R-harmonious groups

TL;DR

This paper introduces R-harmonious groups, a class characterized by a special permutation property of their non-identity elements, and proves that all odd-order groups not divisible by 3 are R-harmonious.

Contribution

It defines R-harmonious groups and establishes a significant class of such groups, expanding understanding of their structural properties.

Findings

Every group of odd order not divisible by 3 is R-harmonious.

The study employs cyclic and split extensions to analyze R-harmonious groups.

Abstract

A group is R-harmonious if there exists a permutation $g_1,g_2,\ldots, g_{n-1}$ of the non-identity elements of $G$ such that the consecutive products $g_1g_2$, $g_2g_3$, $\ldots, g_{n-1}g_1$ also form a permutation of the non-identity elements, where $n=|G|$. We investigate R-harmonious groups via cyclic and split extensions. Among our results, we prove that every group of odd-order not divisible by 3 is R-harmonious.