@article {
author = {Bubboloni, Daniela and Praeger, Cheryl E. and Spiga, Pablo},
title = {Conjectures on the normal covering number of the finite symmetric and alternating groups},
journal = {International Journal of Group Theory},
volume = {3},
number = {2},
pages = {57-75},
year = {2014},
publisher = {University of Isfahan},
issn = {2251-7650},
eissn = {2251-7669},
doi = {10.22108/ijgt.2014.3781},
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.},
keywords = {Covering,symmetric group,alternating group},
url = {https://ijgt.ui.ac.ir/article_3781.html},
eprint = {https://ijgt.ui.ac.ir/article_3781_473bef8f89b0aadff416858b62a9ac31.pdf}
}