R-harmonious groups

A group is R-harmonious if there exists a permutation $g_1,g_2,\dots ,g_{n-1}$ of the non-identity elements of $G$ such that the consecutive products $g_1{g}_2$, $g_2{g}_3$, $\dots ,g_{n-1}{g}_1$ also form a permutation of the non-identity elements, where $n=\mathrm{\mid }G\mathrm{\mid }$. 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.