Abstract
Let FG be the semisimple group algebra of a finite group G over a finite field F. In this article, we obtain a sufficient condition for which G does not have a normal complement in the unit group of FG. In particular, we have studied the normal complement problem for semisimple group algebras of dihedral groups, quaternion groups and groups of order \(p^n\), where \(n=3,4\) and p is an odd prime.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
E. Artin. Geometric Algebra. Wiley Classics Library. Wiley, 2011.
G. Bakshi, S. Gupta, and I.B.S. Passi. The structure of finite semisimple metacyclic group algebras. Journal of the Ramanujan Mathematical Society, 28(2):141–158, 2013.
R. K. Dennis. The structure of the unit group of group rings. In Ring theory II (Proceedings of Second Conference, University Oklahoma, Norman, Okla., 1975), Lecture Notes in Pure and Applied Mathematics, volume 26, pages 103–130, 1977.
S. Gupta and S. Maheshwary. Finite semisimple group algebra of a normally monomial group. International Journal of Algebra and Computation, 29(01):159–177, 2019.
L. R. Ivory. A note on normal complements in mod \(p\) envelopes. Proc. Amer. Math. Soc., 79(1):9–12, 1980.
D. L. Johnson. The modular group-ring of a finite \(p\)-group. Proc. Amer. Math. Soc., 68(1):19–22, 1978.
K. Kaur, M. Khan, and T. Chatterjee. A note on normal complement problem. J. Algebra Appl., 16(1):1750011, 11, 2017.
S. Kaur. On the normal complement problem in modular and semisimple group algebras. Communications in Algebra, 0(0):1–7, 2022.
S. Kaur and M. Khan. The normal complement problem and the structure of the unitary subgroup. Communications in Algebra, 48(8):3628–3636, 2020.
L. E. Moran and R. N. Tench. Normal complements in \({\rm mod}\ p\)-envelopes. Israel J. Math., 27(3-4):331–338, 1977.
R. Sandling. The modular group algebra of a central-elementary-by-abelianp-group. Archiv der Mathematik, 52(1):22–27, 1989.
H. Setia and M. Khan. The normal complement problem in group algebras. Communications in Algebra, 50(1):287–291, 2022.
H. Setia and M. Khan. Normal complement problem over a finite field of characteristic 2. Communications in Algebra, 51(3):977–982, 2023.
R. K. Sharma and G. Mittal. On the unit group of a semisimple group algebra \(\mathbb{F}_q\text{ SL }(2,\mathbb{Z}_5)\). Mathematica Bohemica, 147(1):1–10, 2022.
H. N. Ward. Some results on the group algebra of a group over a prime field. In Seminar on finite groups and related topics, pages 13–19. Harvard University, 1960.
Acknowledgements
We thank the referee for his/her valuable comments and suggestions which have helped us to improve the article.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Bakshi Gurmeet Kaur.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Khan, M., Setia, H. A note on the normal complement problem in semisimple group algebras. Indian J Pure Appl Math (2024). https://doi.org/10.1007/s13226-024-00556-w
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13226-024-00556-w