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dc.contributor.authorAraújo, João
dc.contributor.authorBentz, Wolfram
dc.contributor.authorCameron, Peter Jephson
dc.identifier.citationAraújo , J , Bentz , W & Cameron , P J 2019 , ' Orbits of primitive k -homogenous groups on ( n - k )-partitions with applications to semigroups ' , Transactions of the American Mathematical Society , vol. 371 , no. 1 , pp. 105-136 .
dc.identifier.otherPURE: 249954068
dc.identifier.otherPURE UUID: 9f4d8b99-4c99-4360-91ab-569d0d5c7e40
dc.identifier.otherScopus: 85062177357
dc.identifier.otherORCID: /0000-0003-3130-9505/work/58055605
dc.identifier.otherWOS: 000452652400004
dc.descriptionThis work was developed within FCT project CEMAT-CIÊNCIAS (UID/Multi/04621/2013).en
dc.description.abstractThe purpose of this paper is to advance our knowledge of two of the most classic and popular topics in transformation semigroups: automorphisms and the size of minimal generating sets. In order to do this, we examine the k-homogeneous permutation groups (those which act transitively on the subsets of size k of their domain X) where |X|=n and k<n/2. In the process we obtain, for k-homogeneous groups, results on the minimum numbers of generators, the numbers of orbits on k-partitions, and their normalizers in the symmetric group. As a sample result, we show that every finite 2-homogeneous group is 2-generated. Underlying our investigations on automorphisms of transformation semigroups is the following conjecture: If a transformation semigroup S contains singular maps, and its group of units is a primitive group G of permutations, then its automorphisms are all induced (under conjugation) by the elements in the normalizer of G in the symmetric group. For the special case that S contains all constant maps, this conjecture was proved correct, more than 40 years ago. In this paper, we prove that the conjecture also holds for the case of semigroups containing a map of rank 3 or less. The effort in establishing this result suggests that further improvements might be a great challenge. This problem and several additional ones on permutation groups, transformation semigroups and computational algebra, are proposed in the end of the paper.
dc.relation.ispartofTransactions of the American Mathematical Societyen
dc.rights© 2017, American Mathematical Society. This work has been made available online in accordance with the publisher’s policies. This is the author created, accepted version manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at /
dc.subjectPermutation groupen
dc.subjectTransformation semigroupen
dc.subjectQA Mathematicsen
dc.titleOrbits of primitive k-homogenous groups on (n-k)-partitions with applications to semigroupsen
dc.typeJournal articleen
dc.contributor.institutionUniversity of St Andrews. Pure Mathematicsen
dc.contributor.institutionUniversity of St Andrews. Centre for Interdisciplinary Research in Computational Algebraen
dc.description.statusPeer revieweden

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