Bicyclic commutator quotients with one non-elementary component

TL;DR

This paper investigates the structure of certain 3-class groups in number fields and classifies related finite 3-groups to understand the length of their Hilbert 3-class field towers.

Contribution

It provides a classification of low-order bicyclic 3-groups with specific commutator quotients and links these classifications to properties of number fields' class field towers.

Findings

Classification of finite 3-groups with specific invariants

Insights into the length of Hilbert 3-class field towers for imaginary quadratic fields

Connection between group invariants and capitulation types

Abstract

For any number field K with non-elementary 3-class group Cl(3,K) = C(3^e) x C(3), e >= 2, the punctured capitulation type kappa(K) of K in its unramified cyclic cubic extensions Li, 1 <= i <= 4, is an orbit under the action of S3 x S3. By means of Artin's reciprocity law, the arithmetical invariant kappa(K) is translated to the punctured transfer kernel type kappa(G2) of the automorphism group G2 = Gal(F(3,2,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) x C(3), 2 <= e <= 6, according to the algebraic invariant kappa(G), admits conclusions concerning the length of the Hilbert 3-class field tower F(3,infty,K) of imaginary quadratic number fields K.