Advertisement

Siberian Mathematical Journal

, Volume 60, Issue 1, pp 148–152 | Cite as

Generalized Rigid Metabelian Groups

  • N. S. RomanovskiiEmail author
Article
  • 4 Downloads

Abstract

We study the generalized rigid groups (r-groups), in the metabelian case in more detail. The periodic r-groups are described. We prove that each divisible metabelian r-group decomposes as a semidirect product of two abelian subgroups, each metabelian r-group independently embeds into a divisible metabelian r-group, and the intersection of each collection of divisible subgroups of a metabelian r-group is divisible too.

Keywords

soluble group metabelian group divisible group 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Romanovskii N. S., “Equational Noetherianness of rigid soluble groups,” Algebra and Logic, vol. 48, No. 2, 147–160 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Myasnikov A. and Romanovskii N., “Krull dimension of solvable groups,” J. Algebra, vol. 324, No. 10, 2814–2831 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Romanovskii N. S., “Divisible rigid groups,” Algebra and Logic, vol. 47, No. 6, 426–434 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Myasnikov A. G. and Romanovskii N. S., “Logical aspects of the theory of divisible rigid groups,” Dokl. Math., vol. 90, No. 3, 697–698 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Romanovskii N. S., “Divisible rigid groups. Algebraic closedness and elementary theory,” Algebra and Logic, vol. 56, No. 5, 395–408 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Romanovskii N. S., “Irreducible algebraic sets over divisible decomposed rigid groups,” Algebra and Logic, vol. 48, No. 6, 449–464 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Romanovskiy N. S., “Hilbert’s Nullstellensatz in algebraic geometry over rigid soluble groups,” Izv. Math., vol. 79, No. 5, 1051–1063 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Myasnikov A. G. and Romanovskii N. S., “Model–theoretic aspects of the theory of divisible rigid soluble groups,” Algebra and Logic, vol. 56, No. 1, 82–84 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Myasnikov A. G. and Romanovskii N. S., “Divisible rigid groups. II. Stability, saturation, and elementary submodels,” Algebra and Logic, vol. 57, No. 1, 29–38 (2018).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Romanovskii N. S., “Generalized rigid groups: Definitions, basic properties, and problems,” Sib. Math. J., vol. 59, No. 4, 705–709 (2018).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Romanovskii N. S., “Decomposition of a group over an Abelian normal subgroup,” Algebra and Logic, vol. 55, No. 4, 315–326 (2016).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirsk State UniversityNovosibirskRussia

Personalised recommendations