If \(G\) is a permutation group acting on a set \(\Omega\) with no fixed points in \(\Omega\) then \(G\) is said to have finite movement \(m\) if \(m=\text{move}(G)=\sup\{|\Gamma^g\setminus\Gamma|\mid\Gamma\subseteq\Omega,\;g\in G\}\) exists. By a theorem of C. E. Praeger \(|\Omega|=n\) is finite and so \(G\) is finite if \(G\) has finite movement \(m\) [J. Algebra 144, No. 2, 436-442 (1991; Zbl 0744.20004)].
In the present paper Praeger’s result is sharpened in the following way: Theorem 1.1. Let \(p\) be a prime, \(p\geq 5\), and suppose \(G\) has finite movement \(m\). Then \(n=|\Omega|\leq 4m-p+3\) if \(G\) is not a \(2\)-group and \(p\) is the least odd prime number dividing \(|G|\).
It is not known whether the bound given in Theorem 1.1 is attained. A more refined bound is given in Theorem 1.2 which is attained for an infinite family of groups.