Completely Reachable Automata, Primitive Groups and the State Complexity of the Set of Synchronizing Words

TL;DR

This paper explores the relationship between primitive permutation groups, automata reachability, and the complexity of synchronizing words, introducing new group classifications and characterizing their properties.

Contribution

It provides a new characterization of primitive permutation groups via completely reachable automata and introduces sync-maximal and k-reachable groups, expanding understanding of automata group structures.

Findings

Sync-maximal groups lie between 2-homogeneous and primitive groups.

k-reachable groups with 6 ≤ k ≤ n-6 are either alternating or symmetric.

New characterizations connect automata reachability with permutation group properties.

Abstract

We give a new characterization of primitive permutation groups tied to the notion of completely reachable automata. Also, we introduce sync-maximal permutation groups tied to the state complexity of the set of synchronizing words of certain associated automata and show that they are contained between the $2$-homogeneous and the primitive groups. Lastly, we define $k$-reachable groups in analogy with synchronizing groups and motivated by our characterization of primitive permutation groups. But the results show that a $k$-reachable permutation group of degree $n$ with $6 \le k \le n - 6$ is either the alternating or the symmetric group.