Skip to main content
Log in

Solvable Groups of Finite Metabelian Rank

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

This article has been updated

Abstract

Studying solvable nonabelian groups of finite metabelian rank, we construct an example of a solvable group of finite metabelian rank and infinite special rank. Also, we describe the structure of solvable groups of finite metabelian rank.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

Change history

  • 29 January 2021

    A typo in the author’s name is corrected in the HTML version.

References

  1. Maltsev A. I., “On groups of finite rank,” Mat. Sb., vol. 22, no. 2, 351–352 (1948).

    MathSciNet  Google Scholar 

  2. Maltsev A. I., “On some classes of infinite solvable groups,” Mat. Sb., vol. 28, no. 3, 567–588 (1951).

    Google Scholar 

  3. Glushkov V. M., “On some questions of the theory of nilpotent and locally nilpotent groups without torsion,” Mat. Sb., vol. 30, no. 1, 79–104 (1952).

    MathSciNet  Google Scholar 

  4. Smirnov D. M., “On automorphisms of soluble groups,” Dokl. Akad. Nauk SSSR, vol. 84, no. 5, 891–894 (1953).

    Google Scholar 

  5. Smirnov D. M., “On groups of automorphisms of soluble groups,” Mat. Sb., vol. 32, no. 2, 365–384 (1953).

    MathSciNet  Google Scholar 

  6. Smirnov D. M., “The two classes of soluble groups of finite rank,” Ivanovo Gos. Ped. Inst. Uchen. Zap., vol. 18, 67–74 (1958).

    Google Scholar 

  7. Charin V. S., “On the theory of locally nilpotent groups,” Mat. Sb., vol. 29, no. 2, 433–454 (1951).

    MathSciNet  Google Scholar 

  8. Charin V. S., “On groups of automorphisms of some classes of soluble groups,” Ukrain. Mat. Zh., vol. 5, no. 4, 363–369 (1953).

    MATH  Google Scholar 

  9. Charin V. S., “On locally solvable groups of finite rank,” Mat. Sb., vol. 41, no. 1, 37–48 (1957).

    MathSciNet  Google Scholar 

  10. Charin V. S., “Solvable groups of type \( A_{4} \),” Mat. Sb., vol. 52, no. 3, 895–914 (1960).

    MathSciNet  MATH  Google Scholar 

  11. Charin V. S., “Solvable groups of type \( A_{3} \),” Mat. Sb., vol. 54, no. 4, 489–499 (1961).

    MathSciNet  Google Scholar 

  12. Kargapolov M. I., “On soluble groups of finite rank,” Algebra i Logika, vol. 1, no. 5, 37–44 (1962).

    MathSciNet  Google Scholar 

  13. Merzlyakov Yu. I., “Locally solvable groups of finite rank,” Algebra i Logika, vol. 3, no. 2, 5–16 (1964).

    MathSciNet  Google Scholar 

  14. Merzlyakov Yu. I., “Locally solvable groups of finite rank. II,” Algebra and Logic, vol. 8, no. 6, 391–393 (1969).

    Article  MathSciNet  Google Scholar 

  15. Baer R. and Heineken H., “Radical groups with finite abelian subgroup rank,” Illinois J. Math., vol. 16, no. 4, 533–580 (1972).

    Article  MathSciNet  Google Scholar 

  16. Gorchakov Yu. M., “On the existence of abelian subgroups of infinite rank in locally solvable groups,” Dokl. Akad. Nauk SSSR, vol. 156, no. 1, 17–20 (1964).

    MathSciNet  MATH  Google Scholar 

  17. Shunkov V. P., “On locally finite groups of finite rank,” Algebra and Logic, vol. 10, no. 2, 127–142 (1971).

    Article  MathSciNet  Google Scholar 

  18. Robinson D., “A new treatment of soluble groups with finiteness conditions on their Abelian subgroups,” Uspekhi Mat. Nauk, vol. 34, no. 1, 197–215 (1979).

    MathSciNet  MATH  Google Scholar 

  19. Kargapolov M. I. and Merzlyakov Yu. I., Fundamentals of the Theory of Groups, Springer, New York, Heidelberg, and Berlin (1979).

    Book  Google Scholar 

  20. Robinson D. J. S., “A note on groups of finite rank,” Comp. Math., vol. 21, no. 3, 240–246 (1969).

    MathSciNet  MATH  Google Scholar 

  21. Dashkova O. Yu., “Solvable groups of finite non-Abelian rank,” Ukrainian Math. J., vol. 42, no. 2, 140–144 (1990).

    Article  MathSciNet  Google Scholar 

  22. Dashkova O. Yu., “Locally almost solvable groups of finite non-Abelian rank,” Ukrainian Math. J., vol. 42, no. 4, 421–425 (1990).

    Article  MathSciNet  Google Scholar 

  23. Dashkova O. Yu., “Locally nilpotent groups of finite non-Abelian sectional rank,” Ukrainian Math. J., vol. 47, no. 4, 524–527 (1995).

    Article  MathSciNet  Google Scholar 

  24. Dashkova O. Yu., “Solvable groups of finite non-Abelian sectional rank,” Ukrainian Math. J., vol. 48, no. 3, 467–470 (1996).

    Article  MathSciNet  Google Scholar 

  25. Dashkova O. Yu., “Groups of finite non-Abelian sectional rank,” Ukrainian Math. J., vol. 49, no. 10, 1494–1500 (1997).

    Article  MathSciNet  Google Scholar 

  26. Dashkova O. Y., Dixon M. R., and Kurdachenko L. A., “Linear groups with rank restrictions on the subgroups of infinite central dimension,” J. Pure Appl. Algebra, vol. 208, no. 3, 785–795 (2007).

    Article  MathSciNet  Google Scholar 

  27. Dashkova O. Yu., “Solvable infinite-dimensional linear groups with restrictions on the nonabelian subgroups of infinite rank,” Sib. Math. J., vol. 49, no. 6, 1023–1033 (2008).

    Article  MathSciNet  Google Scholar 

  28. Dashkova O. Yu., “Locally soluble infinite-dimensional linear groups with restrictions on non-Abelian subgroups of infinite ranks,” Algebra and Logic, vol. 47, no. 5, 340-347 (2008).

    Article  Google Scholar 

  29. Dashkova O. Yu., Groups of Finite Metabelian Rank [Russian], Inst. Mat., Kiev (1990).

    MATH  Google Scholar 

  30. Dashkova O. Yu., “Some classes of groups of finite metabelian rank,” Mat. Tr., vol. 23, no. 1, 123–136 (2020).

    Article  MathSciNet  Google Scholar 

  31. Huppert B., Endliche Gruppen I, Springer, Berlin etc. (1967).

    Book  Google Scholar 

  32. Hall P., “On the finiteness of certain soluble groups,” in: News in Forensic Science. Mathematic. Vol. 21, Mir, Moscow (1981), 171–206.

  33. Fuchs L., Infinite Abelian Groups. Vol. 2, Academic, New York and London (1977).

    Google Scholar 

  34. Myagkova N. N., “On groups of finite rank,” Izv. Akad. Nauk SSSR Ser. Mat., vol. 13, no. 6, 495–512 (1949).

    MathSciNet  Google Scholar 

  35. Lennox J. C. and Robinson D. J. S., “Soluble products of nilpotent groups,” Rend. Mat. Univ. Padova, vol. 62, 261–280 (1980).

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to O. Y. Dashkova.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dashkova, O.Y. Solvable Groups of Finite Metabelian Rank. Sib Math J 61, 1066–1074 (2020). https://doi.org/10.1134/S0037446620060075

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0037446620060075

Keywords

UDC

Navigation