  Theories of classification distinguish classes with some good structure
theorem from those for which none is possible. Some classes (dense linear
orders, for instance) are non-classifiable in general, but are classifiable
when we consider only countable members. This paper explores such a notion for
classes of computable structures by working out a sequence of examples.
  We follow recent work by Goncharov and Knight in using the degree of the
isomorphism problem for a class to distinguish classifiable classes from
non-classifiable. In this paper, we calculate the degree of the isomorphism
problem for Abelian $p$-groups of bounded Ulm length. The result is a sequence
of classes whose isomorphism problems are cofinal in the hyperarithmetical
hierarchy. In the process, new back-and-forth relations on such groups are
calculated.
