Abstract: I shall give an elegant combinatorial description of the orbits in a finite abelian group under the action of its full automorphism group. This description comes about by considering the degeneracy partial order on the set of orbits: the orbit of x degenerates to the orbit of y if an endomorphism maps x to y. If time permits, I will explain how I was led to these considerations while studying finite Heisenberg groups and the associated Weil representations.