A A note on the Jacobson radical $J(\mathbb F_{2^n}D_{2m})$
Abstract
UDC 512.5
The dimension of the Jacobson radical $J(\mathbb F_{2^n}D_{2m})$ and the algebraic structure of $\dfrac{\mathbb F_{2^n} D_{{2^{k}} {3}}} {J(\mathbb F_{2^n}D_{2^{k} {3}})}$ are determined. Here, $D_{2m}$ represents the dihedral group of order $2m,$ $\mathbb F_{2^n}$ represents any finite field of characteristic $2$, and $F_{2^n}D_{2m}$ represents the group algebra of the group $D_{2m}$ over the field $\mathbb F_{2^n}.$
References
A. A. Bovdi, Group rings. A textbook, Uzgorod. State Univ. (1974).
R. A. Ferraz, Simple components of center of $FG/J (FG)$, Comm. Algebra, 36, № 9, 3191–3199 (2008). DOI: https://doi.org/10.1080/00927870802103503
J. Gildea, The structure of $F_{5^{k}}D_{20}$, Int. Electron. J. Algebra, 8, 153–160 (2010).
J. Gildea, L. Creedon, The structure of the unit group of the group algebra $F_{2^{k}}D_{8}$, Canad. Math. Bull., 54, № 2, 237–243 (2011). DOI: https://doi.org/10.4153/CMB-2010-098-5
G. Karpilovsky, The Jacobson radical of group algebras, North-Holland, Amsterdam (1987).
G. Karpilovsky, Structure of blocks of group algebras, Pitman Monographs and Surveys in Pure and Applied Mathematics, 33, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York (1987).
G. Karpilovsky, Unit groups of classical rings, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York (1988).
G. Karpilovsky, Unit groups of group rings, Pitman Monographs and Surveys in Pure and Applied Mathematics, 47, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York (1989).
Yogesh Kumar, R. K. Sharma, J. B. Srivastava, Unit group of the group algebra $FS_5$, Acta Math. Acad. Paedagog. Nyhazi, 33, 187–193 (2017).
P. Kumari, M. Sahai, R. K. Sharma, Jordan regular units in rings and group rings, Ukr. Math. J., 75, № 3, 351–363 (2023). DOI: https://doi.org/10.37863/umzh.v75i3.1130
S. Maheshwari, R. K. Sharma, A note on units in $F_qSL (2,3)$, Ukr. Math. J., 73, № 8, 1331–1337 (2022). DOI: https://doi.org/10.1007/s11253-022-01994-7
N. Makhijani, R. K. Sharma, J. B. Srivastava, Units in $F_{2^{k}}D_{2n}$, Int. J. Group Theory, 3, № 3, 25–34 (2014).
N. Makhijani, R. K. Sharma, J. B. Srivastava, The unit group of $F_q D_{30}$, Serdica Math. J., 21, 185–198 (2015).
G. Mittal, R. K. Sharma, A short note on Wedderburn decomposition of a group algebra, Examples and Counterexamples, 2, Article 100105 (2023). DOI: https://doi.org/10.1016/j.exco.2023.100105
Copyright (c) 2024 Yogesh Kumar Yogesh Kumar
This work is licensed under a Creative Commons Attribution 4.0 International License.
