Transitive permutation groups with bounded movement
A permutation group \(G\) on a finite set \(\Omega\) is said to satisfy the ``bounded movement'' condition with respect to \(m\) (in short: \(BM(m)\)) if and only if the following implication holds: If \(g \in G\) and \(\Delta \subseteq \Omega\) satisfy \(\Delta \cap \Delta^ g = \emptyset\), then \(| \Delta| \leq m\). A previous result of the second author [J. Algebra 144, 436-442 (1991; Zbl 0744.20004)] states that if \(G\) acts transitively on \(\Omega\) and satisfies \(BM(m)\) then \(| \Omega| \leq 3m\) (Theorem 1). In the present paper the groups \(G\) meeting this bound are studied. It is conjectured that if \(G\) is transitive on \(\Omega\), satisfies \(BM(m)\) and \(| \Omega| = 3m\), one of (a) \(G = S_ 3\), \(m = 1\); (b) \(G = A_ 4, A_ 5\), \(m = 2\); (c) \(G\) is a 3- group, \(m = 3^ r\) holds. The main result is Theorem 3: Let \(m\) be the least integer for which the conjecture fails and let \(G\) be a minimal counterexample of degree \(3m\). Then \(G\) is a simple group acting primitively on \(\Omega\). Hence the proof of the conjecture is reduced to the study of simple groups acting primitively of degree \(3m\). It has been shown by the second author (loc. cit.) that in this case \(m = 2^ \varepsilon 3^ r 5^{\varepsilon'}\) where \(\varepsilon, \varepsilon' = 0\) or 1 and \(r \geq 0\) holds (Theorem 2). Proofs of Theorem 1 and Theorem 2 are included for convenience.
scientific article; zbMATH DE number 645106