R-harmonious groups

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.