Research output: Contribution to journal › Article › peer-review

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

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.

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

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.

Original language | English |
---|---|

Pages (from-to) | 829-867 |

Number of pages | 39 |

Journal | Proceedings of the London Mathematical Society |

Volume | 113 |

Issue number | 6 |

Early online date | 3 Oct 2016 |

DOIs | |

Publication status | Published - Dec 2016 |

- Permutation group, Semigroup, Synchronization, Graph endomorphism

**Discover related content**

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

## 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 journal › Article › peer-review

## Graphs defined on groups

Cameron, P. J., 15 Apr 2021, (E-pub ahead of print) In: International Journal of Group Theory. In PressResearch output: Contribution to journal › Article › peer-review

## Groups generated by derangements

Bailey, R. A., Cameron, P. J., Giudici, M. & Royle, G. F., 15 Apr 2021, In: Journal of Algebra. 572, p. 245-262Research output: Contribution to journal › Article › peer-review

## 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 journal › Article › peer-review

## 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. 1155–1195Research output: Contribution to journal › Article › peer-review

## Proceedings of the London Mathematical Society (Journal)

Kenneth John Falconer (Editor)

1996 → 1999Activity: Publication peer-review and editorial work types › Editor of research journal

## Projective duals to algebraic and tropical hypersurfaces

Ilten, N. & Len, Y., Nov 2019, In: Proceedings of the London Mathematical Society. 119, 5, p. 1234-1278Research output: Contribution to journal › Article › peer-review

## Highest rank of a polytope for

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.*A*_{n}Research output: Contribution to journal › Article › peer-review

## 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 journal › Article › peer-review

## 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 journal › Article › peer-review

ID: 244980137