Let A be a finite automaton. We are concerned with the minimal length of the words that send all states on a unique state (synchronizing words). J. Cerný has conjectured that, if there exists a synchronizing word in A, then there exists such a word with length ≤ (n-1)^2 where n is the number of states of A. As a generalization, we conjecture that, if there exists a word of rank ≤ k in A, there exists such a word with length ≤ (n-k)^2. In this paper we deal only with automata in which a letter induces a circular permutation and prove the following results: (1) the second conjecture is true for (n-1)/2 ≤ k ≤ n, (2) if n is prime, the first conjecture is true, (3) if n is prime and if there exists a letter of rank n - 1, the second conjecture is true.