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Permutable Subgroups in GLn(D) and Applications to Locally Finite Group Algebras

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Abstract

In this paper we study the existence of free nonabelian subgroups in noncentral permutable subgroups of general skew linear groups and locally finite group algebras.

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References

  1. Aaghabali, M., Bien, M.H.: Self-invariant maximal subfields and their connexion with some conjectures in division rings. arXiv:http://arxiv.org/abs/1905.02246 (2019)

  2. Artin, E.: Geometric Algebra. Interscience Publishers, Inc., New York (1957)

    MATH  Google Scholar 

  3. Bateman, J.M.: On the solvability of unit groups of group algebras. Trans. Amer. Math. Soc. 157, 73–86 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bateman, J.M., Coleman, D.B.: Group algebras with nilpotent unit groups. Proc. Amer. Math. Soc. 19, 448–449 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bovdi, A.: Group algebras with an Engel group of units. J. Aust. Math. Soc. 80, 173–178 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bovdi, A.: Group algebras with a solvable group of units. Commun. Algebra 33, 3725–3738 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bovdi, A., Khripta, I.I.: The Engel property of the multiplicative group of a group algebra (in Russian). Mat. Sb. 182, 130–144 (1991). Engl. trans. in Math. USSR Sb. 72,121–134 (1992)

    MATH  Google Scholar 

  8. Bovdi, A., Khripta, I.I.: Group algebras of periodic groups with solvable multiplicative group (in Russian). Mat. Zametki 22, 421–432 (1977). Engl. trans. in Math. Notes. 22,725–731

    MathSciNet  Google Scholar 

  9. Chiba, K.: Free subgroups and free subsemigroups of division rings. J. Algebra 184, 570–574 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  10. Deo, T.T., Bien, M.H., Hai, B.X.: On weakly locally finite division rings. Acta. Math. Vietnam. 44, 553–569 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  11. Giambruno, A., Sehgal, S., Valenti, A.: Group algebras whose units satisfy a group identity. Proc. Amer. Math. Soc. 125, 629–634 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  12. Giambruno, A., Jespers, E., Valenti, A.: Group identities on units of rings. Arch. Math. (Basel) 63, 291–296 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gonçalves, J.Z.: Free subgroups in the group of units of group rings II. J. Number Theory 21, 121–127 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gonçalves, J.Z.: Free groups in subnormal subgroups and the residual nilpotence of the group of units of group rings. Can. Math. Bull. 27, 365–370 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gonçalves, J.Z., Mandel, A.: Semigroup identities on units of group algebras. Arch. Math. (Basel) 57, 539–545 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gonçalves, J.Z., Mandel, A.: Are there free groups in division rings?. Israel J. Math. 53, 69–80 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gonçalves, J.Z., Río, A.́D.: A survey on free subgroups in the group of units of group rings. J. Algebra Appl. 12, 1350004 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gross, F.: Subnormal, core-free, quasinormal subgroups are solvable. Bull. Lond. Math. Soc. 7, 93–95 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hai, B.X., Ngoc, N.K.: A note on the existence of non-cyclic free subgroups in division rings. Arch. Math. 101, 437–443 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kaplansky, I.: A theorem on division rings. Can. J. Math. 3, 290–292 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  21. Khripta, I.I.: The nilpotence of the multiplicative group of a group ring (in Russian). Mat. Zametki 11, 191–200 (1972). Engl. trans. in Math. Notes 11,119–124 (1972)

    MathSciNet  Google Scholar 

  22. Lam, T.Y.: A First Course in Noncommutative Rings. Graduate Texts in Mathematics, vol. 131. Springer-Verlag, New York (1991)

  23. Lichtman, A.I.: On subgroups of the multiplicative group of skew fields. Proc. Amer. Math. Soc. 63, 15–16 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  24. Liu, C.-H.: Some properties on rings with units satisfying a group identity. J. Algebra 232, 226–235 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  25. Liu, C.-H.: Group algebras with units satisfying a group identity. Proc. Amer. Math. Soc. 127, 327–336 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  26. Menal, P.: Private letter to B. Hartley. April 6 (1981)

  27. Ngoc, N.K., Bien, M.H., Hai, B.X.: Free subgroups in almost subnormal subgroups of general skew linear groups. Algebra i Analiz. 28, 220–235 (2017). Engl. trans. in St. Petersburg, Math. J. 28,707–717

    MathSciNet  MATH  Google Scholar 

  28. Passman, D.S.: The Algebraic Structure of Group Rings. Wiley-Interscience, New York (1977)

    MATH  Google Scholar 

  29. Ramezan-Nassab, M.: Group algebras with Engel unit groups. J. Aust. Math. Soc. 101, 244–252 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  30. Ramezan-Nassab, M.: Group algebras with locally nilpotent unit groups. Commun. Algebra 44, 604–612 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  31. Riley, D.M.: Group algebras with units satisfying an Engel identity. Rend. Circ. Mat. Palermo (2) 49, 540–544 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  32. Robinson, D.J.S.: A Course in the Theory of Groups, 2nd edn. Graduate Texts in Mathematics, vol. 80. Springer, New York (1996)

  33. Rotman, J.J.: Advanced Modern Algebra. 3rd edn. Graduate Studies in Mathematics. vol. 165. American Mathematical Society (2015)

  34. Sahai, M.: On the Jacobson radical and unit groups of group algebras. Publ. Mat. 42, 339–346 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  35. Shirvani, M., Wehrfritz, B.A.F.: Skew Linear Groups. London Mathematical Society Lecture Note Series, vol. 118. Cambridge University Press, Cambridge (1986)

  36. Stonehewer, S.E.: Permutable subgroups of infinite groups. Math. Z. 125, 1–16 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  37. Thompson, J.G.: An example of core-free quasinormal subgroups of p-groups. Math. Z. 96, 226–227 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  38. Tits, J.: Free subgroups in linear groups. J. Algebra 20, 250–270 (1972)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the referee for his/her careful reading and comments. The second and third authors were funded by Vietnam National University HoChiMinh City (VNUHCM) under grant number B2020-18-02.

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, University of Science, Ho Chi Minh City, Vietnam

    Le Qui Danh, Mai Hoang Bien & Bui Xuan Hai

  2. Vietnam National University, Ho Chi Minh City, Vietnam

    Le Qui Danh, Mai Hoang Bien & Bui Xuan Hai

  3. Department of Mathematics, Mechanics, and Informatics, University of Architecture Ho Chi Minh City, 196 Pasteur Str., Dist. 3, Ho Chi Minh City, Vietnam

    Le Qui Danh

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  1. Le Qui Danh
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  3. Bui Xuan Hai
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Danh, L.Q., Bien, M.H. & Hai, B.X. Permutable Subgroups in GLn(D) and Applications to Locally Finite Group Algebras. Vietnam J. Math. 51, 277–288 (2023). https://doi.org/10.1007/s10013-021-00513-8

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  • DOI: https://doi.org/10.1007/s10013-021-00513-8

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