How do you choose a random finite abelian group?

A d x d integer matrix M gives a linear map from Z^d to Z^d. The cokernel of M is Z^d/Im(M). If det(M) is nonzero, then the cokernel is a finite abelian group of order det(M) and rank at most d.

What do these groups ‘look like’? How often are they cyclic? What can we say about their p-Sylow subgroups? What happens if instead of looking at all matrices, we only consider symmetric ones? We will discuss distributions on finite abelian p-groups, focusing on ones that come from cokernels of families of random matrices. We will explain how these distributions are related to questions from number theory about ideal class groups, elliptic curves, and sublattices of Z^d.