Schur sigma-Groups of Scholz-Taussky Type F

TL;DR

This paper classifies specific Schur sigma-groups of Scholz-Taussky type F, linking their algebraic properties to the Galois groups of certain 3-class field towers over imaginary quadratic fields.

Contribution

It determines the smallest Schur sigma-groups with given properties and provides evidence of their realization as Galois groups of class field towers over imaginary quadratic fields.

Findings

Identification of smallest Schur sigma-groups for specified parameters.

Evidence of these groups being realized as second 3-class groups of imaginary quadratic fields.

Connection established between group-theoretic structures and number field Galois groups.

Abstract

For finite metabelian 3-groups M with elementary bicyclic commutator quotient M/M' = C3*C3, coclass cc(M) in {4,6}, and transfer kernel type F, the smallest Schur sigma-groups S with second derived quotient S/S" = M are determined. Evidence is provided of arithmetical realizations of these groups by second 3-class groups M = Gal(F(3,2,K)/K), respectively 3-class field tower groups S = Gal(F(3,infty,K)/K), of imaginary quadratic number fields K=Q(sqrt{d}).