On (2, 3, *t*)-Generations for the Rudvalis Group Ru

Author(s)
Faryad Ali

Affiliation(s)

Department of Mathematics and Statistics, Faculty of Science, Al Imam Mohammad Ibn Saud Islamic University, Riyadh, Saudi Arabia.

Department of Mathematics and Statistics, Faculty of Science, Al Imam Mohammad Ibn Saud Islamic University, Riyadh, Saudi Arabia.

ABSTRACT

A group *G* is
said to be(2,3,*t*) -generated if it can be generated by an involution *x* and an element *y* so that *0*(*y*)=3 and *0*(*xy*)=*t*. In the present article, we determine all (2,3,*t*)-generations for the Rudvalis sporadic simple group Ru, where t is any divisor of .

Cite this paper

F. Ali, "On (2, 3,*t*)-Generations for the Rudvalis Group Ru," *Applied Mathematics*, Vol. 4 No. 9, 2013, pp. 1290-1295. doi: 10.4236/am.2013.49174.

F. Ali, "On (2, 3,

References

[1] G. A. Miller, “On the Groups Generated by Two Operators,” Bulletin of the AMS—American Mathematical Society, Vol. 7, No. 10, 1901, pp. 424-426. doi:10.1090/S0002-9904-1901-00826-9

[2] A. M. Macbeath, “Generators of Linear Fractional Groups,” Proceedings of Symposia in Pure Mathematics, Vol. 12, No. 1, 1969, pp. 14-32.

[3] A. J. Woldar, “On Hurwitz Generation and Genus Actions of Sporadic Groups,” Illinois Journal of Mathematics, Vol. 33, No. 3, 1989, pp. 416-437.

[4] J. Moori, “(2, 3, p)-Generations for the Fischer Group F22,” Communications in Algebra, Vol. 22, No. 11, 1994, pp. 4597-4610. doi:10.1080/00927879408825089

[5] S. Ganief and J. Moori, “(2, 3, t)-Generations for the Janko Group J3,” Communications in Algebra, Vol. 23, No. 12, 1995, pp. 4427-4437. doi:10.1080/00927879508825474

[6] F. Ali and M. A. F. Ibrahim, “On the Simple Sporadic Group He Generated by (2, 3, t) Generators,” Bulletin of the Malaysian Mathematical Sciences Society, Vol. 35, No. 3, 2012, pp. 745-753.

[7] M. A. Al-Kadhi and F. Ali, “(2, 3, t)-Generations for the Conway Group Co3,” International Journal of Algebra, Vol. 4, No. 25-28, 2010, pp. 1341-1353.

[8] M. A. Al-Kadhi and F. Ali, “(2, 3, t)-Generations for the Conway Group Co2,” Journal of Mathematics and Statistics, Vol. 8, No. 3, 2012, pp. 339-341. doi:10.3844/jmssp.2012.339.341

[9] M. R. Darafsheh and A. R. Ashrafi, “Generating Pairs for the Sporadic Group Ru,” Journal of Applied Mathematics and Computing, Vol. 12, No. 1-2, 2003, pp. 143-154. doi:10.1007/BF02936188

[10] M. D. E. Conder, R. A. Wilson and A. J. Woldar, “The Symmetric Genus of Sporadic Groups,” Proceedings of the American Mathematical Society, Vol. 116, No. 3, 1992, pp. 653-663. doi:10.1090/S0002-9939-1992-1126192-2

[11] R. A. Wilson, “The Geometry and Maximal Subgroups of the Simple Groups of A. Rudvalis and J. Tits,” Proceedings London Mathematical Society, Vol. 48, No. 3, 1984, pp. 533-563.

[12] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, “ of Finite Groups,” Oxford University Press, Clarendon, Oxford, 1985.

