Orbits of primitive k-homogenous groups on (n-k)-partitions with applications to semigroups
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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 in the end of the paper.
Araújo , J , Bentz , W & Cameron , P J 2017 , ' Orbits of primitive k -homogenous groups on ( n - k )-partitions with applications to semigroups ' Transactions of the American Mathematical Society . DOI: 10.1090/tran/7274
Transactions of the American Mathematical Society
© 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 www.ams.org / https://doi.org/10.1090/tran/7274
This work was developed within FCT project CEMAT-CIÊNCIAS (UID/Multi/04621/2013).
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