Co-periodicity isomorphisms between forests of finite p-groups

TL;DR

This paper explores co-periodicity isomorphisms in forests of finite p-groups, revealing how algebraic invariants transform and demonstrating that certain coclass subtrees have finite informational content, with implications for understanding the structure of these groups.

Contribution

It introduces co-periodicity isomorphisms between coclass forests, reducing the complexity of the entire metabelian skeleton to finite data, a novel insight in the study of finite p-groups.

Findings

Behavior of invariants under isomorphisms described with simple laws

Finite information content in coclass subtrees with metabelian mainline

Evidence of co-periodicity reducing data complexity in forests

Abstract

Based on a general theory of descendant trees of finite p-groups and the virtual periodicity isomorphisms between the branches of a coclass subtree, the behavior of algebraic invariants of the tree vertices and their automorphism groups under these isomorphisms is described with simple transformation laws. For the tree of finite 3-groups with elementary bicyclic commutator quotient, the information content of each coclass subtree with metabelian mainline is shown to be finite. As a striking novelty in this paper, evidence is provided of co-periodicity isomorphisms between coclass forests which reduce the information content of the entire metabelian skeleton and a significant part of non-metabelian vertices to a finite amount of data.