Advertisement

Monatshefte für Mathematik

, Volume 187, Issue 4, pp 603–615 | Cite as

Finiteness conditions for the non-abelian tensor product of groups

  • R.  Bastos
  • I. N. Nakaoka
  • N. R.  Rocco
Article
  • 102 Downloads

Abstract

Let G, H be groups that act compatibly on each other and consider the non-abelian tensor product \(G \otimes H\). We prove that the set of all tensors \(T_{\otimes }(G,H)=\{g\otimes h{:}\,g \in G,\,h\in H\}\) is finite if and only if the non-abelian tensor product \(G \otimes H\) is finite. Further, we examine a finiteness criterion for \(G \otimes H\), when G and H are FC-groups. We also establish a sufficient condition for a finitely generated non-abelian tensor product to be finite. Some finiteness conditions for G in terms of certain torsion elements of the non-abelian tensor square \(G \otimes G\) are also studied.

Keywords

Local properties Locally finite groups Non-abelian tensor product of groups 

Mathematics Subject Classification

20E25 20F50 20J06 

References

  1. 1.
    Bastos, R., Rocco, N.R.: The non-abelian tensor square of residually finite groups. Monatsh. Math. 183, 61–69 (2017)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Blyth, R.D., Fumagalli, F., Morigi, M.: Some structural results on the non-abelian tensor square of groups. J. Group Theory 13, 83–94 (2010)MathSciNetCrossRefGoogle 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)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Brown, R., Loday, J.-L.: Van Kampen theorems for diagrams of spaces. Topology 26, 311–335 (1987)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Donadze, G., Ladra, M., Thomas, V.Z.: On some closure properties of the non-abelian tensor product. J. Algebra 472, 399–413 (2017)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Ellis, G.: The non-abelian tensor product of finite groups is finite. J. Algebra 111, 203–205 (1987)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Ellis, G., Leonard, F.: Computing Schur multipliers and tensor products of finite groups. Proc. R. Ir. Acad. 95A, 137–147 (1995)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Ladra, M., Thomas, V.Z.: Two generalizations of the nonabelian tensor product. J. Algebra 369, 96–113 (2012)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Macedoǹska, O.: On difficult problems and locally graded groups. J. Math. Sci. (N.Y.) 142, 1949–1953 (2007)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Nakaoka, I.N.: Non-abelian tensor products of solvable groups. J. Group Theory 3, 157–167 (2000)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Parvizi, M., Niroomand, P.: On the structure of groups whose exterior or tensor square is a \(p\)-group. J. Algebra 352, 347–353 (2012)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Robinson, D.J.S.: A Course in the Theory of Groups, 2nd edn. Springer, New York (1996)CrossRefGoogle Scholar
  13. 13.
    Rocco, N.R.: On a construction related to the non-abelian tensor square of a group. Bull. Braz. Math. Soc. 22, 63–79 (1991)CrossRefGoogle Scholar
  14. 14.
    Rocco, N.R.: A presentation for a crossed embedding of finite solvable groups. Commun. Algebra 22, 1975–1998 (1994)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Rosenlicht, M.: On a result of Baer. Proc. Am. Math. Soc. 13, 99–101 (1962)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Shumyatsky, P.: Applications of Lie ring methods to group theory. In: Costa, R., Grishkov, A., Guzzo Jr., H., Peresi, L.A. (eds.) Non-associative Algebra and its Applications, Lecture Notes in Pure and Applied Mathematics, vol. 211, pp. 373–395. Dekker, New York (2000)Google Scholar
  17. 17.
    Thomas, V.Z.: The non-abelian tensor product of finite groups is finite: a homology-free proof. Glasgow Math. J. 52, 473–477 (2010)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Visscher, M.P.: On the nilpotency class and solvability lenght of nonabelian tensor products of groups. Arch. Math. 73, 161–171 (1999)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Zelmanov, E.: The solution of the restricted Burnside problem for groups of odd exponent. Math. USSR Izv. 36, 41–60 (1991)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Zelmanov, E.: The solution of the restricted Burnside problem for 2-groups. Math. Sb. 182, 568–592 (1991)Google Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidade de BrasíliaBrasíliaBrazil
  2. 2.Departamento de MatemáticaUniversidade Estadual de MaringáMaringáBrazil

Personalised recommendations