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

Let gamma(sn) be the minimum number of proper subgroups hi, i =1,...,ell, of the symmetric group sn such that each element in sn lies in someconjugate of one of the hi. in this paper we conjecture thatgamma(sn) =(n/2)(1-1/p_1) (1-1/p_2) + 2, where p1, p2 are the two smallest primes in the factorization of n and n isneither a prime power nor a product of two primes. support for the conjectureis given by a previous result for the case where n has at most two distinct prime divisors. we give further evidence by confirming the conjecture for certain 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(an), when n is even, and provide a similar amount of evidence.