Abstract
We improve on recent estimates for the probability of generating the alternating and symmetric groups A n and S n . In particular, we find the sharp lower bound if the probability is given by a quadratic in n −1. This leads to improved bounds on the largest number h(A n ) such that a direct product of h(A n ) copies of A n can be generated by two elements.
Similar content being viewed by others
References
Dixon J.D.: The probability of generating the symmetric group. Math. Z 110, 199–205 (1969)
J. D. Dixon, Asymptotics of generating the symmetric and alternating groups, Electron. J. Combin. 12 (2005), Research paper 56, 5 pp.
Hall P.: The Eulerian function of a group. J. Math. Oxford 7, 133–141 (1936)
Liebeck M.W., Shalev A.: Simple groups, probabilistic methods, and a conjecture of Kantor and Lubotszky. J. Algebra 184, 31–57 (1996)
Maróti A., Tamburini M.C.: Bounds for the probability of generating the symmetric and alternating groups. Arch. Math. (Basel) 96(2), 115–121 (2011)
Menezes N.E., Quick M., Roney-Dougal C.M.: The probability of generating a finite simple group. Israel J. Math 198, 371–392 (2013)
N. E. Menezes, Random generation and chief length of finite groups, PhD thesis, University of St Andrews (2013).
Author information
Authors and Affiliations
Corresponding author
Additional information
The research of the first author is supported by the Australian Research Council grant DP120100446. This work was done whilst the second author was visiting The University of Western Australia as a Cheryl E. Praeger Visiting Research Fellow. The authors would like to thank the anonymous referee for their extremely helpful remarks.
Rights and permissions
About this article
Cite this article
Morgan, L., Roney-Dougal, C.M. A note on the probability of generating alternating or symmetric groups. Arch. Math. 105, 201–204 (2015). https://doi.org/10.1007/s00013-015-0796-8
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00013-015-0796-8