 (1998).

    MathSciNet  Google Scholar 

  6. E. I. Bunina, “Elementary equivalence of Chevalley groups over local rings,” Usp. Mat. Nauk, 61, No. 2, 349–350 (2006).

    MathSciNet  MATH  Google Scholar 

  7. E. I. Bunina and A. V. Mikhalev, “Elementary equivalence of endomorphism rings Abelian p-groups,” J. Math. Sci., 137, No. 6, 5212–5274 (2006).

    Article  MathSciNet  Google Scholar 

  8. E. I. Bunina and A. V. Mikhalev, “Elementary equivalence of categories of modules over rings, endomorphism rings, and automorphism groups of modules,” J. Math. Sci., 137, No. 6, 5275–5335 (2006).

    Article  MathSciNet  Google Scholar 

  9. M. Erdélyi, “Direct summands of Abelian torsion groups,” Acta Univ. Debrecen, 2, 145–149 (1955).

    MathSciNet  Google Scholar 

  10. L. Fuchs, “On the structure of Abelian p-groups,” Acta Math. Acad. Sci. Hung., 4, 267–288 (1953).

    Article  MATH  Google Scholar 

  11. L. Fuchs, “Notes on Abelian groups. I,” Ann. Univ. Sci. Budapest, 2, 5–23 (1959); II, Acta Math. Acad. Sci. Hung., 11, 117–125 (1960).

    MATH  Google Scholar 

  12. L. Fuchs, Infinite Abelian Groups, Tulane Univ. New Orleans (1970).

    MATH  Google Scholar 

  13. C. C. Chang and H. G. Keisler, Model Theory, Amer. Elsevier, New York (1973).

    MATH  Google Scholar 

  14. L. Yu. Kulikov, “To the theory of Abelian groups of arbitrary power,” Mat. Sb., 9, 165–182 (1941).

    MATH  Google Scholar 

  15. L. Yu. Kulikov, “To the theory of Abelian groups of arbitrary power,” Mat. Sb., 16, 129–162 (1945).

    Google Scholar 

  16. L. Yu. Kulikov, “Generalized primary groups. I,” Tr. Mosk. Mat. Obshch., 1, 247–326 (1952); II, Tr. Mosk. Mat. Obshch., 2, 85–167 (1953).

    MathSciNet  Google Scholar 

  17. A. I. Maltsev, “On elementary properties of linear groups,” in: Problems of Mathematics and Mechanics [in Russian], Novosibirsk (1961), pp. 110–132.

  18. H. Prüfer, “Untersuchungen über die Zerlegbarkeit der abzählbaren primären Abelschen Gruppen,” Math. Z., 17, 35–61 (1923).

    Article  MathSciNet  Google Scholar 

  19. S. Shelah, “Interpreting set theory in the endomorphism semi-group of a free algebra or in the category,” Ann. Sci. Univ. Clermont Math., 13, 1–29 (1976).

    Google Scholar 

  20. T. Szele, “On direct decomposition of Abelian groups,” J. London Math. Soc., 28, 247–250 (1953).

    Article  MathSciNet  MATH  Google Scholar 

  21. T. Szele, “On the basic subgroups of Abelian p-groups,” Acta Math. Acad. Sci. Hungar., 5, 129–141 (1954);

    Article  MathSciNet  MATH  Google Scholar 

  22. V. Tolstykh, “Elementary equivalence of infinite-dimensional classical groups,” Ann. Pure Appl. Logic, 105, 103–156 (2000).

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to E. I. Bunina.

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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 15, No. 7, pp. 81–112, 2009.

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Bunina, E.I., Roizner, M.A. Elementary equivalence of the automorphism groups of Abelian p-groups. J Math Sci 169, 614–635 (2010). https://doi.org/10.1007/s10958-010-0063-2

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