**Borel Isomorphism Relations of****Countable Reduced Abelian p-groups**

** Abstract**

This paper covers two major results. The first one states that any algorithm that
can determine whether two arbitrarily given countable reduced 2-groups are isomorphic
is as complicated as the process of computing their Ulm invariants, namely, it has
to go through a transfinite iteration of unbounded countable length. In the language
of descriptive set theory, this can be stated precisely as "the set {(G1, G2): G1,
G2 are isomorphic reduced 2-groups} is relatively D11 to the set { (G1, G2): G1,
G2 are reduced 2-groups} but is not relatively Borel".

The second theorem denies the possibility of finding a Borel process to construct
isomorphisms between any two given isomorphic countable reduced 2-groups.

**Introduction**

H. Ulm proved in 1933 that the structure of a reduced countable Abelian p-group is
completely determined up to isomorphism by a sequence of invariants called the Ulm
invariants. The original methods he invented for the computation of these invariants
and the construction of isomorphisms require a transfinite iteration whose length,
depending on the group, can be any arbitrarily large countable ordinal. One may therefore
ask whether there is an alternative algorithm that requires only transfinite recursions
with bounded countable lengths. More precisely, if each countable p-group is coded
by an element in the Cantor space, can we find a Borel partial function from the Cantor
space into itself that would compute the rank and Ulm invariants of any reduced countable
Abelian p-group? Can we find a Borel procedure that can construct an isomorphism between
any two given isomorphic reduced countable Abelian p-groups? In this paper, we prove
that the answers to both questions are unfortunately negative.

We start our investigation with the search for a minimal substructure of a p-group
that generates the whole group and also retains the characteristics of the group.
Unless otherwise stated, all groups in this paper are assumed to be Abelian and the
group operation is addition.

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[paper 1]