New perspectives of the power-commutator-structure: Coclass trees of CF-groups and related BCF-groups

TL;DR

This paper explores the structure of certain finite 3-groups using coclass trees, revealing a periodic bifurcation pattern that allows systematic construction of these groups via p-group generation algorithms.

Contribution

It introduces a new perspective on the coclass structure of CF- and BCF-groups, demonstrating their construction through a chain of bifurcations from a single root.

Findings

Identification of two coclass trees with specific rank distributions.

Proof of construction of all group vertices via p-group descendants.

Establishment of periodic bifurcation patterns in group structures.

Abstract

Let e>1 be an integer. Among the finite 3-groups G with bicyclic commutator quotient G/G' ~ C(3^e) * C(3), having one non-elementary component with logarithmic exponent e, there exists a unique pair of coclass trees with distinguished rank distribution rho ~ (2,2,3;3). One tree T(e)(M(e,1)) consists of CF-groups with coclass e, and the other tree T(e+1)(M(e+1,1)) consists of BCF-groups with coclass e+1. It is proved that, due to a chain of periodic bifurcations, the vertices of all pairs (T(e),T(e+1)) with e>2 can be constructed as p-descendants of the single root M(3,1) of order 729 by means of the p-group generation algorithm by Newman and O'Brien.