Covering the symmetric groups with proper subgroups

Let G be a group that is a set-theoretic union of finitely many proper subgroups. Cohn defined σ(G) to be the least integer m such that G is the union of m proper subgroups. Tomkinson showed that σ(G) can never be 7, and that it is always of the form q+1 (q a prime power) for solvable groups G. In this paper we give exact or asymptotic formulas for σ(Sn). In particular, we show that σ(Sn)⩽2n-1, while for alternating groups we find σ(An)⩾2n-2 unless n=7 or 9. An application of this result is also given.