CROWNS IN PROFINITE GROUPS AND APPLICATIONS

If G is a finite soluble group then the crown of a complemented chief factor H/K of G is a specific normal factor of G which is associated with all complemented chief factors of G that are G-isomorphic to H/K. This idea was first introduced by W. Gaschütz to construct the prefrattini subgroups, and was subsequently generalized to all finite groups by J. Lafuente . In the present paper we generalize the notion of crown to profinite groups and show that when G is a finitely generated profinite group one can associate with G a certain infinite formal Dirichlet series, which is then studied using the notion of crown.