Siberian Mathematical Journal

, Volume 47, Issue 4, pp 634–642 | Cite as

First-order definability and algebraicity of the sets of annihilating and generating collections of elements for some relatively free solvable groups

  • Ch. K. Gupta
  • E. I. Timoshenko


We study the first-order definable, Diophantine, and algebraic subsets in the set of all ordered sets generating a group or generating a group as a normal subgroup for some relatively free solvable groups.


generating collection annihilating collection Diophantine set algebraic set first-order definable set solvable group nilpotent group 


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  1. 1.
    Mal’tsev A. I., “On the equation zxyx −1 y −1 z −1 = aba −1 b −1 in a free group,” Algebra i Logika, 1, No. 5, 45–50 (1962).MathSciNetzbMATHGoogle Scholar
  2. 2.
    Baumslag G., Myasnikov A. G., and Remeslennikov V. N., “Algebraic geometry over groups. I. Algebraic sets and ideal theory,” J. Algebra, 219, No. 1, 16–79 (1999).zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Roman’kov V. A., “Test elements for free solvable groups of rank 2,” Algebra and Logic, 40, No. 2, 106–111 (2001).MathSciNetCrossRefGoogle Scholar
  4. 4.
    Remeslennikov V. N. and Sokolov V. G., “Some properties of Magnus embeddings,” Algebra i Logika, 9, No. 5, 566–578 (1970).zbMATHMathSciNetGoogle Scholar
  5. 5.
    Neumann H., Varieties of Groups [Russian translation], Mir, Moscow (1969).Google Scholar
  6. 6.
    Rhemtulla A. and Akhavan-Malayeri M., “Commutator length of abelian-by-nilpotent groups,” Glasgow Math. Z., 40, No. 1, 117–121 (1998).zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Timoshenko E. I., “On universal theories of metabelian groups and the Shmel’kin embedding,” Siberian Math. J., 42, No. 5, 981–986 (2001).zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Mal’tsev A. I., “On free solvable groups,” Dokl. Akad. Nauk SSSR, 130, No. 3, 495–498 (1960).Google Scholar
  9. 9.
    Ershov Yu. L., Decidability Problems and Constructive Models [in Russian], Nauka, Moscow (1980).Google Scholar
  10. 10.
    Penzin Yu. G., “Undecidability of the integer theory with addition and the coprimality predicate, ” in: Abstracts: 3 All-Union Conference on Mathematical Logic [in Russian], Novosibirsk, 1974, pp. 149–153.Google Scholar
  11. 11.
    Bachmuth S., “Automorphisms of free metabelian groups,” Trans. Amer. Math. Soc., 118, 93–104 (1965).zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Gupta C. K. and Timoshenko E. I., “Automorphic and endomorphic reducibility and primitive endomorphisms of free metabelian groups,” Comm. Algebra, 25, 3057–3070 (1997).zbMATHMathSciNetGoogle Scholar
  13. 13.
    Gupta Ch. K. and Timoshenko E. I., “Test rank for some free polynilpotent groups,” Algebra and Logic, 42, No. 1, 20–28 (2003).zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • Ch. K. Gupta
    • 1
  • E. I. Timoshenko
    • 2
  1. 1.Manitoba UniversityWinnipegCanada
  2. 2.Novosibirsk State University of Architecture and BuildingNovosibirskRussia

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