On some designs constructed from the groups PSL2(q), q=53,61,64

Document Type : Research Paper

Author

Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan, Iran

Abstract

In this paper, we use the primitive permutation representations of the simple groups PSL2(53), PSL2(61) and PSL2(64) and construct 1-designs by the Key-Moori Method 1.
It is shown that the groups PSL2(53), PSL2(53):2, PSL2(61), PSL2(61):2, PSL2(64), PSL2(64):2, PSL2(64):3 and PSL2(64):6 appear as the full automorphism groups of these obtained designs.

Keywords


[1] E. F. Assmus, Jr. and J. D. Key, Designs and Their Codes, Cambridge University Press, Cambridge, (1992).
[2] W. Bosma and J. Cannon, Handbook of Magma Functions, Department of Mathematics, University of
Sydney, (1994), http://www.magma.maths.usyd.edu.au/magma/.
[3] T. Beth, D. Jungnickel and H. Lenz, Design Theory Vol. 1, (second edition), Cambridge University Press, Cambridge, (1999).
[4] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, Atlas of Finite Groups, Oxford University Press, Oxford, (1985).
[5] M. R. Darafsheh, Designs from the Group PSL2(q), q Even, Des. Codes Cryptogr. 39 (2006), pp. 311-316.
[6] M. R. Darafsheh, A.R. Ashra and M. Khademi, Some Designs Related to Group Actions, Ars. Combin. 86 (2008), pp. 65-75.
[7] M. R. Darafsheh, A.R. Ashra and M. Khademi, On Designs Constructed by Group Actions, J. Combin. Math. Combin. Comput. 70 (2009), pp. 235-245.
[8] M. R. Darafsheh, A. Iranmanesh and R. Kahkeshani, Designs from the Groups PSL2(q) for Certain q, Quaest. Math. 32 (2009), pp. 1-10.
[9] R. Kahkeshani, 1-Designs from the Group PSL2(59) and Their Automorphism Groups, Math. Interdisc. Res. 3 (2018), pp. 147-158.
[10] J. D. Key and J. Moori, Designs, Dodes and Graphs from the Janko Groups J1 and J2, J. Combin. Math. Combin. Comput. 40 (2002), pp. 143-159.
[11] J. D. Key and J. Moori, Correction to: Codes, Designs and Graphs from the Janko Groups J1 and J2, J. D. Key and J. Moori, JCMCC 40 (2002), pp. 143-159, J. Combin. Math. Combin. Comput. 64 (2008),
pp. 153.
[12] J. D. Key, J. Moori and B.G. Rodrigues, On some Designs and Codes from Primitive Representations of some Finite Simple Groups, J. Combin. Math. Combin. Comput. 45 (2002), pp. 3-19.
[13] J. Moori, Finite Groups, Designs and Codes, Information Security, Coding Theory and Related Combinatorics, NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur. Vol. 29 (2011), pp. 202-230.
[14] B. Rodrigues, Codes of Designs and Graphs from Finite Simple Groups, Ph.D. Thesis, University of Natal, South Africa, (2003).
[15] J. J. Rotman, An Introduction to the Theory of Groups, (fourth edition), Springer-Verlag, Berlin, (1995).
[16] M. Suzuki, Group Theory I, Springer-Verlag, Berlin, (1982).