The k-star Property for Permutation Groups

Rosemary Clough, Cheryl E. Praeger, Csaba Schneider

Abstract

For an integer k at least 2, a permutation group G has the k-star property if, for every k-subset of points, G contains an element that fixes it setwise but not pointwise. This property holds for all k-transitive, generously k-transitive, and almost generously k-transitive permutation groups. Study of the k-star property was motivated by recent work on the case k = 3 by P. M. Neumann and the second author. The paper focuses on intransitive groups with the k-star property, studying properties of their transitive constituents, and relationships between the k-star and m-star properies for $k \neq m$. Several open problems are posed.