  The goal of this paper is to prove theorems that elucidate the
Cohen-Lenstra-Martinet conjectures for the distributions of class groups of
number fields, and further the understanding of their implications. We start by
giving a simpler statement of the conjectures. We show that the probabilities
that arise are inversely proportional the to number of automorphisms of
structures slightly larger than the class groups. We find the moments of the
Cohen-Lenstra-Martinet distributions and prove that the distributions are
determined by their moments. In order to apply these conjectures to class
groups of non-Galois fields, we prove a new theorem on the capitulation kernel
(of ideal classes that become trivial in a larger field) to relate the class
groups of non-Galois fields to the class groups of Galois fields. We then
construct an integral model of the Hecke algebra of a finite group, show that
it acts naturally on class groups of non-Galois fields, and prove that the
Cohen-Lenstra-Martinet conjectures predict a distribution for class groups of
non-Galois fields that involves the inverse of the number of automorphisms of
the class group as a Hecke-module.
