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Primitive groups, graph endomorphisms and synchronization

Aroújo, João; Bentz, Wolfram; Araújo, João; Cameron, Peter J.; Royle, Gordon; Schaefer, Artur

Authors

João Aroújo

João Araújo

Peter J. Cameron

Gordon Royle

Artur Schaefer



Abstract

© 2016 London Mathematical Society. Let Ω be a set of cardinality n, G be a permutation group on Ω and f : Ω → Ω be a map that is not a permutation. We say that G synchronizes f if the transformation semigroup 〈G, f〉 contains a constant map, and that G is a synchronizing group if G synchronizes every non-permutation. A synchronizing group is necessarily primitive, but there are primitive groups that are not synchronizing. Every non-synchronizing primitive group fails to synchronize at least one uniform transformation (that is, transformation whose kernel has parts of equal size), and it had previously been conjectured that this was essentially the only way in which a primitive group could fail to be synchronizing, in other words, that a primitive group synchronizes every non-uniform transformation. The first goal of this paper is to prove that this conjecture is false, by exhibiting primitive groups that fail to synchronize specific non-uniform transformations of ranks 5 and 6. As it has previously been shown that primitive groups synchronize every non-uniform transformation of rank at most 4, these examples are of the lowest possible rank. In addition, we produce graphs with primitive automorphism groups that have approximately √n n non-synchronizing ranks, thus refuting another conjecture on the number of non-synchronizing ranks of a primitive group. The second goal of this paper is to extend the spectrum of ranks for which it is known that primitive groups synchronize every non-uniform transformation of that rank. It has previously been shown that a primitive group of degree n synchronizes every non-uniform transformation of rank n - 1 and n - 2, and here this is extended to n - 3 and n - 4. In the process, we will obtain a purely graph-theoretical result showing that, with limited exceptions, in a vertex-primitive graph the union of neighbourhoods of a set of vertices A is bounded below by a function that is asymptotically √|A|. Determining the exact spectrum of ranks for which there exist non-uniform transformations not synchronized by some primitive group is just one of several natural, but possibly difficult, problems on automata, primitive groups, graphs and computational algebra arising from this work; these are outlined in the final section.

Publication Date 2016-12
Journal Proceedings of the London Mathematical Society
Print ISSN 0024-6115
Electronic ISSN 1460-244X
Publisher London Mathematical Society
Peer Reviewed Peer Reviewed
Volume 113
Issue 6
Pages 829-867
Institution Citation Aroújo, J., Bentz, W., Cameron, P. J., Royle, G., & Schaefer, A. (2016). Primitive groups, graph endomorphisms and synchronization. Proceedings of the London Mathematical Society, 113(6), 829-867. https://doi.org/10.1112/plms/pdw040
DOI https://doi.org/10.1112/plms/pdw040
Keywords Primitive groups
Publisher URL http://plms.oxfordjournals.org/content/early/2016/10/03/plms.pdw040.abstract?sid=2384eab0-9b26-4f7f-a20c-1ff0e807e562
Copyright Statement ©2016 University of Hull
Additional Information This is the accepted version of the following article: João Araújo, Wolfram Bentz, Peter J. Cameron, Gordon Royle, and Artur Schaefer. Primitive groups, graph endomorphisms and synchronization. Proc. London Math. Soc. first published online October 3, 2016 doi:10.1112/plms/pdw040 , which has been published in final form at http://plms.oxfordjourn.../2016/10/03/plms.pdw040

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