Transitive permutation groups of minimal movement
The ``movement'' of a permutation group refers to the largest number of points which can be moved to a disjoint set by some element of the group. It was shown by \textit{C. E. Praeger} [J. Algebra 144, No. 2, 436-442 (1991; Zbl 0744.20004)] that a transitive permutation group of movement \(m\) has degree at most \(3m\). Equality is possible here (movement \(m\) on a set of \(3m\) points) if the group has exponent three, or if it is \(S_3\) (\(m=1\)), \(A_4\) (\(m=2\)), or \(A_5\) (\(m=2\)). \textit{A. Gardiner} and \textit{C. E. Praeger} [J. Algebra 168, No. 3, 798-803 (1994; Zbl 0816.20005)] conjectured that these are the only cases of equality, and they proved that a minimal counterexample is either a 3-group or a simple group acting primitively. The present paper completes the proof of this conjecture by considering first 3-groups and then simple groups. In treating 3-groups, the authors prove that a transitive \(p\)-group whose fixed-point-free elements all have order \(p\) must have exponent \(p\) if \(p=2\) or 3; they ask as an open question whether this holds for larger primes \(p\). For simple groups the authors use the classification of finite simple groups and prove a series of lemmas to eliminate the various families of simple groups.
scientific article; zbMATH DE number 883792