Construction d'une famille de codes associés à certains groupes finis. (Construction of a family of codes associated to certain finite groups)
The author shows the optimality of the Perrin conjecture extending Schreier's formula for subgroups of free groups to free monoids. A code on an alphabet A is a subset X of \(A^*\) freely generating \(X^*\). Suppose Q is a set, \(q\in Q\), G is a group acting on Q and H is the subgroup of G fixing q. For any surjective homomorphism f: \(A^*\to G\), \(f^{-1}(H)\) is free and the generating code will be denoted by \(Z<G,H,f>\). Theorem 1: Let G be a noncyclic group of degree d and let X be a finite subset of \(Z<G,H,f>\). If G is a syntactic group of \(X^*\), then X has at least \(d+1\) elements. Theorem 2: Let G be an abelian group of rank 2 and degree d. Then there is a code X on \(\{\) a,b\(\}\) of \(d+1\) elements so that G is a syntactic group of \(X^*\).
scientific article; zbMATH DE number 4093791