Unit Group of the Group Algebra $\mathbb{F}_qGL(2,7)$
DOI:
https://doi.org/10.52737/18291163-2024.16.3-1-14Keywords:
Unit Group, Group Algebra, General Linear Group, Wedderburn DecompositionAbstract
In this paper, we consider the general linear group $GL(2, 7)$ of $2 \times 2$ invertible matrices over the finite field of order $7$ and compute the unit group of the semisimple group algebra $\mathbb{F}_qGL(2,7)$, where $\mathbb{F}_q$ is a finite field. For the computation of the unit group, we need the Wedderburn decomposition of $\mathbb{F}_qGL(2,7)$, which is determined by first computing the Wedderburn decomposition of the group algebra $\mathbb{F}_q(PSL(3, 2)\rtimes C_2)$. Here $PSL(3,2)$ is the projective special linear group of degree 3 over a finite field of 2 elements.
References
S. Ansari and M. Sahai, Units in $F (C_n × Q_{12})$ and $F (C_n × D_{12})$. Int. Elect. J. Algebra, 34 (2023), pp. 182-196. https://doi.org/10.24330/ieja.1299278
N. Arvind and S. Panja, The unit group of $K_qSL(3, 2)$, $p ≥ 11$. Preprint: https://arxiv.org/abs/2106.07261 (2021).
Y. Bai, Y. Li and J. Peng, Unit groups of finite group algebras of abelian groups of order 17 to 20. AIMS Mathematics, 6 (2021), no. 7, pp. 7305-7317. https://doi.org/10.3934/math.2021428
G.K. Bakshi, S. Gupta and I.B.S. Passi, The algebraic structure of finite metabelian group algebras. Comm. Algebra, 43 (2015), no. 6, pp. 2240-2257. https://doi.org/10.1080/00927872.2014.888566
Z. Balogh, The structure of the unit group of some group algebras. Miskolc Math. Notes, 21 (2020), no. 2, pp. 615-620. https://doi.org/10.18514/mmn.2020.3406
Z. Balogh and A. Bovdi, On units of group algebras of 2-groups of maximal class. Commun. Algebra, 32 (2004), no. 8, pp. 3227-3245. https://doi.org/10.1081/agb-120039288
A. Bovdi, The group of units of a group algebra of characteristic $p$. Publ. Math. Debrecen, 52 (1998), no. 1-2, pp. 193-244. https://doi.org/10.5486/pmd.1998.2009
A. Bovdi, Modular group algebras whose group of unitary units is locally nilpotent. Acta Sci. Math., 88 (2022), pp. 571-576. https://doi.org/10.1007/s44146-022-00037-8
R. Ferraz, Simple components of center of $FG/J(FG)$. Comm. Algebra, 36 (2008), pp. 3191-3199.
GAP: Groups, Algorithm and Programming - a system for computational discrete algebra, Ver. 4.12.2, 2022.
G. Gardam, A counterexample to the unit conjecture for group rings. Ann. of Math., 194 (2021), no. 3, pp. 967-979. https://doi.org/10.4007/annals.2021.194.3.9
M. Hill, Irreducible representation of $GL_2(F_q)$. Preprint: https://people.math.harvard.edu/archive/126_fall_98/papers/mahill.pdf (1998).
T. Hurley, Convolutional codes from units in matrix and group rings. Int. J. Pure Appl. Math., 50 (2009), pp. 431-463.
S. Inam, S. Kanwal and R. Ali, A new encryption scheme based on groupring. Contemp. Math., 2 (2021), no. 2, pp. 103-112. https://doi.org/10.37256/cm.222021611
S. Maheshwari and R.K. Sharma, The unit group of the group algebra $F_qSL(2,Z_3)$. J. Algebra Comb. Discrete Appl., 3 (2015), pp. 1-6.
N. Makhijani and R.K. Sharma, The unit group of algebra of circulant matrices. Int. J. Group Theory, 3 (2014), pp. 13-16.
N. Makhijani, R.K. Sharma and J.B. Srivastava, The unit group of some special semisimple group algebras. Quaest. Math., 39 (2016), no. 1, pp. 9-28. https://doi.org/10.2989/16073606.2015.1024410
C.P. Milies and S.K. Sehgal, An Introduction to group rings, Springer Dordrecht, 2002.
G. Mittal, S. Kumar and S. Kumar, A quantum secure ID-based cryptographic encryption based on group rings. Sadhana, 47 (2022), Article no. 35. https://doi.org/10.1007/s12046-022-01806-5
G. Mittal and R.K. Sharma, Computation of the Wedderburn decomposition of semisimple group algebras of groups up to order 120. Annales Math. et Info., 59 (2023), pp. 54-70. https://doi.org/10.33039/ami.2023.07.001
G. Mittal and R.K. Sharma, Computation of Wedderburn decomposition of groups algebras from their subalgebra. Bull. Korean Math. Soc., 59 (2022), no. 3, pp. 781-787.
G. Mittal and R.K. Sharma, Unit group of semisimple group algebras of some non-metabelian groups of order 120. Asian-European J. Math., 15 (2022), no. 3, Article no. 2250059. https://doi.org/10.1142/s1793557122500590
M. Sahai and S. Ansari, Group of units of finite group algebras for groups of order 24. Ukrainian Math. J., 75 (2023), no. 2, pp. 215-229. https://doi.org/10.37863/umzh.v75i2.6680
R.K. Sharma and G. Mittal, On the unit group of a semisimple group algebra $F_qSL(2,Z_5)$. Math. Bohemica, 147 (2022), no. 1, pp. 1-10. https://doi.org/10.21136/mb.2021.0104-20
R.K. Sharma, J.B. Srivastava and M. Khan, The unit group of $mathbb{F}S_3$. Acta Math. Acad. Paedagog. Nyhazi. (N.S.), 23 (2007), pp. 129-142.
N.U. Sivaranjani, E. Nandakumar, G. Mittal and R.K. Sharma, On the unit group of the semisimple group algebra $K_qGL(2,Z_5)$. J. of Inter. Math., 27 (2024), no. 1, pp. 121-134. https://doi.org/10.47974/jim-1733
N.U. Sivaranjani, E. Nandakumar, G. Mittal and R.K. Sharma, Units of the semisimple group algebras $F_qSL(2, 8)$ and $F_qSL(2, 9)$. Annales Math. et Info., 59 (2023), pp. 117-130. https://doi.org/10.33039/ami.2023.12.002
G. Tang, Y. Wei and Y. Li, Unit groups of group algebras of some small groups. Czech. Math. J., 64 (2014), pp. 149-157. https://doi.org/10.1007/s10587-014-0090-0
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