  A notion of degeneration of elements in groups is introduced. It is used to
parametrize the orbits in a finite abelian group under its full automorphism
group by a finite distributive lattice. A pictorial description of this lattice
leads to an intuitive self-contained exposition of some of the basic facts
concerning these orbits, including their enumeration. Given a partition
$\lambda$, the lattice parametrizing orbits in a finite abelian p-group of type
$\lambda$ is found to be independent of p. The order of the orbit corresponding
to each parameter, which turns out to be a polynomial in p, is calculated. The
description of orbits is extended to subquotients by certain characteristic
subgroups. Each such characteristic subquotient is shown to have a unique
maximal orbit.
