Homogeneity in infinite permutation groups (Q1061217)

If G is a permutation group on a set X, then G is said to be k- homogeneous (respectively k-transitive) if it is transitive on the set of unordered (respectively ordered) k-subsets of X. It is shown that if X is infinite, \(k\geq 5\), and G is (k-1)-transitive but not k-transitive, then the following hold: (i) G is not \((k+r)\)-homogeneous for any \(r\geq 3\); (ii) if G is \((k+2)\)-homogeneous, then the group induced by G on any k- subset of X is the alternating group \(A_ k\). This improves an earlier bound due to Hodges. The proof uses well-known upper bounds on the orders of finite primitive permutation groups which are derived from the classification of the finite simple groups.