Periodic Schur sigma-groups of non-elementary bicyclic type

TL;DR

This paper constructs infinitely many large Schur sigma-groups with specific non-elementary bicyclic quotients, revealing their structure, origins from descendant trees, and realization as 3-class field tower groups of imaginary quadratic fields.

Contribution

It introduces a method to generate infinite families of Schur sigma-groups with non-elementary bicyclic quotients using descendant trees, and connects these groups to 3-class field towers of quadratic fields.

Findings

Infinite families of Schur sigma-groups constructed

Groups realized as 3-class field tower groups of quadratic fields

Explicit descriptions of group structures and ranks

Abstract

Infinitely many large Schur sigma-groups G with non-elementary bicyclic commutator quotient G/G' = C(3^e) x C(3), e >= 2, are constructed as periodic sequences of vertices in descendant trees of finite 3-groups. A single root gives rise to pairs of metabelian groups G with logarithmic order lo(G) = 4+e for e >= 3. Three roots are ancestors of pairs of non-metabelian groups G with moderate rank distribution rho(G) = (2,2,3;3) and lo(G) = 7+e for e >= 5. Twentyseven roots produce sextets of non-metabelian groups G with elevated rank distribution rho(G) = (3,3,3;3) and lo(G) = 19+e for e >= 9. The soluble length of non-metabelian groups is always sl(G) = 3. The groups can be realized as 3-class field tower groups Gal(F(3,infty,K)/K) of imaginary quadratic number fields K = Q(d^1/2) with fundamental discriminants d < 0.