  In number theory, great efforts have been undertaken to study the
Cohen-Lenstra probability measure on the set of all finite abelian $p$-groups.
On the other hand, group theorists have studied a probability measure on the
set of all partitions induced by the probability that a randomly chosen
$n\times n$-matrix over $\FF_p$ is contained in a conjucagy class associated
with this partitions, for $n \to \infty$.
  This paper shows that both probability measures are identical. As a
consequence, a multitide of results can be transferred from each theory to the
other one. The paper contains a survey about the known methods to study the
probability measure and about the results that have been obtained so far, from
both communities.
