Abstract
In this paper we introduce a method of sequencing the elements of a finite group that gives rise to a complete mapping of the group. Our definition was motivated by the concept of a harmonious graph invented by Graham and Sloane. Our concept has several connections to graph theory and as an application we complete the characterization of elegant cycles begun by Chang, Hsu, and Rogers. Our definitions are also variations of the notion of an
R-sequenceable group first introduced by Ringel in his solution of the map coloring problem for all compact 2-dimensional manifolds except the sphere and expanded upon by Friedlander, Gordon, and Miller.