On the structure of some group codes

In this paper, we study the following problem: Which characteristics does a codeC possess when the syntactic monoidsyn(C ^*) of the star closureC ^* ofC is a group? For a codeC, if the syntactic monoidsyn(C ^*) is a group, then we callC a group code. This definition of a group code is different from the one in [1] (see [1], 46–47). Schützenberger had characterized the structure of finite group codes and had proved thatC is a finite group code if and only ifC is a full uniform code (see [5], [8]). Fork-prefix andk-suffix codes withk≥2,k-infix,k-outfix,p-infix,s-infix, right semaphore codes and left semaphore codes, etc., we obtain similar results. It is proved that the above mentioned codes are group codes if and only if they are uniform codes.