A Classification of Free and Free-Like Nilpotent Groups

2024/01/08 by Adam M. Moubarak, Moubarak, Adam
Computer Science · Engineering · Mathematics · #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Metric Geometry (math.MG) #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2401.04277

openalex publication_date 2024/01/08 · openalex created_date 2024/01/13 · openalex updated_date 2026/07/28

Abstract

Suppose G is a T-group (finitely generated torsion-free nilpotent) with centralizers outside of the derived subgroup being abelian of rank equal to rank(Z₁)+1. This includes the class of free nilpotent groups N_r,c of a given rank r and class c. It is shown that the upper and lower central series coincide in such groups and from this that they are metabelian. We then prove that all such groups arise as semidirect products of free abelian groups with respect to representation [G,G]→ UT(n,ℤ) by automorphisms constructed from integer powers of elements in defining relations we call integral weights of G.

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