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The \(k\)-star property for permutation groups. (English) Zbl 1152.20001

Summary: 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 properties for \(k\neq m\). Several open problems are posed.

MSC:

20B05 General theory for finite permutation groups
20B15 Primitive groups
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