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 βn of a cover of the symmetric group Sn by abelian subgroups and the maximal cardinality αn of a set of pairwise noncommuting elements of Sn. 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 βn = αn for all n⩾0.