Holmstrom Cohen-Lenstra heuristics

arXiv:0912.4977 The Global Cohen-Lenstra Heuristic from arXiv Front: math.NT by Johannes Lengler The Cohen-Lenstra heuristic is a universal principle that assigns to each group a probability that tells how often this group should occur “in nature”. The most important, but not the only, applications are sequences of class groups, which behave like random sequences of groups with respect to the so-called Cohen-Lenstra probability measure

So far, it was only possible to define this probability measure for finite abelian $p$-groups. We prove that it is also possible to define an analogous probability measure on the set of \emph{all} finite abelian groups when restricting to the $\Sigma$-algebra on the set of all finite abelian groups that is generated by uniform properties, thereby solving a problem that was open since 1984.

nLab page on Cohen-Lenstra heuristics