Genus number and l-rank of genus group of cyclic extensions of degree l

Let l be an odd prime, and k an algebraic number field of a finite degree. Let S be a finite product of distinct prime ideals g of k such that Ng1 (mod l). Let I(s) (resp. P(s)) denote the group of ideals (resp. principal ideals) of k prime to S, and let P_S denote the ray modulo S. In this paper we prove that the order (resp. the l-rank) of I(S)/P(S)^lP_S is expressed by the decomposition groups of prime factors of S in a Galois extension K_o (resp. K_r) over k. As an application of this, some results about genus theory are obtained.