Normal coverings of finite symmetric and alternating groups

In this paper we investigate the minimum number of maximal subgroups H i , i = 1 , … , k of the symmetric group S n (or the alternating group A n ) such that each element in the group S n (respectively A n ) lies in some conjugate of one of the H i . We prove that this number lies between a ϕ ( n ) and bn for certain constants a , b , where ϕ ( n ) is the Euler phi-function, and we show that the number depends on the arithmetical complexity of n. Moreover in the case where n is divisible by at most two primes, we obtain an upper bound of 2 + ϕ ( n ) / 2 , and we determine the exact value for S n when n is odd and for A n when n is even.