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

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

Abstract

© 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.

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(1), 105-136. https://doi.org/10.1090/tran/7274

• 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
• Publisher: American Mathematical Society
• Peer Reviewed: Peer Reviewed
• Volume: 371
• Issue: 1
• Pages: 105-136
• 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
• Public URL: https://hull-repository.worktribe.com/output/453260
• 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.
• Contract Date: Jul 4, 2017