Symmetric harmoniousness of odd-order groups

TL;DR

This paper proves that all odd-order groups have a symmetric harmonious permutation property, where a permutation of elements results in a permutation of their consecutive products with a specific inverse symmetry.

Contribution

It establishes that every odd-order group is symmetric harmonious, introducing a new structural property and applying it to find new R*-sequenceable groups.

Findings

Every odd-order group is symmetric harmonious.

Constructs specific permutations with inverse symmetry.

Provides new examples of R*-sequenceable groups.

Abstract

We prove that every odd-order group is symmetric harmonious: there exists a permutation $g_0,g_1,\ldots, g_{\ell-1}$ of elements of $G$ such that the consecutive products $g_0g_1,g_1g_2,\ldots, g_{\ell-1}g_0$ also form a permutation of elements of $G$ and $g_{\ell-i}=g_i^{-1}$ for all $1\leq i \leq \ell-1$. We apply this result to obtain new examples of R*-sequenceable groups.