Differences of Bijections

When is a function from an abelian group to itself expressible as a difference of two bijections? Answering this question for finite cyclic groups solves a problem about juggling. A theorem of Marshall Hall settles the question for finite abelian groups, and a forgotten theorem of László Fuchs settles the question for infinite abelian groups. After explicating these theorems, we extend the problem by examining expressibility as a difference of injections or surjections. We also extend the question beyond the realm of group theory, where the expressibility of a function translates to questions about partial transversals in Latin squares.