BCF-groups with elevated rank distribution

TL;DR

This paper constructs infinite families of large Schur sigma-groups with specific properties, including elevated rank distribution and particular transfer kernel types, many of which are realized as 3-class field tower groups of imaginary quadratic fields.

Contribution

It introduces new infinite families of Schur sigma-groups with elevated rank distribution and constructs their realizations as 3-class field tower groups of imaginary quadratic fields.

Findings

Constructed infinitely many large Schur sigma-groups with specific properties.

Realized these groups as 3-class field tower groups of imaginary quadratic fields for certain parameters.

Described the structure of their metabelianizations and lower central factors.

Abstract

Infinitely many large Schur sigma-groups G with logarithmic order lo(G)=19+e, non-elementary bicyclic commutator quotient G/G' ~ C(3^e) x C(3), e >= 2, elevated rank distribution rho(G)=(3,3,3;3), punctured transfer kernel type kappa(G) ~ (144;4) and soluble length sl(G)=3 are constructed. Up to e <= 4, they are realized as 3-class field tower groups Gal(F(3,infty,K)/K) of imaginary quadratic number fields K=Q(d^1/2), d<0. Their metabelianizations M=G/G'' are BCF-groups with lo(M)=8+e and bicyclic third lower central factor gamma3(M)/gamma4(M) ~ C(3) x C(3).