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Transactions of the American Mathematical Society

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Orbits of primitive $ k$-homogenous groups on $ (n-k)$-partitions with applications to semigroups


Authors: João Araújo, Wolfram Bentz and Peter J. Cameron
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 20B30, 20B35, 20B15, 20B40, 20M20, 20M17
DOI: https://doi.org/10.1090/tran/7274
Published electronically: May 3, 2018
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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 $ \vert X\vert=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.

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Additional Information

João Araújo
Affiliation: Universidade Aberta and CEMAT-CIÊNCIAS, Departamento de Matemática, Faculdade de Ciências, Universidade de Lisboa, 1749-016, Lisboa, Portugal
Email: jaraujo@ptmat.fc.ul.pt

Wolfram Bentz
Affiliation: School of Mathematics & Physical Sciences, University of Hull, Kingston upon Hull, HU6 7RX, United Kingdom
Email: w.bentz@hull.ac.uk

Peter J. Cameron
Affiliation: School of Mathematics and Statistics, University of St Andrews, St Andrews, Fife KY16 9SS, United Kingdom
Email: pjc20@st-andrews.ac.uk

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.
Received by editor(s): December 28, 2015
Received by editor(s) in revised form: January 17, 2017, and January 29, 2017
Published electronically: May 3, 2018
Additional Notes: The first author is the corresponding author.
This work was developed within FCT project CEMAT-CIÊNCIAS (UID/Multi/04621/2013).
Article copyright: © Copyright 2018 American Mathematical Society

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