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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Bubboloni, D., Praeger, C. E., & Spiga, P. (2014). Conjectures on the normal covering number of the finite symmetric and alternating groups. International Journal of Group Theory, 3(2), 57-75. doi: 10.22108/ijgt.2014.3781
MLA
Daniela Bubboloni; Cheryl E. Praeger; Pablo Spiga. "Conjectures on the normal covering number of the finite symmetric and alternating groups". International Journal of Group Theory, 3, 2, 2014, 57-75. doi: 10.22108/ijgt.2014.3781
HARVARD
Bubboloni, D., Praeger, C. E., Spiga, P. (2014). 'Conjectures on the normal covering number of the finite symmetric and alternating groups', International Journal of Group Theory, 3(2), pp. 57-75. doi: 10.22108/ijgt.2014.3781
VANCOUVER
Bubboloni, D., Praeger, C. E., Spiga, P. Conjectures on the normal covering number of the finite symmetric and alternating groups. International Journal of Group Theory, 2014; 3(2): 57-75. doi: 10.22108/ijgt.2014.3781
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