Skip to content

Research at St Andrews

Primitive groups, graph endomorphisms and synchronization

Research output: Contribution to journalArticlepeer-review

DOI

Open Access permissions

Open

Author(s)

João Araújo, Wolfram Bentz, Peter Jephson Cameron, Gordon Royle, Artur Schaefer

School/Research organisations

Abstract

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 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.
Close

Details

Original languageEnglish
Pages (from-to)829-867
Number of pages39
JournalProceedings of the London Mathematical Society
Volume113
Issue number6
Early online date3 Oct 2016
DOIs
Publication statusPublished - Dec 2016

    Research areas

  • Permutation group, Semigroup, Synchronization, Graph endomorphism

Discover related content
Find related publications, people, projects and more using interactive charts.

View graph of relations

Related by author

  1. Forbidden subgraphs of power graphs

    Manna, P., Cameron, P. J. & Mehatari, R., 5 Jun 2021, (Accepted/In press) In: Electronic Journal of Combinatorics.

    Research output: Contribution to journalArticlepeer-review

  2. Graphs defined on groups

    Cameron, P. J., 15 Apr 2021, (E-pub ahead of print) In: International Journal of Group Theory. In Press

    Research output: Contribution to journalArticlepeer-review

  3. Groups generated by derangements

    Bailey, R. A., Cameron, P. J., Giudici, M. & Royle, G. F., 15 Apr 2021, In: Journal of Algebra. 572, p. 245-262

    Research output: Contribution to journalArticlepeer-review

  4. Undirecting membership in models of Anti-Foundation

    Adam-Day, B. & Cameron, P. J., Apr 2021, In: Aequationes Mathematicae. 95, 2, p. 393-400 8 p.

    Research output: Contribution to journalArticlepeer-review

  5. The existential transversal property: a generalization of homogeneity and its impact on semigroups

    Araújo, J., Bentz, W. & Cameron, P. J., Feb 2021, In: Transactions of the American Mathematical Society. 374, 2, p. 1155–1195

    Research output: Contribution to journalArticlepeer-review

Related by journal

  1. Proceedings of the London Mathematical Society (Journal)

    Kenneth John Falconer (Editor)

    19961999

    Activity: Publication peer-review and editorial work typesEditor of research journal

Related by journal

  1. Projective duals to algebraic and tropical hypersurfaces

    Ilten, N. & Len, Y., Nov 2019, In: Proceedings of the London Mathematical Society. 119, 5, p. 1234-1278

    Research output: Contribution to journalArticlepeer-review

  2. Highest rank of a polytope for An

    Cameron, P. J., Fernandes, M. E., Leemans, D. & Mixer, M., 4 Jul 2017, In: Proceedings of the London Mathematical Society. 115, 1, p. 135-176 42 p.

    Research output: Contribution to journalArticlepeer-review

  3. On regularity and the word problem for free idempotent generated semigroups

    Dolinka, I., Gray, R. D. & Ruskuc, N., 3 Mar 2017, In: Proceedings of the London Mathematical Society. 114, 3, p. 401-432 32 p.

    Research output: Contribution to journalArticlepeer-review

  4. The Assouad dimensions of projections of planar sets

    Fraser, J. M. & Orponen, T., Feb 2017, In: Proceedings of the London Mathematical Society. 114, 2, p. 374-398 25 p.

    Research output: Contribution to journalArticlepeer-review

ID: 244980137

Top