Upper bounds on the uniform spreads of the sporadic simple groups

Document Type : Research Paper

Authors

1 Mathematics, Faculty of Science, University of Qom. Qom, Iran

2 Department of Mathematics, Faculty of Science, Iran University of Science & Technology, Tehran, Iran

Abstract

‎‎A finite group $G$ has uniform spread $k$ if there exists a fixed conjugacy class $C$ of elements in $G$ with the property that‎ ‎for any $k$ nontrivial elements $s_1, s_2,‎\ldots‎,s_k$ in $G$ there exists $y\in C$ such that $G = \langle s_i,y\rangle$ for $i=1, 2,‎\ldots,k$‎. ‎Further‎, ‎the exact uniform spread of $G$ is the largest $k$ such that $G$ has the uniform spread $k$‎. ‎In this paper we give upper bounds on the exact uniform spreads of thirteen sporadic simple groups‎.

Keywords

Main Subjects


[1] R. Abb ott, J. Bray, S. Linton, S. Nickerson, S. Norton, R. Parker, I. Suleiman, J. Tripp, P. Walsh and R. Wilson,
ATLAS of Finite Group Representations - Version 3. Available online http://brauer.maths.qmul.ac.uk/Atlas/
v3/.
[2] M. Aschbacher and R. M. Guralnick, Some applications of the rst cohomology group, J. Algebra , (1984) 90
446{460.
[3] G. J. Binder, The two-element bases of the symmetric group, Izv. Vyss. Ucebn. Zaved. Mathematika , (1970) 90
9{11.
[4] J. D. Bradley and P. E. Holmes, Improved b ounds for the spread of sp oradic groups, LMS J. Comput. Math. ,
(2007) 10 132{140.
[5] J. D. Bradley and J. Mo ori, On the exact spread of sp oradic simple groups, Comm. Algebra , (2007) 35 2588{2599.
[6] J. L. Brenner and J. Wiegold, Two-generator groups I, Michigan Math. J. , (1975) 22 53{64.
[7] T. Breuer, R. M. Guralnick and W. M. Kantor, Probabilistic generation of nite simple groups I I, J. Algebra ,
(2008) 320 443{494.
[8] W. Bosma and J. J. Cannon, Handbook of Magma functions , Scho ol of Mathematics and Statistics, University of
Sydney, Sydney, 1995. http://magma.maths.usyd.edu.au/magma/faq/citing .
[9] T. Burness and S. Guest, On the uniform spread of almost simple linear groups, Nagoya Math. J. , (2013) 209
35{109.
[10] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, An ATLAS of Finite Groups , Oxford:
Oxford University Press, 1985.
[11] B. Fairbairn, New Upp er Bounds On the spreads of the sp oradic simple groups, Comm. Algebra , (2012) 40 1872{
1877.
[12] B. Fairbairn, The exact spread of M 23 is 8064, Int. J. Group Theory , (2012) 1 1-2.
[13] S. Ganief and J. Mo ori, On the spread of the sp oradic simple groups, Comm. Algebra , (2001) 29 3239{3255.
[14] R. M. Guralnick and W. M. Kantor, Probabilistic generation of nite simple groups, J. Algebra , (2000) 234
743{792.
[15] R. M. Guralnick and A. Shalev, On the spread of nite simple groups, Combinatorica , (2003) 23 73{87.
[16] S. Harp er, On the uniform spread of almost simple symplectic and orthogonal groups, J. Algebra , (2017) 490
330{371.
[17] G. A. Miller, On the groups generated by two op erators, Bul l. Amer. Math. Soc. , (1901) 7 424{426.
[18] R. Steinb erg, Generators for simple groups, Canad. J. Math. , (1962) 14 277{283.
[19] The GAP Group, GAP Groups, Algorithms, and Programming , (2014) Version 4.7.5, http://www.gap- system.
org.
[20] R. A. Wilson, The maximal subgroups of the baby monster I, J. Algebra , (1999) 211 1{14.
[21] A. Woldar, The exact spread of the Mathieu group M 11 , J. Group Theory , (2007) 10 167{171.