Bicyclic commutator quotients with one non-elementary component

Abstract

. For any number field K with non-elementary 3-class group Cl 3 ( K ) ≃ C 3 e × C 3 , e > 2, the punctured capitulation type κ ( K ) of K in its unramified cyclic cubic extensions L i , 1 6 i 6 4, is an orbit under the action of S 3 × S 3 . By means of Artin’s reciprocity law, the arithmetical invariant κ ( K ) is translated to the punctured transfer kernel type κ ( G 2 ) of the automorphism group G 2 = Gal(F 23 ( K ) /K ) of the second Hilbert 3-class field of K . A classification of finite 3-groups G with low order and bicyclic commutator quotient G/G ′ ≃ C 3 e × C 3 , 2 6 e 6 6, according to the algebraic invariant κ ( G ), admits conclu-sions concerning the length of the Hilbert 3-class field tower F ∞ 3 ( K ) of imaginary quadratic number fields K .