Commuting probabilities of finite groups.

The commuting probability of a finite group is defined to be the probability that two randomly chosen group elements commute. Let P ⊂ (0,1] be the set of commuting probabilities of all finite groups. We prove that every point of P is nearly an Egyptian fraction of bounded complexity. As a corollary, we deduce two conjectures of Keith Joseph from 1977: all limit points of P are rational, and P is well ordered by >. We also prove analogous theorems for bilinear maps of abelian groups. [ABSTRACT FROM AUTHOR]