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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.

Authors

João Araújo

Peter J. Cameron

Abstract

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 e↵ort 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.

Journal Article Type Article
Publication Date May 3, 2018
Journal Transactions of the American Mathematical Society
Print ISSN 0002-9947
Electronic ISSN 0002-9947
Publisher American Mathematical Society
Peer Reviewed Peer Reviewed
Volume 371
Pages 105-136
Institution Citation 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, 105-136. https://doi.org/10.1090/tran/7274
DOI https://doi.org/10.1090/tran/7274
Keywords Transformation semigroups; Regular semigroups; Permutation groups; Primitive groups; Homogeneous groups; Rank of semigroups; Automorphisms of semigroups
Publisher URL http://www.ams.org/journals/tran/2019-371-01/S0002-9947-2018-07274-9/
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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