## Abstract

This paper introduces the concept of orbit-homogeneity of permutation groups: a group G is orbit-t-homogeneous if two sets of cardinality t lie in the same orbit of G whenever their intersections with each G-orbit have the same cardinality. For transitive groups, this coincides with the usual notion of t-homogeneity. This concept is also compatible with the idea of partition transitivity introduced by Martin and Sagan. Further, this paper shows that any group generated by orbit-t-homogeneous subgroups is orbit-t-homogeneous, and that the condition becomes stronger as t increases up to [n/2], where n is the degree. So any group G has a unique maximal orbit-t-homogeneous subgroup Ω_{t}(G), and Ω_{t}(G) ≤ Ω_{t} _{t-1}(G). Some structural results for orbit-t-homogeneous groups, and a number of examples, are also given.

Original language | English |
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Pages (from-to) | 587-596 |

Number of pages | 10 |

Journal | Bulletin of the London Mathematical Society |

Volume | 38 |

Issue number | 4 |

DOIs | |

Publication status | Published - 1 Jan 2006 |