Sync-Maximal Permutation Groups Equal Primitive Permutation Groups

TL;DR

This paper proves that sync-maximal permutation groups are exactly the primitive groups, providing a new characterization of primitive groups through automata theory and exploring stronger definitions of sync-maximality.

Contribution

It establishes the precise equivalence between sync-maximal groups and primitive groups, solving an open problem and offering a novel perspective on group classification.

Findings

Sync-maximal groups are exactly the primitive groups.

Provides a new automata-theoretic characterization of primitive groups.

Explores stronger variants of sync-maximality.

Abstract

The set of synchronizing words of a given $n$-state automaton forms a regular language recognizable by an automaton with $2^n - n$ states. The size of a recognizing automaton for the set of synchronizing words is linked to computational problems related to synchronization and to the length of synchronizing words. Hence, it is natural to investigate synchronizing automata extremal with this property, i.e., such that the minimal deterministic automaton for the set of synchronizing words has $2^n - n$ states. The sync-maximal permutation groups have been introduced in [{\sc S. Hoffmann}, Completely Reachable Automata, Primitive Groups and the State Complexity of the Set of Synchronizing Words, LATA 2021] by stipulating that an associated automaton to the group and a non-permutation has this extremal property. The definition is in analogy with the synchronizing groups and analog to a characterization of primitivity obtained in the mentioned work. The precise relation to other classes of groups was mentioned as an open problem. Here, we solve this open problem by showing that the sync-maximal groups are precisely the primitive groups. Our result gives a new characterization of the primitive groups. Lastly, we explore an alternative and stronger definition than sync-maximality.