Sur la Classification et le Denombrement des Sous-groupes du Groupe Modulaire et de leurs Classes de Conjugaison

Abstract

In this article we give a classification of the sub-groups in PSL(2,Z) and of the conjugacy classes of these sub-groups by the mean of an combinatorial invariant: some trivalent diagrams (dotted or not). We give explicit formulae enabling to count the number of isomorphism classes of these structures and of some of their variations, as function of the number of their arcs. Until now, the counting of non-dotted diagrams was an open problem, for it gives also the number of unrooted combinatorial maps, triangular or general respectively. The article ends with the description of a high performance algorithm to enumerate those structures witch is built upon an unexpected factoring of the cycle index series of the considered combinatorial species.