Languages Recognized by Finite Supersoluble Groups

In this paper, we give two descriptions of the languages recognized by finite supersoluble groups. We first show that such a language belongs to the Boolean algebra generated by the modular products of elementary commutative languages. An elementary commutative language is defined by a condition specifying the number of occurrences of each letter in its words, modulo some fixed integer. Our second characterization makes use of counting functions computed by transducers in strict triangular form. Eilenberg’s variety theorem [7] is a powerful tool for classifying regular languages. It states that, given a variety of finite monoids V, the class of languages V whose syntactic monoid belongs to V is a variety of languages, that is, a class of regular languages closed under finite union, complement, left and right quotients and inverse of morphisms. Further, the correspondence V → V between varieties of finite monoids and varieties of languages is one-to-one and onto. Eilenberg’s theorem can be used in both ways: given a variety of languages, one can look for the corresponding variety of monoids, or, given a variety of monoids, one can seek for a combinatorial description of the corresponding variety of languages. Examples abound in the literature: for instance, aperiodic monoids correspond to star-free languages, J -trivial languages to piecewise testable languages, etc. We refer the reader to [10] for a survey. It is therefore natural to ask for a nice characterization of the variety of languages corresponding to the variety of groups. The answer to this frequently asked question is unfortunately negative: there is no known satisfactory answer to this question. The reason is hidden in the complexity of finite groups since a solution would probably require a description of the languages recognized by each finite simple group . . . However, solutions are known for some important subvarieties: abelian groups [7], p-groups [7, 17, 18], nilpotent groups [7, 16] and soluble groups [14, 18]. The The authors acknowledge support from the AutoMathA programme of the European Science Foundation. The second author was supported by the Grant AINV07/093 from the Conselleria d’Empresa, Universitat i Ciència de la Generalitat Valenciana and the third author was supported by the Grant PR2007-0164 from MEC of Spain. LIAFA, Université Paris VII and CNRS, Case 7014, 75205 Paris Cedex 13, France. Dpt. de Matemàtica Aplicada, Universitat d’Alacant, Sant Vicent del Raspeig, Ap. Correus 99, E – 03080 Alacant.