Conjectures on the normal covering number of finite symmetric and alternating groups

Document Type: Research Paper

Authors

1 University of Firenze

2 The University of Western Australia

3 University of Milano-Bicocca

Abstract

Let $\gamma(S_n)$ be the minimum number of proper subgroups‎ ‎$H_i,\ i=1‎, ‎\dots‎, ‎l $ of the symmetric group $S_n$ such that each element in $S_n$‎ ‎lies in some conjugate of one of the $H_i.$ In this paper we‎ ‎conjecture that $$\gamma(S_n)=\frac{n}{2}\left(1-\frac{1}{p_1}\right)‎ ‎\left(1-\frac{1}{p_2}\right)+2,$$ where $p_1,p_2$ are the two smallest primes‎ ‎in the factorization of $n\in\mathbb{N}$ and $n$ is neither a prime power nor‎ ‎a product of two primes‎. ‎Support for the conjecture is given by a previous result‎ ‎for $n=p_1^{\alpha_1}p_2^{\alpha_2},$ with $(\alpha_1,\alpha_2)\neq (1,1)$‎. ‎We give further evidence by confirming the conjecture for integers‎
‎of the form $n=15q$ for an infinite set of primes $q$‎, ‎and by reporting on a‎ ‎$ Magma$ computation‎. ‎We make a similar conjecture for $\gamma(A_n)$‎, ‎when $n$ is even‎, ‎and provide a similar amount of evidence‎.

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W. Bosma, J. Cannon and C. Playoust (1997). The Magma algebra system. I. The user language. J. Symbolic Comput.. 24, 235-265
R. Brandl, D. Bubboloni and I. Hupp (2001). Polynomials with roots mod $p$ for all pri-mes $p$. J. Group Theory. 4, 233-239
J. R. Britnell and A. Maroti Normal coverings of linear groups. Algebra and Number Theory, (to appear).
D. Bubboloni (1998). Coverings of the Symmetric and Alternating Groups. Quaderno del Dipartimento di Matematica ``U. Dini`` Firenze, preprint. Available at .{http://arxiv.org/abs/1009.3866.}. 7
D. Bubboloni, F. Luca and P. Spiga (2012). Compositions of n satisfying some coprimality conditions. J. Number Theory. 132, 2922-2946
D. Bubboloni and C. E. Praeger (2011). Normal coverings of finite symmetric and alternating groups. J. Combin. Theory Ser. A. 118, 2000-2024
D. Bubboloni, C. E. Praeger and P. Spiga (2013). Normal coverings and pairwise generation of finite alternating and symmetric groups. J. Algebra. 390, 199-215
P. J. Cameron (1999). Permutation Groups. London Mathematical Society Student Texts, Cambridge University Press, Cambridge. 45
R. Guralnick and K. Margaard (1998). On the Minimal Degree of a Primitive Permutation Group. J. Algebra. 207, 127-145
K. Tetrad Kourovka Notebook: Unsolved Problems in Group Theory. 18th Edition, to appear.
P. Muller (2013). Permutation groups with a cyclic two-orbits subgroup and monodromy groups of Laurent polynomials. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5). XII (2), 369-438
H. Wielandt (1964). Finite Permutation Groups. Academic Press, New York-London.