Regular orbits of cyclic subgroups in permutation representations of certain simple groups (Q1858239)

Let \(G\) be a group and \(\Omega\) a set. If \(G\) acts on \(\Omega\), then a \(G\)-orbit is regular if its cardinality is \(|G|\). The authors study regular orbits of cyclic subgroups of finite simple groups. The main theorem of this paper is the following theorem: Theorem 1.1. Let \(G\) be a known finite simple group, not isomorphic to an alternating group \(A_n\), which admits a doubly transitive permutation representation. Then every cyclic subgroup \(H\subset G \) has a regular orbit in any non-trivial permutation representation of \(G\). The authors [in J. Algebra 226, No. 1, 451-478 (2000; Zbl 0957.20013)] proved this theorem for \(\text{PSL}(n,q)\).