The quasiprimitive almost elusive groups (Q6593812)

Let \(G\) be a nontrivial permutation group on a set \(\Omega\), an element \(g \in G\) is a derangement if \(\omega^{g} \not =\omega\) for every \(\omega \in \Omega\). If \(G\) is finite and transitive on \(\Omega\), a classical theorem of Jordan (which is an easy consequence of the orbit counting lemma), shows that \(G\) always contains a derangement. An extension of Jordan's theorem due to \textit{B. Fein}, \textit{W. M. Kantor} and \textit{M. Schacher} [J. Reine Angew. Math. 328, 39-57 (1981; Zbl 0457.13004)], shows that \(G\) always contains a derangement of prime power order (the proof requires the classification of finite simple groups). Interestingly, it is not possible to guarantee the existence in \(G\) of a derangement of prime order, as shown by the action of the Mathieu group \(\mathrm{M}_{11}\) on the laterals of the right cosets of a maximal subgroup isomoprhic to \(\mathrm{PSL}_{2}(11)\).\N\NA finite transitive group \(G\) is called elusive if it contains no derangements of prime order. Although a complete classification of the elusive groups is currently out of reach, a large class of examples were classified by \textit{M. Giudici} in [J. Lond. Math. Soc., II. Ser. 67, No. 1, 73--84 (2003; Zbl 1050.20002)].\N\NIn [J. Algebra 594, 519--543 (2022; Zbl 1523.20004)], \textit{T. C. Burness} and the author introduced the family of almost elusive groups, which contain a unique conjugacy class of derangements of prime order. In the paper under review, the author completes the classification of the quasiprimitive almost elusive groups.