Primitive groups, graph endomorphisms and synchronization

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.

Between primitive and 2-transitive: Synchronization and its friends

“…In the paper [11], the authors extend this result to show that a primitive group of degree n synchronizes any map of rank n − 4 or greater. The nonbasic group S 3 ≀ S 2 , the automorphism group of the 3 × 3 square grid, has degree 9 and fails to synchronize a map of rank 3 (the grid graph has clique number and chromatic number 3); so this result is within one of best possible.…”

Imprimitive permutations in primitive groups

“…For example, ((2, 2), (2, 1, 1), (1,1)) is a clustering of (2, 2, 2, 1, 1, 1, 1). The clusters are partitions of 4, 4 and 2 respectively.…”

“…This is shown by the clustering ((1, 5), (10, 10, 10, 10, 10, 10)). For the partition (1, 5) = (1 × 1, 1 × 5) is an ic-partition of type (1,6), and (10, . .…”

“…In many practical situations we know that a primitive group contains a given permutation and we want to know which group it can be; in some other practical situations we know the group and would like to know if it contains a permutation of some given type. For example, a group G ≤ S n is said to be nonsynchronizing if it is contained in the automorphism group of a non-trivial primitive graph with complete core, that is, with clique number equal to chromatic number (see for example, [1,2,3,5,6,9,30]).…”

“…The largest part (10) has 5 among its divisors, and the remaining parts can be rearranged in a set {2, 3} whose sum is 5. Another example is P = (1,2,5,7,17,19,23,111). The largest part (111) has 37 among its divisors, and the other parts can be rearranged in sets adding up to that number: {{2, 5, 7, 23}, {1, 17, 19}}.…”