The unit group of group algebra $F_qSL(2;Z_3)$

Abstract

Let $\F_q$ be a finite field of characteristic $p$ having $q$ elements, where $q = p^k$ and $p\ge 5$.  Let $ SL(2,\Z_3)$ be the special linear group of $2\times2$ matrices with determinant $1$ over $\Z_3$. In this note we establish the structure of the unit group of $\F_q SL(2,\Z_3)$.

Keywords

• Group algebra, Unit group, Finite field

References

1. L. Creedon, J. Gildea, The structure of the unit group of the group algebra F2kD, Canad. Math. Bull. 54(2) (2011) 237–243.
2. R. A. Ferraz, Simple components of the center of FG/J(FG), Comm. Algebra 36(9) (2008) 3191–3199.
3. J. Gildea, The structure of the unit group of the group algebra FkA4, Czechoslovak Math. J. 61(2) (2011) 531–539.
4. J. Gildea, F. Monaghan, Units of some group algebras of groups of order 12 over any finite field of characteristic 3, Algebra Discrete Math. 11(1) (2011) 46–58
5. P. R. Helm, A presentation for SL(2, Zpr), Comm. Algebra 10(15) (1982) 1683–1688.
6. T. Hurley, Group rings and ring of matrices, Int. J. Pure Appl. Math. 31(3) (2006) 319–335.
7. T. Hurley, Convolutional codes from units in matrix and group rings, Int. J. Pure Appl. Math. 50(3) (2009) 431–463
8. R. Lidl, H. Niederreiter, Introduction to finite fields and their applications, Cambridge University Press, New York, 2000.

9. N. Makhijani, Units in finite group algebras, IIT Delhi, 2014.
10. N. Makhijani, R. K. Sharma, J. B. Srivastava, A note on units in FpmD2pn, Acta Math. Acad. Paedagog. Nyházi. 30(1) (2014) 17–25.
11. N. Makhijani, R. K. Sharma, J. B. Srivastava, The unit group of algebra of circulant matrices, Int. J. Group Theory. 3(4) (2014) 13–16
12. N. Makhijani, R. K. Sharma, J. B. Srivastava, The unit group of Fq[D30], Serdica Math. J. 41(2-3) (2015) 185–198.
13. C. P. Milies, S. K. Sehgal, An introduction to group rings, Kluwer Academic Publishers, 2002.
14. S. Perlis, G. L. Walker, Abelian group algebras of finite order, Trans. Amer. Math. Soc. 68(3) (1950) –426
15. R. K. Sharma, J. B. Srivastava, M. Khan, The unit group of FA4, Publ. Math. Debrecen 71(1-2) (2007) 21–26.
16. R. K. Sharma, J. B. Srivastava, M. Khan, The unit group of FS3, Acta Math. Acad. Paedagog. Nyházi. 23(2) (2007) 129–142.
17. R. K. Sharma, P. Yadav, Unit group of algebra of circulant matrices, Int. J. Group Theory. 2(4) (2013) 1–6
18. R. K. Sharma, P. Yadav, The unit group of ZpQ8, Algebras Groups Geom. 25(4) (2008) 425–429.
19. G. Tang, Y. Wei, Nanning, Y. Li, Units group of group algebras of some small groups, Czechoslovak Math. J. 64(1) (2014) 149–157.