On the groups of codes with empty kernel

An internal factor of a word x is a word v such that x = uvw for some nonempty words u, w. The kernel of a set X of words is the set of words of X which are internal factors of words of X. Let φ be the syntactic morphism of the submonoid X∗ generated by X. We prove that if X is a code with empty kernel, the groups contained in the image by φ of the complement of the set of internal factors of the words of X are cyclic. This generalizes a result announced by Schützenberger in 1964.