Permutation 3-groups with no fixed-point-free elements. (Q2854025)

Every finite transitive permutation group contains a fixed-point-free element of prime power order, [see \textit{B. Fein, W. M. Kantor} and \textit{M. Schacher}, J. Reine Angew. Math. 328, 39-57 (1981; Zbl 0457.13004)]. The author refutes a related conjecture for \(p=3\), by constructing for \(n\geq 3\) a permutation group which is a \(3\)-group without fixed-point-free elements, having four orbits of size \(3^{2n-1}\) and one orbit of size \(3^{2n}\). These groups are obtained from a suitable subgroup of \(\mathrm{PGL}(2,\mathbb Z_3[\sqrt 2])\) by reducing modulo \(3^n\); [see also \textit{E. Crestani} and \textit{P. Spiga}, Isr. J. Math. 180, 413--424 (2010; Zbl 1217.20001)] for \(p\geq 5\).