The symmetric genus spectrum of abelian groups

Let S denote the set of positive integers that appear as the symmetric genus of a finite abelian group and let So denote the set of positive integers that appear as the strong symmetric genus of a finite abelian group. The main theorem of this paper is that S = S-0. As a result, we obtain a set of necessary and sufficient conditions for an integer g to belong to S. This also shows that S has an asymptotic density and that it is approximately 0.3284.