Mathematical Notes

, Volume 64, Issue 6, pp 798–803 | Cite as

Center of some solvable groups with one defining relation

  • E. I. Timoshenko


Necessary and sufficient conditions for the center of a metabelian group with one defining relation to be nontrivial are found. The center of such a group is described. The center of a group of the formF/N′〈g〉 F is studied under certain conditions. By means of a new technique, the recent result of A. F. Krasnikov and the author on the center of a group of the above form is sharpened.

Key words

metabelian group free group center of a group n-solvable group defining relations 


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  1. 1.
    Unsolved Problems in Group Theory. The Kourovka Notebook [in Russian], 13th edition, Novosibirsk (1995).Google Scholar
  2. 2.
    E. I. Timoshenko, “Center of a group with one defining relation in the variety of 2-solvable groups,”Sibirsk. Mat. Zh. [Siberian Math. J.],14, No. 6, 1351–1355 (1973).Google Scholar
  3. 3.
    M. I. Kargapolov,Selected Works. Groups [in Russian], Nauka, Novosibirsk (1991).Google Scholar
  4. 4.
    A. F. Krasnikov and E. I. Timoshenko, “Two remarks on the groups of the formF/[N, N],” in:Dep. VINITI No.4567-B85 [in Russian], VINITI, Moscow (1985).Google Scholar
  5. 5.
    S. Bachmuth, “Automorphisms of free metabelian groups,”Trans. Amer. Math. Soc.,118, 93–104 (1965).MATHMathSciNetGoogle Scholar
  6. 6.
    A. I. Mal'tsev, “On the equationzxyx −1 y −1 z −1=aba −1 b −1 in a free group,”Algebra i Logika [Algebra and Logic],1, No. 5, 45–50 (1962).MATHMathSciNetGoogle Scholar
  7. 7.
    S. Bachmuth, E. Formanek, and H. Y. Mochizuki, “IA-automorphisms of two-generator torsion-free groups,”J. Algebra,40, No. 1, 19–30 (1976).MathSciNetGoogle Scholar
  8. 8.
    E. I. Timoshenko, “On the primitive elements of the free groups of the variety 803-01,”Mat. Zametki [Math. Notes],61, No. 6, 884–889 (1997).MATHMathSciNetGoogle Scholar

Copyright information

© Kluwer Academic/Plenum Publishers 1999

Authors and Affiliations

  • E. I. Timoshenko
    • 1
  1. 1.Novosibirsk State Academy of ConstructionUSSR

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