On the groups of codes with empty kernel. (Q982209)

A word \(v\in A^*\) is an internal factor of a word \(x\in A^*\) iff \(x=uvw\) for some nonempty words \(u,w\). The kernel of a set \(X\subset A^*\) is the set of words from \(X\) which are internal factors of some word from \(X\). It is shown, that if \(X\) is a code with empty kernel, \(F\) the set of internal factors of words from \(X\) and \(\varphi\) the syntactic morphism of the submonoid \(X^*\), then any group \(G\) contained in \(\varphi(A^*\setminus F)\) is cyclic. A subclass of codes with empty kernel are semaphore codes, thus this is a generalization of a result of \textit{M. P. Schützenberger} [Inf. Control 7, 23-26 (1964; Zbl 0122.15004)].