A note on the groups of some codes

The starting point of this note is the article On the synchronizing properties of certain prefix codes published by Schützenberger in 1964 [2]. This article presents a family of maximal prefix codes called J -codes, that we call semaphore codes. The main result is that a semaphore code is always of the form X where X is a synchronized semaphore code and n ≥ 1. Two proofs are presented of this result. The first one is a direct combinatorial proof, which is hard to follow. The second one uses intermediary results which are interesting in their own (this proof is reproduced in [1]). First, it is proved (Remark 1) that the group of a semaphore code is a regular permutation group. At the end of the paper, it is claimed that the group is cyclic, but the proof is not correct (we shall come back to this point). Next, it is proved (Property 2) that if the group G of a maximal prefix code X is regular (the statement says abelian but the proof uses only the fact that it is regular), then there is a decomposion X = Y ◦ Z ◦ T such that Y, T are synchronizing and Z is a group code with G(Z) = G. A third result (Remark 2) shows that if X is a semaphore code such that X = Y ◦ Z ◦ T with Z a regular group code, then X = U with U a synchronized semaphore code and n ≥ 1. We prove here a result which is a generalization of the fact that the group of a semaphore code is regular. Namely, we prove that, for a semaphore code X ⊂ A, any group in the syntactic monoid of X which meets the image of AXA is regular (Theorem 1). We actually prove this result for codes which more general than semaphore codes, in the sense that they need not be maximal. However, we have not been able to prove that all these groups are cyclic, as claimed in [2]. We discuss this conjecture in Section 4.