[13] The GAP Group, “ -Groups, Algorithms and Programming, Version 4.3,” St Andrews, Aachen, 2002. http://www.gap-system.org

[14] R. A. Wilson, et al., “A World-Wide-Web,” Atlas of Group Representations, 2006.

http://brauer.maths.qmul.ac.uk/Atlas/v3

[15] M. D. E. Conder, “Random Walks in Large Finite Groups,” The Australasian Journal of Combinatorics, 4, No. 1, 1991, pp. 49-57.

[16] W. Bosma, J. Cannon and C. Playoust, “The Magma Algebra System. I. The User language,” Journal of Symbolic Computation, Vol. 24, No. 3, 1997, pp. 235-265.

[1] G. A. Miller, “On the Groups Generated by Two Operators,” Bulletin of the AMS—American Mathematical Society, Vol. 7, No. 10, 1901, pp. 424-426. doi:10.1090/S0002-9904-1901-00826-9

[2] A. M. Macbeath, “Generators of Linear Fractional Groups,” Proceedings of Symposia in Pure Mathematics, Vol. 12, No. 1, 1969, pp. 14-32.

[3] A. J. Woldar, “On Hurwitz Generation and Genus Actions of Sporadic Groups,” Illinois Journal of Mathematics, Vol. 33, No. 3, 1989, pp. 416-437.

[4] J. Moori, “(2, 3, p)-Generations for the Fischer Group F22,” Communications in Algebra, Vol. 22, No. 11, 1994, pp. 4597-4610. doi:10.1080/00927879408825089

[5] S. Ganief and J. Moori, “(2, 3, t)-Generations for the Janko Group J3,” Communications in Algebra, Vol. 23, No. 12, 1995, pp. 4427-4437. doi:10.1080/00927879508825474

[6] F. Ali and M. A. F. Ibrahim, “On the Simple Sporadic Group He Generated by (2, 3, t) Generators,” Bulletin of the Malaysian Mathematical Sciences Society, Vol. 35, No. 3, 2012, pp. 745-753.

[7] M. A. Al-Kadhi and F. Ali, “(2, 3, t)-Generations for the Conway Group Co3,” International Journal of Algebra, Vol. 4, No. 25-28, 2010, pp. 1341-1353.

[8] M. A. Al-Kadhi and F. Ali, “(2, 3, t)-Generations for the Conway Group Co2,” Journal of Mathematics and Statistics, Vol. 8, No. 3, 2012, pp. 339-341. doi:10.3844/jmssp.2012.339.341

[9] M. R. Darafsheh and A. R. Ashrafi, “Generating Pairs for the Sporadic Group Ru,” Journal of Applied Mathematics and Computing, Vol. 12, No. 1-2, 2003, pp. 143-154. doi:10.1007/BF02936188

[10] M. D. E. Conder, R. A. Wilson and A. J. Woldar, “The Symmetric Genus of Sporadic Groups,” Proceedings of the American Mathematical Society, Vol. 116, No. 3, 1992, pp. 653-663. doi:10.1090/S0002-9939-1992-1126192-2

[11] R. A. Wilson, “The Geometry and Maximal Subgroups of the Simple Groups of A. Rudvalis and J. Tits,” Proceedings London Mathematical Society, Vol. 48, No. 3, 1984, pp. 533-563.

[12] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, “ of Finite Groups,” Oxford University Press, Clarendon, Oxford, 1985.

[13] The GAP Group, “ -Groups, Algorithms and Programming, Version 4.3,” St Andrews, Aachen, 2002. http://www.gap-system.org

[14] R. A. Wilson, et al., “A World-Wide-Web,” Atlas of Group Representations, 2006.

http://brauer.maths.qmul.ac.uk/Atlas/v3

[15] M. D. E. Conder, “Random Walks in Large Finite Groups,” The Australasian Journal of Combinatorics, 4, No. 1, 1991, pp. 49-57.

[16] W. Bosma, J. Cannon and C. Playoust, “The Magma Algebra System. I. The User language,” Journal of Symbolic Computation, Vol. 24, No. 3, 1997, pp. 235-265.