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The lower central series of the unit group of an integral group ring

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Abstract

The aim of this article is to draw attention towards various natural but unanswered questions related to the lower central series of the unit group of an integral group ring.

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References

  1. S. R. Arora, A.W. Hales, and I.B.S. Passi, Jordan decomposition and in integral group rings, Comm. Algebra 21 (1993), no. 1, 25–35.

    Article  MathSciNet  Google Scholar 

  2. A. Bächle, S. Maheshwary, and L. Margolis, Abelianization of the unit group of an integral group ring, Pac. J. Math. 312(2021), no. 2, 309–334.

  3. G. K. Bakshi, S. Maheshwary, and I. B. S. Passi, Integral group rings with all central units trivial, J. Pure Appl. Algebra 221 (2017), no. 8, 1955–1965.

    Article  MathSciNet  Google Scholar 

  4. G. K. Bakshi, S. Maheshwary, and I. B. S. Passi, Group rings and the RS-property, Comm. Algebra 47 (2019), no. 3, 969–977.

    Article  MathSciNet  Google Scholar 

  5. R. A. Ferraz, Simple components and central units in group algebras, J. Algebra 279 (2004), no. 1, 191–203.

    Article  MathSciNet  Google Scholar 

  6. B. Hartley and P. F. Pickel, Free subgroups in the unit groups of integral group rings, Canad. J. Math. 32 (1980), no. 6, 1342–1352.

    Article  MathSciNet  Google Scholar 

  7. S. Maheshwary and I. B. S. Passi, The upper central series of the unit groups of integral group rings: a survey, 2018, Group Theory and Computation, Indian Statistical Institute Series, Springer, pp. 175–195.

  8. S. Maheshwary, and I. B. S. Passi, Units and augmentation powers in integral group rings, J. Group Theory, 23(2020), no. 6, 931–944.

  9. I. Musson and A. Weiss, Integral group rings with residually nilpotent unit groups, Arch. Math. (Basel) 38 (1982), no. 6, 514–530.

    Article  MathSciNet  Google Scholar 

  10. I. B. S. Passi, Group rings and their augmentation ideals, Lecture Notes in Mathematics, vol. 715, Springer, Berlin, 1979.

  11. J. Ritter and S. K. Sehgal, Trivial units in\(RG\), Math. Proc. R. Ir. Acad. 105A (2005), no. 1, 25–39.

    Article  MathSciNet  Google Scholar 

  12. S. K. Sehgal, Units in integral group rings, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 69, Longman Scientific & Technical, Harlow, 1993, With an appendix by Al Weiss.

  13. R. K. Sharma and S. Gangopadhyay, On chains in units of\({\bf Z}A_4\), Math. Sci. Res. Hot-Line 4 (2000), no. 9, 1–33.

    MathSciNet  Google Scholar 

  14. R. K. Sharma and S. Gangopadhyay, On units in\({\bf Z}D_8\), PanAmer. Math. J. 11 (2001), no. 1, 1–9.

    MathSciNet  Google Scholar 

  15. R. K. Sharma, S. Gangopadhyay, and V. Vetrivel, On units in\({\bf Z}S_3\), Comm. Algebra 25 (1997), no. 7, 2285–2299.

    Article  MathSciNet  Google Scholar 

  16. S. K. Sehgal and H. J. Zassenhaus, Integral group rings with nilpotent unit groups, Comm. Algebra 5 (1977), no. 2, 101–111.

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The author’s research is supported by DST, India (INSPIRE/04/2017/000897). This research was also supported in part by the International Centre for Theoretical Sciences (ICTS) during a visit for participating in the program- Group Algebras, Representations and Computation (Code: ICTS/Prog-garc2019/10).

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Correspondence to Sugandha Maheshwary.

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Communicated by Gadadhar Misra.

Dedicated to Prof. Passi on his 80th birthday.

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Maheshwary, S. The lower central series of the unit group of an integral group ring. Indian J Pure Appl Math 52, 709–712 (2021). https://doi.org/10.1007/s13226-021-00184-8

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  • DOI: https://doi.org/10.1007/s13226-021-00184-8

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