Journal of Pure and Applied Algebra | 2019
Classification of metabelian 2-groups G with Gab ≃ (2,2n),n ≥ 2, and rank d(G′)=2; Applications to real quadratic number fields
Abstract
Abstract We characterize all finite metabelian 2-groups G whose abelianizations G ab are of type ( 2 , 2 n ) , with n ≥ 2 , and for which their commutator subgroups G ′ have rank = 2 . This is given in terms of the order of the abelianizations of the maximal subgroups and the structure of the abelianizations of those normal subgroups of index 4 in G. We then translate these group theoretic properties to give a characterization of number fields k with 2-class group Cl 2 ( k ) ≃ ( 2 , 2 n ) , n ≥ 2 , such that the rank of Cl 2 ( k 1 ) = 2 where k 1 is the Hilbert 2-class field of k. In particular, we apply all this to real quadratic number fields whose discriminants are a sum of two squares.