Bubboloni, D., Praeger, C., Spiga, P. (2014). Conjectures on the normal covering number of finite symmetric and alternating groups. International Journal of Group Theory, 3(2), 57-75.

Daniela Bubboloni; Cheryl E. Praeger; Pablo Spiga. "Conjectures on the normal covering number of finite symmetric and alternating groups". International Journal of Group Theory, 3, 2, 2014, 57-75.

Bubboloni, D., Praeger, C., Spiga, P. (2014). 'Conjectures on the normal covering number of finite symmetric and alternating groups', International Journal of Group Theory, 3(2), pp. 57-75.

Bubboloni, D., Praeger, C., Spiga, P. Conjectures on the normal covering number of finite symmetric and alternating groups. International Journal of Group Theory, 2014; 3(2): 57-75.

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

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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