Minimal covers of Sn by abelian subgroups and maximal subsets of pairwise noncommuting elements

Abstract

We give best possible asymptotic upper and lower bounds for the minimal cardinality β n of a cover of the symmetric group S n by abelian subgroups and the maximal cardinality α n of a set of pairwise noncommuting elements of S n . We show that the average values of β n (n − 2) ! and of α n (n − 2) ! are bounded above and below by positive constants. Finally, we show that the sequence { β n α n } is bounded and that if it converges, then β n = α n for all n⩾0.