Normal coverings and pairwise generation of finite alternating and symmetric groups

Abstract

The normal covering number γ(G) of a finite, non-cyclic group G is the least number of proper subgroups such that each element of G lies in some conjugate of one of these subgroups. We prove that there is a positive constant c such that, for G a symmetric group Sym(n) or an alternating group Alt(n), γ(G)⩾cn. This improves results of the first two authors who had earlier proved that aφ(n)⩽γ(G)⩽2n/3, for some positive constant a, where φ is the Euler totient function. Bounds are also obtained for the maximum size κ(G) of a set X of conjugacy classes of G=Sym(n) or Alt(n) such that any pair of elements from distinct classes in X generates G, namely cn⩽κ(G)⩽2n/3.