On the movement of transitive permutation \(p\)-groups (Q1579870)

If \(G\) is a permutation group on a set \(\Omega\) with no fixed points in \(\Omega\), \(G\) is said to have bounded movement if the cardinalities \(|\Gamma^g\setminus\Gamma|\) are bounded above for all \(g\in G\) and \(\Gamma\subseteq\Omega\), and the maximum of these cardinalities \(|\Gamma^g\setminus\Gamma|\) is called the movement of \(G\). In [J. Algebra 144, No. 2, 436-442 (1991; Zbl 0744.20004)], \textit{C. E. Praeger} found a bound (depending on \(m\) and \(p\)) on \(|\Omega|\) for a transitive group \(G\) (which is not a 2-group) having movement \(m\), where \(p\) is the least odd prime dividing \(|G|\). In [\textit{A. Hassani}, \textit{M. Khayaty}, \textit{E. I. Khuhuro} and \textit{C. E. Praeger}, J. Algebra 214, No. 1, 317-337 (1999; Zbl 0922.20006)], the groups attaining this bound were classified in three families. The groups being in one of these families are believed by the author to be of exponent \(p\). The author cannot prove it, but he discusses the problem, giving some sufficient conditions (linked to Hughes subgroups) for \(G\) to have exponent \(p\).