Cycle types in infinite permutation groups (Q1895618)

The main result of the paper is Theorem 1.1: Let \(G\) be a transitive permutation group of an infinite set \(\Omega\) and let \(k\) be a positive integer or \(k = \aleph_0\). Suppose that \(G\) contains an element \(g\) whose cycle decomposition consists of a single infinite cycle, a finite positive number of finite cycles of length at least 2 and at least \(k\) fixed points. (i) If \(k\) is finite, then \(G\) is \((k+1)\)-transitive on \(\Omega\). (ii) If \(k\) is infinite and \((G, \Omega)\) is primitive, then \(G\) is highly transitive. The proof of (ii) is quite hard and makes use of the recent classification of infinite primitive Jordan groups.