  Let p be a fixed prime. An Abelian p-group is an Abelian group (not
necessarily finitely generated) in which every element has for its order some
power of p. The countable Abelian p-groups are classified by Ulm's theorem, and
Khisamiev characterized the Abelian p-groups with computable copies. A
computable structure A is said to be $\Delta^0_\alpha$ categorical if for any
computable structure B isomorphic to A there is a $\Delta^0_\alpha$ function
witnessing that the two are isomorphic.
  The present paper seeks to characterize $\Delta^0_\alpha$ categoricity for
Abelian p-groups, and results of this kind are given for broad classes of
Abelian p-groups and values of $\alpha$. The remaining open cases are
exhaustively described.

Logic;Group Theory