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Journal of Mathematical Sciences

, Volume 193, Issue 4, pp 586–590 | Cite as

A criterion of elementary equivalence of automorphism groups of reduced Abelian p-groups

  • M. A. Roizner
Article
  • 34 Downloads

Abstract

Consider reduced Abelian p-groups (p ≥ 3) A 1 and A 2. In this paper, we prove that the automorphism groups Aut A 1 and Aut A 2 are elementary equivalent if and only if the groups A 1 and A 2 are equivalent in second-order logic bounded by the cardinalities of the basic subgroups of A 1 and A 2.

Keywords

Automorphism Group Endomorphism Ring Associative Ring Chevalley Group Basic Subgroup 
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References

  1. 1.
    C. I. Beidar and A. V. Mikhalev, “On Malcev’s theorem on elementary equivalence of linear groups,” Contemp. Math., 131, 29–35 (1992).MathSciNetCrossRefGoogle Scholar
  2. 2.
    E. I. Bunina, “Elementary equivalence of unitary linear groups over fields,” Fundam. Prikl. Mat., 4, No. 4, 1265–1278 (1998).MathSciNetMATHGoogle Scholar
  3. 3.
    E. I. Bunina, “Elementary equivalence of unitary linear groups over rings and skewfields,” Russ. Math. Surv., 53, No. 2, 137–138 (1998).MathSciNetCrossRefGoogle Scholar
  4. 4.
    E. I. Bunina, “Elementary equivalence of Chevalley groups,” Russ. Math. Surv., 56, No. 1, 157–158 (2001).MathSciNetCrossRefGoogle Scholar
  5. 5.
    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).MathSciNetCrossRefGoogle Scholar
  6. 6.
    E. I. Bunina and A. V. Mikhalev, “Elementary equivalence of endomorphism rings Abelian p-groups,” J. Math. Sci., 137, No. 6, 5212–5274 (2006).MathSciNetCrossRefGoogle Scholar
  7. 7.
    E. I. Bunina and M. A. Roizner, “Elementary equivalence of the automorphism groups of Abelian p-groups,” J. Math. Sci., 169, No. 5, 614–635 (2010).MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    C. C. Chang and H. G. Keisler, Model Theory, Am. Elsevier, New York (1973).MATHGoogle Scholar
  9. 9.
    L. Fuchs, Infinite Abelian Groups, Vols. I, II, Tulane Univ., New Orleans, Louisiana (1970).Google Scholar
  10. 10.
    L. Kaloujnine, “Sur les groupes abéliens primaires sans éléments de hauteur infinie,” C. R. Acad. Sci. Paris, 225, 713–715 (1947).MathSciNetGoogle Scholar
  11. 11.
    L. Ya. Kulikov, “Generalized primary groups. I, II,” Tr. Mosk. Mat. Obshch., 1, 247–326 (1952), 2, 85–167 (1953).MathSciNetGoogle Scholar
  12. 12.
    A. I. Maltsev, “On elementary properties of linear groups,” in: Problems of Mathematics and Mechanics [in Russian], pp. 110–132 (1961).Google Scholar
  13. 13.
    M. A. Roizner, Elementary Equivalence of the Automorphism Groups of Reduced Abelian p-Groups, arXiv:math.GR/1207.1951v1.Google Scholar
  14. 14.
    T. Szele, “On the basic subgroups of abelian p-groups,” Acta Math. Acad. Sci. Hungar., 5, 129–141 (1954), Math. Soc., 28, 247–250 (1953).MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    V. Tolstykh, “Elementary equivalence of infinite-dimensional classical groups,” Ann. Pure Appl. Logic, 105, 103–156 (2000).MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Moscow State UniversityMoscowRussia

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