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Vol. 49 No. 4 (2023)
Serdica Mathematical Journal

Articles

A short note on the Wedderburn components of a semisimple finite group algebra

https://doi.org/10.55630/serdica.2023.49.241-250

Published 21-12-2023

  • Gaurav Mittal
    • ,
  • Rajendra Kumar Sharma

Gaurav Mittal
Indian Institute of Technology Roorkee

Rajendra Kumar Sharma
Indian Institute of Technology Delhi

Abstract

One of the classical problems in the subject of group algebras is that of deducing Wedderburn decomposition of a finite semisimple group algebra. In this short note, we discuss how to check whether a matrix ring over a finite field is a Wedderburn component of the Wedderburn decomposition of a group algebra or not. Finally, we formulate an open problem in this direction.

2020 Mathematics Subject Classification:

20C05

Keywords

Wedderburn decomposition, unit group, finite field

Full text

Author Details

Gaurav Mittal

Department of Mathematics
Indian Institute of Technology Roorkee
Roorkee, India
e-mail: gaurav.mittaltwins@gmail.com

Rajendra Kumar Sharma

Department of Mathematics
Indian Institute of Technology Delhi
New Delhi, India
e-mail: rksharmaiitd@gmail.com

References

  1. N. Arvind, S. Panja. Unit group of some finite semi-simple group algebras. J. Egyptian Math. Soc. 30, 1 (2022) Paper No. 17, 9 pp.
  2. N. Arvind, S. Panja. Unit group FqSL(3, 2), p ≥ 11, Preprint, 2021, arXiv:2106.07261v1.
  3. G. K. Bakshi, S. Gupta, I. B. S. Passi. The algebraic structure of finite metabelian group algebras. Comm. Algebra 43, 6 (2015), 2240–2257.
  4. O. Broche, A. del Río. Wedderburn decomposition of finite group algebras. Finite Fields Appl. 13, 1 (2007), 71–79, 2007.
  5. C. Dietzel, G. Mittal. Summands of finite group algebras. Czechoslovak Math. J. 71(146), 4 (2021), 1011–1014.
  6. R. A. Ferraz. Simple components of the center of FG/J(FG). Comm. Algebra 36, 9 (2008), 3191–3199.
  7. S. Gupta, S. Maheshwary. Finite semisimple group algebra of a normally monomial group. Internat. J. Algebra Comput. 29, 1 (2019), 159–177.
  8. R. Lidl, H. Niederreiter. Introduction to finite fields and their applications. New York, Cambridge University Press, 2000.
  9. S. Maheshwari, R. K. Sharma. The unit group of group algebra FqSL(2, Z3), J. Algebra Comb. Discrete Struct. Appl. 3, 1 (2016), 1–6.
  10. N. Makhijani, R. K. Sharma, J. B. Srivastava. A note on the structure of FpkA5/J(FpkA5). Acta Sci. Math. (Szeged) 82, 1–2 (2016), 29–43.
  11. C. P. Milies, S. K. Sehgal. An Introduction to Group Rings. Dordrecht/Boston/London, Kluwer Academic Publishers, 2002.
  12. G. Mittal, R. K. Sharma. On unit group of finite group algebras of non-metabelian groups up to order 72. Math. Bohem. 146, 4 (2021), 429–455.
  13. G. Mittal, R. K. Sharma. On unit group of finite group algebras of non-metabelian groups of order 108. J. Algebra Comb. Discrete Struct. Appl. 8, 2 (2021), 59–71.
  14. G. Mittal, R. K. Sharma. Unit group of semisimple group algebras of some non-metabelian groups of order 120. Asian-Eur. J. Math. 15, 3 (2022), Paper No.2250059, 11 pp.
  15. G. Mittal, R. K. Sharma. The unit groups of semisimple group algebras of some non-metabelian groups of order 144, Math. Bohemica, Online first, 2022, 16 pp, DOI: 10.21136/MB.2022.0067-22.
  16. G. Mittal, R. K. Sharma. Wedderburn decomposition of a semisimple group algebra FqG from a subalgebra of factor group of G. Int. Electron. J. Algebra 32 (2022), 91–100.
  17. G. Mittal, R. K. Sharma. Computation of Wedderburn decomposition of groups algebras from their subalgebra. Bull. Korean Math. Soc. 59, 3 (2022), 781–787.
  18. G. Mittal, R. K. Sharma. A short note on Wedderburn decomposition of a group algebra. Ex. Countex. 3 (2023), 100105.
  19. G. Olteanu. Computing the Wedderburn decomposition of group algebras by the Brauer-Witt theorem. Math. Comp. 76, 258 (2007), 1073–1087.
  20. S. Perlis, G. L. Walker. Abelian group algebras of finite order. Trans. Amer. Math. Soc. 68 (1950), 420–426.
  21. R. K. Sharma, G. Mittal. On the unit group of a semisimple group algebra FqSL(2, Z5). Math. Bohem. 147, 1 (2022), 1–10.
  22. R. K. Sharma, J. B. Srivastava, M. Khan. The unit group of FS3. Acta Math. Acad. Paedagog. Nyházi. (N.S.) 23, 2 (2007), 129—142.
  23. R. Waxman. On the Wedderburn component of a group algebra. Mathematics Stack Exchange, https://www.math.stackexchange.com/q/4682495.