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Orbits of primitive k-homogenous groups on (N − k)-partitions with applications to semigroups

Araújo, João; Bentz, Wolfram; Cameron, Peter J.


João Araújo

Wolfram Bentz

Peter J. Cameron


© 2018 American Mathematical Society. The 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 at the end of the paper.


Araújo, J., Bentz, W., & Cameron, P. J. (2018). Orbits of primitive k-homogenous groups on (N − k)-partitions with applications to semigroups. Transactions of the American Mathematical Society, 371(1), 105-136.

Journal Article Type Article
Acceptance Date May 9, 2017
Online Publication Date May 24, 2017
Publication Date May 3, 2018
Deposit Date Jul 4, 2017
Publicly Available Date Jul 4, 2017
Journal Transactions of the American Mathematical Society
Print ISSN 0002-9947
Electronic ISSN 1088-6850
Publisher American Mathematical Society
Peer Reviewed Peer Reviewed
Volume 371
Issue 1
Pages 105-136
Keywords Transformation semigroups; Regular semigroups; Permutation groups; Primitive groups; Homogeneous groups; Rank of semigroups; Automorphisms of semigroups
Public URL
Publisher URL
Additional Information This is the author's accepted manuscript of an article accepted for publication in Transactions of the American mathematical society.


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