# Permutations with a distinct divisor property

## TL;DR

This paper characterizes abelian groups with the distinct divisor property (DDP) and constructs non-abelian DDP groups, revealing new structural insights into permutation properties related to group elements.

## Contribution

It provides a complete characterization of abelian DDP groups and introduces a method to construct non-abelian DDP groups using group extensions.

## Key findings

- Abelian DDP groups have a unique element of order 2.
- Existence of infinitely many non-abelian DDP groups.
- Construction method via group extensions.

## Abstract

A finite group of order $n$ is said to have the distinct divisor property (DDP) if there exists a permutation $g_1,\ldots, g_n$ of its elements such that $g_i^{-1}g_{i+1} \neq g_j^{-1}g_{j+1}$ for all $1\leq i<j<n$. We show that an abelian group is DDP if and only if it has a unique element of order 2. We also describe a construction of DDP groups via group extensions by abelian groups and show that there exist infinitely many non abelian DDP groups.