Covering the symmetric groups with proper subgroups

摘要

Let G be a group that is a set-theoretic union of finitely many proper subgroups. Cohn defined sigma(G) to be the least integer m such that G is the union of m proper subgroups. Tomkinson showed that sigma(G) can never be 7, and that it is always of the form q + 1 (q a prime power) for solvable groups. In this paper we give exact or asymptotic formulas for sigma(S-n). In particular, we show that sigma(S-n) <= 2(n-1), while for alternating groups we find sigma(A(n)) >= 2(n-2) unless n = 7 or 9. An application of this result is also given. (c) 2004 Elsevier Inc. All rights reserved.