Skip to main content
SpringerLink
Log in

The Lattice of Fully Invariant Subgroups of a Cotorsion Group

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

In this monograph, we discuss some problems of the theory of Abelian groups. In particular, the lattice of fully invariant subgroups of a cotorsion group is studied in detail. The currently available results related to this topic are presented.

The purpose of this monograph is to acquaint undergraduates, senior students, and doctoral candidates with the state of the art in this field.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. P. S. Aleksandrov, Introduction to Set Theory and General Topology [in Russian], Nauka, Moscow (1977).

    Google Scholar 

  2. R. Baer, “Types of elements and characteristic subgroups of Abelian groups,” Proc. London Math. Soc. (2), 39, 481–514 (1935).

    Article  MathSciNet  Google Scholar 

  3. S. Bazzoni and L. Salce, “An independence result on cotorsion theories over valuation domains,” J. Algebra, 243, No. 1, 294–320 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  4. K. M. Benabdallah, B. J. Eisenstadt, J. M. Irwin, and E. W. Poluianov, “The structure of large subgroups of primary Abelian groups,” Acta Math. Acad. Sci. Hungar., 21, 421–435 (1970).

    Article  MATH  MathSciNet  Google Scholar 

  5. G. Birkhoff, Lattice Theory, Am. Math. Soc., Providence, Rhode Island (1967).

    MATH  Google Scholar 

  6. A. I. Borevich and I. R. Shafarevich, Number Theory, Pure Appl. Math., 20, Academic Press, New York–London (1966).

    Google Scholar 

  7. A. R. Chekhlov, “On projective invariant subgroups of Abelian groups,” Vestn. Tomsk. Gos. Univ. Mat. Mekh., No. 1, 31-36 (2009).

  8. A. L. S. Corner, “The independence of Kaplansky’s notions of transitivity and full transitivity,” Q. J. Math. Oxford Ser. (2), 27, No. 105, 15–20 (1976).

    Article  MATH  MathSciNet  Google Scholar 

  9. D. Cutler, J. Irwin, J. Pfaendtner, and T. Snabb, “pω+n-Projective Abelian p-groups having big direct sum of cyclic summands,” Lect. Notes Math., 1006, 523–533 Springer-Verlag, Berlin (1983).

    Google Scholar 

  10. D. Cutler and C. Missel, “The structure of C-decomposable pω+n-projective Abelian p-groups,” Commum. Algebra, 12, Nos. 3–4, 301–319 (1984).

    Article  MATH  MathSciNet  Google Scholar 

  11. P. V. Danchev and B. Goldsmith, “On the socles of fully invariant subgroups of Abelian p-groups,” Arch. Math. (Basel), 92, No. 3, 191–199 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  12. Yu. B. Dobrusin, “Torsion-free quasi-pure injective Abelian groups,” in: Abelian Groups and Modules [in Russian], 136, Tomsk. Gos. Univ., Tomsk (1980), pp. 45–69.

    Google Scholar 

  13. M. Dugas and S. Shelah, “E-Transitive groups in L,” in: Abelian Group Theory (Perth, 1987), Contemp. Math., 87, Am. Math. Soc., Providence, Rhode Island (1989), pp. 191–199.

  14. S. Files, “On transitive mixed Abelian groups,” in: Abelian Groups and Modules (Colorado Springs, CO, 1995), Lect. Notes Pure Appl. Math., 182, Marcel Dekker, New York (1996), pp. 243–251.

  15. S. Files and B. Goldsmith, “Transitive and fully transitive groups,” Proc. Am. Math. Soc., 126, No. 6, 1605–1610 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  16. L. Fuchs, Infinite Abelian Groups, Vol. I, Pure Appl. Math., 36, Academic Press, New York–London (1970).

  17. L. Fuchs, Infinite Abelian Groups, Vol. II, Pure Appl. Math., 36-II, Academic Press, New York–London (1973).

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

  19. R. Göbel, “The characteristic subgroups of the Baer–Specker group,” Math. Z., 140, 289–292 (1974).

    Article  MATH  MathSciNet  Google Scholar 

  20. R. Göbel, S. Shelah, and S. L. Wallutis, “On the lattice of cotorsion theories,” J. Algebra, 238, No. 1, 292–313 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  21. S. Ya. Grinshpon and P. A. Krylov, “Fully invariant subgroups, full transitivity, and homomorphism groups of Abelian groups,” J. Math. Sci., 128, No. 3, 2894–2997 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  22. S. Ya. Grinshpon and V. M. Misyakov, “On fully transitive Abelian groups,” in: Abelian Groups and Modules (1986), pp. 12–27.

