Orbit-homogeneity in permutation groups

Peter J. Cameron*, Alexander W. Dent

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)

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 Ωt(G), and Ωt(G) ≤ Ωt t-1(G). Some structural results for orbit-t-homogeneous groups, and a number of examples, are also given.

Original languageEnglish
Pages (from-to)587-596
Number of pages10
JournalBulletin of the London Mathematical Society
Volume38
Issue number4
DOIs
Publication statusPublished - 1 Jan 2006

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