The structure of tiles in $\mathbb{Z}_{p^n}\times \mathbb{Z}_q$ and $\mathbb{Z}_{p^n}\times \mathbb{Z}_p$

TL;DR

This paper characterizes tiles in specific finite abelian groups using geometric methods involving p-homogeneous trees, offering a visual criterion for understanding their structure.

Contribution

It introduces a geometric characterization of tiles in certain finite abelian groups through the concept of p-homogeneous trees, providing new visual insights.

Findings

Geometric characterization of tiles in  groups

Use of p-homogeneous trees for visualization

Provides criteria for tile structure

Abstract

In this paper, we provide a geometric characterization of tiles in the finite abelian groups \( \mathbb{Z}_{p^n} \times \mathbb{Z}_q \) and \( \mathbb{Z}_{p^n} \times \mathbb{Z}_p \) using the concept of a \( p \)-homogeneous tree, which provides an intuitively visualizable criterion.