Covering the symmetric groups with proper subgroups

Abstract: Let G be a group that is a set-theoretic union of finitely many proper subgroups. Cohn defined to be the least integer m such that G is the union of m proper subgroups. Tomkinson showed that can never be 7, and that it is always of the form (q a prime power) for solvable groups G. In this paper we give exact or asymptotic formulas for . In particular, we show that , while for alternating groups we find unless or 9. An application of this result is also given. [Copyright &y& Elsevier]