Limit points in the range of the commuting probability function on finite groups. (Q2842797)

Let \(G\) be a finite group. The author denotes by \(\text{Pr}(G)\) the commuting probability of \(G\), i.e., the probability that a uniformly random ordered pair of elements of \(G\) commute. It is well-known that \(\text{Pr}(G)=k(G)/|G|\), where \(k(G)\) is the number of conjugacy classes of \(G\). The author is interested in three conjectures studied in an unpublished 1969 Ph.D. thesis by \textit{K. S. Joseph} and published in [Am. Math. Mon. 84, 550-551 (1977; Zbl 0372.60013)].NEWLINENEWLINE If \(\mathcal G\) denotes the family of all finite groups, these conjectures are that (1) every limit point of the set \(\text{Pr}(\mathcal G)\) is rational, (2) if \(l\) is a limit point of \(\text{Pr}(\mathcal G)\), then there exists an \(\varepsilon>0\) (depending on \(l\)) such that \(\text{Pr}(\mathcal G)\cap(l-\varepsilon,l)=\emptyset\), and (3) \(\text{Pr}(\mathcal G)\cup\{0\}\) is a closed subset of \(\mathbb R\).NEWLINENEWLINE While Conjecture (3) remains elusive at this time, in the paper the author provides some new evidence in favor of Conjectures (1) and (2) by proving that their conclusions are true for any limit point of \(\text{Pr}(\mathcal G\)) which is greater than \(\tfrac{2}{9}\).