Normal coverings of finite symmetric and alternating groups

In this paper we investigate the minimum number of maximal subgroups Hi, i = 1, ... k of the symmetric group S(n) (or the alternating group An) 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 phi(n) and bn for certain constants a, b. where phi(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 +phi(n)/2, and we determine the exact value for S(n) when n is odd and for A(n) when n is even. (C) 2011 Elsevier Inc. All rights reserved.