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Archiv der Mathematik

, Volume 107, Issue 2, pp 127–133 | Cite as

Non-abelian tensor square of finite-by-nilpotent groups

  • Raimundo Bastos
  • Noraí R. Rocco
Article

Abstract

Let G be a group. We denote by \({\nu(G)}\) an extension of the non-abelian tensor square \({G \otimes G}\) by \({G \times G}\). We prove that if G is finite-by-nilpotent, then the non-abelian tensor square \({G \otimes G}\) is finite-by-nilpotent. Moreover, \({\nu(G)}\) is nilpotent-by-finite (Theorem A). Also we characterize BFC-groups in terms of \({\nu(G)}\) among the groups G in which the derived subgroup is finitely generated (Theorem B).

Mathematics Subject Classification

Primary 20E34 20F24 Secondary 20F14 

Keywords

Structure theorems Derived series FC-groups 

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References

  1. 1.
    Blyth R.D., Morse R.F.: Computing the nonabelian tensor square of polycyclic groups. J. Algebra 321, 2139–2148 (2009)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Brown R., Loday J.-L.: Van Kampen theorems for diagrams of spaces. Topology 26, 311–335 (1987)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Brown R., Johnson D.L., Robertson E.F.: Some computations of non-abelian tensor products of groups. J. Algebra 111, 177–202 (1987)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Gumber D., Kalra H.: On the converse of a theorem of Schur. Arch. Math. 101, 17–20 (2013)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Ph. Hall, Finite-by-nilpotent groups, Proc. Camb. Philos. Soc. 52 (1956), 611–616Google Scholar
  6. 6.
    Hilton P.: On a theorem of Schur. Int. J. Math. Math. Sci. 28, 455–460 (2001)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    L.-C. Kappe, Nonabelian tensor products of groups: the commutator connection, Proc. Groups St. Andrews 1997 at Bath, London Math. Soc. Lecture Notes 261 (1999), 447–454.Google Scholar
  8. 8.
    B. C. R. Lima and R. N De Oliveira, Weak commutativity between two isomorophic polycyclic groups, J. Group. Theory 19 (2016), 239–248.Google Scholar
  9. 9.
    Moravec P.: The exponents of nonabelian tensor products of groups. J. Pure Appl. Algebra 212, 1840–1848 (2008)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Nakaoka I.N., Rocco N.R.: A survey of non-abelian tensor products of groups and related constructions. Bol. Soc. Paran. Mat. 30, 77–89 (2012)MathSciNetGoogle Scholar
  11. 11.
    Neumann B.H.: Groups with finite classes of conjugate elements. Prof. London Math. Soc. 1, 178–187 (1951)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Niroomand P.: The converse of Schur’s theorem. Arch. Math. 94, 401–403 (2010)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Niroomand P., Parvizi M.: On the structure of groups whose exterior or tensor square is a p-group. J. Algebra 352, 347–353 (2012)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    D. J. S. Robinson, A course in the theory of groups, 2nd edition, Springer-Verlag, New York, 1996.Google Scholar
  15. 15.
    Rocco N.R.: On a construction related to the non-abelian tensor square of a group, Bol. Soc. Brasil Mat. 22, 63–79 (1991)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Rocco N.R.: A presentation for a crossed embedding of finite solvable groups. Comm. Algebra 22, 1975–1998 (1994)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    M K. Yadav, Converse of Schur’s Theorem – A statement, preprint available at arXiv:1212.2710v2 [math.GR].

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidade de BrasíliaBrasiliaBrazil

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