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

Let γ(Sn) be the minimum number of proper subgroups Hi, i = 1,..., l of the symmetric group Sn such that each element in Sn lies in some conjugate of one of the Hi. In this paper we conjecture that where p1; p2 are the two smallest primes in the factorization of n ∈ 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 = p1 α1 p2 α2, with (α1, α2) ≠ (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 γ(An), when n is even, and provide a similar amount of evidence. © 2013 University of Isfahan.