We give new characterizations of primitive permutation groups related to the notion of completely reachable automata and to the property of an automaton having a set of synchronizing words of maximal possible state complexity. Without reference to automata, given a finite permutation group on n elements we show that the following properties (and variants thereof) are equivalent: (1) the permutation group is primitive, (2) in the transformation monoid generated by the permutation group and any mapping whose image has size $n-1$ 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 permutation group and any mapping whose image has size $n-1$ there exists, for every two distinct and non-empty subsets, an element mapping precisely one to a singleton set. Then, we state a new characterization of k-homogeneity, state a sufficient condition for reachability of all subsets of size at most $n-2$ when adjoining a function of rank $n-2$ and define k-reachable permutation groups motivated by our characterizations 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. 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.

我们给出了与完全可达自动机的概念以及具有一组最大可能状态复杂性的同步字的自动机的属性相关的原语置换群的新特征。在不参考自动机的情况下，给定n 个元素上的有限置换群，我们表明以下属性（及其变体）是等效的：（1）置换群是本原的，（2）在由置换群生成的变换幺半群中，以及任何其图像具有大小的映射$n-1$对于每个非空子集，存在一个将整个排列域映射到该子集的元素，（3）在由排列组生成的变换幺半群和其图像具有大小的任何映射中$n-1$对于每两个不同且非空的子集，存在一个元素将其精确地映射到单例集。然后，我们提出了k均匀性的新表征，给出了所有大小子集最多可达性的充分条件$n-2$当邻接秩函数时$n-2$并根据我们对原始排列组的表征来定义k可达排列组。但结果表明，n次k 可达置换群$6\le k\le n-6$是交替基团或对称基团。最后，我们将我们的结果应用于自动机，并表明自动机的变换幺半群包含一个原始排列组和一个从其图像中精确排除一个状态的映射是完全可达的，并且具有以下属性：同步词集的最小自动机具有最大可能的状态数。