New characterizations of primitive permutation groups with applications to synchronizing automata

Abstract

For a finite permutation group on n elements we show the following (and variants thereof) equivalences: (1) the permutation group is primitive, (2) in the transformation monoid generated by the group and any rank n−1 mapping there exists, for every non-empty subset, an element mapping the whole permutation domain onto this subset, (3) in the transformation monoid generated by the group and any rank n−1 mapping there exists, for every two distinct subsets, an element mapping precisely one to a singleton set. We also investigate further properties related to the reachability of subsets. Lastly, we apply our results to automata and show that automata whose transformation monoids contain a primitive permutation group and a mapping that excludes precisely one state from its image are completely reachable and have the property that a minimal automaton for the set of synchronizing words has the maximal possible number of states.