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 ( S n ) . In particular, we show that σ ( S n ) ⩽ 2 n - 1 , while for alternating groups we find σ ( A n ) ⩾ 2 n - 2 unless n = 7 or 9. An application of this result is also given.