  23. D. K. Harrison, “Infinite Abelian groups and homological methods,” Ann. Math. (2), 69, 366–391 (1959).

    Article  MATH  Google Scholar 

  24. P. Hill, “On transitive and fully transitive primary groups,” Proc. Am. Math. Soc., 22, 414–417 (1969).

    Article  MATH  Google Scholar 

  25. P. Hill and C. Megibben, “On the theory and classification of Abelian p-groups,” Math. Z., 190, No. 1, 17–38 (1985).

    Article  MATH  MathSciNet  Google Scholar 

  26. P. Hill and C. Megibben, “Extensions of torsion-complete groups,” Proc. Am. Math. Soc., 44, 259–262 (1974).

    Article  MATH  MathSciNet  Google Scholar 

  27. J. M. Irwin and E. A. Walker, “On N-high subgroups of Abelian groups,” Pac. J. Math., 11, 1363–1374 (1961).

    Article  MATH  MathSciNet  Google Scholar 

  28. I. Kaplansky, Infinite Abelian Groups, The University of Michigan Press, Ann Arbor (1969).

    MATH  Google Scholar 

  29. T. Kemoklidze, “On the full transitivity of a cotorsion hull,” Georgian Math. J., 13, No. 1, 79–84 (2006).

    MATH  MathSciNet  Google Scholar 

  30. T. Kemoklidze, “The lattice of fully invariant subgroups of a cotorsion hull,” Georgian Math. J., 16, No. 1, 89–104 (2009).

    MATH  MathSciNet  Google Scholar 

  31. T. Koyama, “On p-indicators in Ext(Q/Z, T),” Hiroshima Math. J., 8, No. 1, 211–216 (1978).

    MATH  MathSciNet  Google Scholar 

  32. L. Ya. Kulikov, “Generalized primary groups, I,” Tr. Mosk. Mat. Obshch., 1, 247–326 (1952).

    MathSciNet  Google Scholar 

  33. R. C. Linton, “On fully invariant subgroups of primary Abelian groups,” Michigan Math. J., 22, No. 3, 281–284 (1975).

    MathSciNet  Google Scholar 

  34. A. Mader, “The fully invariant subgroups of reduced algebraically compact groups,” Publ. Math. Debrecen, 17, 299–306 (1970).

    MathSciNet  Google Scholar 

  35. W. May and E. Toubassi, “Endomorphisms of Abelian groups and the theorem of Baer and Kaplansky,” J. Algebra, 43, No. 1, 1–13 (1976).

    Article  MATH  MathSciNet  Google Scholar 

  36. C. Megibben, “Large subgroups and small homomorphisms,” Michigan Math. J., 13, 153–160 (1966).

    Article  MATH  MathSciNet  Google Scholar 

  37. V. M. Misyakov, “On the complete transitivity of reduced Abelian groups,” in: Abelian Groups and Modules [in Russian], Nos. 11–12, Tomsk. Gos. Univ., Tomsk (1994), pp. 134–156.

    Google Scholar 

  38. J. D. Moore and E. J. Hewett, “On fully invariant subgroups of Abelian p-groups,” Comment. Math. Univ. St. Paul., 20, 97–106 (1971/72).

  39. A. I. Moskalenko, “Cotorsion hull of a separable p-group,” Algebra Logika, 28, No. 2, 207–226 (1989).

    Article  MathSciNet  Google Scholar 

  40. A. I. Moskalenko, “On pure completely characteristic subgroups of a coperiodic group,” in: Abelian Groups and Modules [in Russian], Nos. 11–12, Tomsk. Gos. Univ., Tomsk (1994), pp. 157–165.

    Google Scholar 

  41. R. J. Nunke, “Modules of extensions over Dedekind rings,” Illinois J. Math., 3, 222–241 (1959).

    MATH  MathSciNet  Google Scholar 

  42. R. J. Nunke, “Purity and subfunctors of the identity,” in: Topics in Abelian Groups. Proc. Symp., New Mexico State Univ., 1962, Scott, Foresman and Co., Chicago, Ill. (1963), pp. 121–171.

    Google Scholar 

  43. R. S. Pierce, “Homomorphisms of primary Abelian groups,” in: Topics in Abelian Groups. Proc. Symp., New Mexico State Univ., 1962, Scott, Foresman and Co., Chicago, Ill. (1963), pp. 215–310.

    Google Scholar 

  44. H. Prüfer, Unendliche abelsche Gruppen von Elementen endlicher ordnung, Dissertation, Berlin (1921).

  45. M. Shiffman, “The ring of automorphisms of an Abelian group,” Duke Math. J., 6, 579–597 (1940).

    Article  MathSciNet  Google Scholar 

  46. L. A. Skornyakov, Elements of Lattice Theory [in Russian], Nauka, Moscow (1982).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. A. Tsereteli Kutaisi State University, Kutaisi, Georgia

    T. Kemoklidze

Authors
  1. T. Kemoklidze
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to T. Kemoklidze.

Additional information

Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 87, Algebra, 2013.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kemoklidze, T. The Lattice of Fully Invariant Subgroups of a Cotorsion Group. J Math Sci 203, 621–751 (2014). https://doi.org/10.1007/s10958-014-2164-9

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-014-2164-9

Keywords

Search

Navigation