On the structure of a poly-$\mathbb{Z}$ group

TL;DR

This paper investigates the structure of poly-$bZ$ groups, presenting a construction method and analyzing their multiplicative and automorphism structures for groups up to rank 3.

Contribution

It introduces a systematic construction process for poly-$bZ$ groups and explores their algebraic properties up to a certain complexity.

Findings

Method for constructing poly-$bZ$ groups via specific maps

Description of multiplicative structures of groups up to $G_3$

Analysis of automorphism groups for these poly-$bZ$ groups

Abstract

In this paper we study a certain class of polycyclic groups. We outline a method for constructing a poly-$\mathbb{Z}$ group $G_n$ by describing a process for selecting maps that are used to extend $G_i$ to $G_{i+1}$ for $1 \leq i \leq n-1$ and describe the multiplicative structure and automorphism groups of some poly-$\mathbb{Z}$ groups up to $G_3$.