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.
Similar content being viewed by others
References
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.
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.
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.
G. K. Bakshi, S. Maheshwary, and I. B. S. Passi, Group rings and the RS-property, Comm. Algebra 47 (2019), no. 3, 969–977.
R. A. Ferraz, Simple components and central units in group algebras, J. Algebra 279 (2004), no. 1, 191–203.
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.
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.
S. Maheshwary, and I. B. S. Passi, Units and augmentation powers in integral group rings, J. Group Theory, 23(2020), no. 6, 931–944.
I. Musson and A. Weiss, Integral group rings with residually nilpotent unit groups, Arch. Math. (Basel) 38 (1982), no. 6, 514–530.
I. B. S. Passi, Group rings and their augmentation ideals, Lecture Notes in Mathematics, vol. 715, Springer, Berlin, 1979.
J. Ritter and S. K. Sehgal, Trivial units in\(RG\), Math. Proc. R. Ir. Acad. 105A (2005), no. 1, 25–39.
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.
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.
R. K. Sharma and S. Gangopadhyay, On units in\({\bf Z}D_8\), PanAmer. Math. J. 11 (2001), no. 1, 1–9.
R. K. Sharma, S. Gangopadhyay, and V. Vetrivel, On units in\({\bf Z}S_3\), Comm. Algebra 25 (1997), no. 7, 2285–2299.
S. K. Sehgal and H. J. Zassenhaus, Integral group rings with nilpotent unit groups, Comm. Algebra 5 (1977), no. 2, 101–111.
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).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Gadadhar Misra.
Dedicated to Prof. Passi on his 80th birthday.
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13226-021-00184-8