Minimal covers of \(S_ n\) by Abelian subgroups and maximal subsets of pairwise noncommuting elements
Let \({\mathcal G}\) be a finite group, \(\alpha\) (\({\mathcal G})\) be the maximal cardinality of a set of pairwise noncommuting elements of \({\mathcal G}\), and \(\beta\) (\({\mathcal G})\) be the minimal cardinality of a cover of \({\mathcal G}\) by abelian subgroups. These numbers were studied recently in particular by \textit{D. R. Mason} [Math. Proc. Camb. Philos. Soc. 83, 205-209 (1978; Zbl 0374.20037)] and \textit{L. Pyber} [J. Lond. Math. Soc., II. Ser. 35, 287-295 (1987; Zbl 0588.20016)]. The present author now concentrates on the case when \({\mathcal G}={\mathcal S}_ n\), the symmetric group on n letters. By lengthy and non-trivial arguments, he shows that the average values of \(\alpha\) (\({\mathcal S}_ n)/(n-2)!\) and \(\beta (S_ n)/(n-2)!\) are bounded above and below by positive constants. It is also proved that the sequence \(\{\) \(\beta\) (\({\mathcal S}_ n)/\alpha ({\mathcal S}_ n)\}\) is bounded, and that its convergence would imply \(\alpha\) (\({\mathcal S}_ n)=\beta ({\mathcal S}_ n)\) for \(n\geq 1\).
scientific article; zbMATH DE number 4089791