The unit group of group algebra $\mathbb{F}_qSL(2;\mathbb{Z}_3)$
Keywords:
Group algebra, Unit group, Finite fieldAbstract
Let \(\mathbb{F}_q\) be a finite field of characteristic \(p\) having \(q\) elements, where \(q = p^k\) and \(p \ge 5\). Let \(SL(2,\mathbb{Z}_3)\) be the special linear group of \(2 \times 2\) matrices with determinant \(1\) over \(\mathbb{Z}_3\). In this note we establish the structure of the unit group of \(\mathbb{F}_qSL(2,\mathbb{Z}_3)\).
Received: 5 August 2015 | Accepted: 20 October 2015
Downloads
References
L. Creedon and J. Gildea, The structure of the unit group of the group algebra $\mathbb{F}_{2^k}D_8$, Canad. Math. Bull. 54(2) (2011) 237-243.
R. A. Ferraz, Simple components of the center of $FG/J(FG)$, Comm. Algebra 36(9) (2008) 3191-3199.
J. Gildea, The structure of the unit group of the group algebra $\mathbb{F}_{2^k}A_4$, Czechoslovak Math. J. 61(2) (2011) 531-539.
J. Gildea and F. Monaghan, Units of the group algebra of the group $C_n \times D_6$ over any finite field of characteristic 3, Int. Electron. J. Algebra 11 (2012) 46-58.
P. R. Helm, A presentation for $SL(2, \mathbb{Z}_{p^r})$, Comm. Algebra 10(15) (1982) 1683-1688.
T. Hurley, Group rings and ring of matrices, Int. J. Pure Appl. Math. 31(3) (2006) 319-335.
T. Hurley, Convolutional codes from units in matrix and group rings, Int. J. Pure Appl. Math. 50(3) (2009) 431-463.
R. Lidl and H. Niederreiter, Introduction to Finite Fields and Their Applications, Cambridge University Press, New York (2000).
N. Makhijani, Units in finite group algebras, Ph.D. thesis, IIT Delhi (2014).
N. Makhijani, R. K. Sharma, and J. B. Srivastava, A note on units in $\mathbb{F}_{p^m}D_{2p^n}$, Acta Math. Acad. Paedagog. Nyházi. 30(1) (2014) 17-25.
N. Makhijani, R. K. Sharma, and J. B. Srivastava, The unit group of algebra of circulant matrices, Int. J. Group Theory 3(4) (2014) 13-16.
N. Makhijani, R. K. Sharma, and J. B. Srivastava, The unit group of $\mathbb{F}_q[D_{30}]$, Serdica Math. J. 41(2-3) (2015) 185-198.
C. P. Milies and S. K. Sehgal, An Introduction to Group Rings, Algebras and Applications 1, Kluwer Academic Publishers, Dordrecht (2002).
S. Perlis and G. L. Walker, Abelian group algebras of finite order, Trans. Amer. Math. Soc. 68(3) (1950) 420-426.
R. K. Sharma, J. B. Srivastava, and M. Khan, The unit group of $FA_4$, Publ. Math. Debrecen 71(1-2) (2007) 21-26.
R. K. Sharma, J. B. Srivastava, and M. Khan, The unit group of $FS_3$, Acta Math. Acad. Paedagog. Nyházi. 23(2) (2007) 129-142.
R. K. Sharma and P. Yadav, Unit group of algebra of circulant matrices, Int. J. Group Theory 2(4) (2013) 1-6.
R. K. Sharma and P. Yadav, The unit group of $\mathbb{Z}_pQ_8$, Algebras Groups Geom. 25(4) (2008) 425-429.
G. Tang, Y. Wei, and Y. Li, Units group of group algebras of some small groups, Czechoslovak Math. J. 64(1) (2014) 149-157.
