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On the commutator subgroup of the unit group of a modular group algebra

  • Tibor Juhász EMAIL logo and Meena Sahai
Published/Copyright: August 5, 2025
Journal of Group Theory
From the journal Journal of Group Theory

Abstract

Let 𝐹 be a field of odd characteristic and 𝐺 a group for which the group algebra F G is modular. In this paper, necessary and sufficient conditions are presented for the commutator subgroup of the unit group of F G to be nilpotent of class at most 2.

Award Identifier / Grant number: K132951

Funding statement: The first author was supported by the National Research, Development and Innovation Office (NKFIH) Grant No. K132951.

Acknowledgements

The authors are grateful to the referee for careful reading of the paper and his/her valuable suggestions that have improved its presentation.

  1. Communicated by: Evgenii I. Khukhro

References

[1] A. Bovdi, Group algebras with a solvable group of units, Comm. Algebra 33 (2005), no. 10, 3725–3738. 10.1080/00927870500243213Search in Google Scholar

[2] F. Catino and E. Spinelli, On the derived length of the unit group of a group algebra, J. Group Theory 13 (2010), no. 4, 577–588. 10.1515/jgt.2010.008Search in Google Scholar

[3] D. B. Coleman and R. Sandling, Mod 2 group algebras with metabelian unit groups, J. Pure Appl. Algebra 131 (1998), no. 1, 25–36. 10.1016/S0022-4049(97)00105-9Search in Google Scholar

[4] A. Giambruno, S. K. Sehgal and A. Valenti, Group identities on units of group algebras, J. Algebra 226 (2000), no. 1, 488–504. 10.1006/jabr.1999.8203Search in Google Scholar

[5] T. Juhász, The derived length of the unit group of a group algebra—the case G = Syl p ( G ) , J. Algebra Appl. 16 (2017), no. 8, Article ID 1750142. 10.1142/S0219498817501420Search in Google Scholar

[6] T. Juhász, G. T. Lee, S. K. Sehgal and E. Spinelli, On the lower bound of the derived length of the unit group of a nontorsion group algebra, Algebr. Represent. Theory 23 (2020), no. 2, 457–466. 10.1007/s10468-019-09855-xSearch in Google Scholar

[7] T. Juhász and M. Sahai, Modular group algebras with centrally metabelian unit groups, J. Algebra 632 (2023), 783–800. 10.1016/j.jalgebra.2023.06.011Search in Google Scholar

[8] T. Juhász and E. Spinelli, Group rings with metabelian unit groups, J. Pure Appl. Algebra 226 (2022), no. 6, Article ID 106946. 10.1016/j.jpaa.2021.106946Search in Google Scholar

[9] I. I. Khripta, The nilpotence of the multiplicative group of a group ring, Math. Notes 11 (1972), 119–124. 10.1007/BF01097929Search in Google Scholar

[10] B. Külshammer and R. K. Sharma, Lie centrally metabelian group rings in characteristic 3, J. Algebra 180 (1996), no. 1, 111–120. 10.1006/jabr.1996.0055Search in Google Scholar

[11] J. Kurdics, On group algebras with metabelian unit groups, Period. Math. Hungar. 32 (1996), no. 1–2, 57–64. 10.1007/BF01879732Search in Google Scholar

[12] G. T. Lee, Group Identities on Units and Symmetric Units of Group Rings, Algebr. Appl. 12, Springer, London, 2010. 10.1007/978-1-84996-504-0Search in Google Scholar

[13] F. Levin and G. Rosenberger, Lie metabelian group rings, Group and Semigroup Rings (Johannesburg 1985), North-Holland Math. Stud. 126, North-Holland, Amsterdam (1986), 153–161. 10.1016/S0304-0208(08)71519-6Search in Google Scholar

[14] C.-H. Liu and D. S. Passman, Group algebras with units satisfying a group identity. II, Proc. Amer. Math. Soc. 127 (1999), no. 2, 337–341. 10.1090/S0002-9939-99-04684-5Search in Google Scholar

[15] D. S. Passman, Group algebras whose units satisfy a group identity. II, Proc. Amer. Math. Soc. 125 (1997), no. 3, 657–662. 10.1090/S0002-9939-97-04024-0Search in Google Scholar

[16] D. J. S. Robinson, A Course in the Theory of Groups, 2nd ed., Grad. Texts in Math. 80, Springer, New York, 1996. 10.1007/978-1-4419-8594-1Search in Google Scholar

[17] R. Rossmanith, Lie centre-by-metabelian group algebras in even characteristic. I, Israel J. Math. 115 (2000), 51–75. 10.1007/BF02810580Search in Google Scholar

[18] R. Rossmanith, Lie centre-by-metabelian group algebras in even characteristic. II, Israel J. Math. 115 (2000), 77–99. 10.1007/BF02810581Search in Google Scholar

[19] R. Rossmanith, Unit groups of Lie centre-by-metabelian group algebras, Interactions Between Ring Theory and Representations of Algebras Lecture Notes Pure Appl. Math. 210, Dekker, New York (2000), 313–319. Search in Google Scholar

[20] M. Sahai, Group algebras with centrally metabelian unit groups, Publ. Mat. 40 (1996), no. 2, 443–456. 10.5565/PUBLMAT_40296_14Search in Google Scholar

[21] M. Sahai, Group algebras satisfying a certain Lie identity, Comm. Algebra 34 (2006), no. 3, 817–828. 10.1080/00927870500441577Search in Google Scholar

[22] M. Sahai, On group algebras K G with U ( K G ) nilpotent of class at most 2, Noncommutative Rings, Group Rings, Diagram Algebras and Their Applications, Contemp. Math. 456, American Mathematical Society, Providence (2008), 165–173. 10.1090/conm/456/08889Search in Google Scholar

[23] M. Sahai and J. B. Srivastava, A note on Lie centrally metabelian group algebras, J. Algebra 187 (1997), no. 1, 7–15. 10.1006/jabr.1997.6354Search in Google Scholar

[24] A. Shalev, Meta-abelian unit groups of group algebras are usually abelian, J. Pure Appl. Algebra 72 (1991), no. 3, 295–302. 10.1016/0022-4049(91)90067-CSearch in Google Scholar

[25] R. K. Sharma and J. B. Srivastava, Lie centrally metabelian group rings, J. Algebra 151 (1992), no. 2, 476–486. 10.1016/0021-8693(92)90123-4Search in Google Scholar

Received: 2024-11-16
Revised: 2025-05-25
Published Online: 2025-08-05

© 2025 Walter de Gruyter GmbH, Berlin/Boston

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