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# Open-problems

Countable Abelian recursive reduced p-groups:

1. Does a reduced Abelian p-group (of finite rank) always have a recursive generating tree?

2. Is the following set
{(G, H ) : H is a maximal reduced subgroup of G }
Borel relative to the set
{(G, H ) ; H is a subgroup of G} ?

3. We know that if G has rank £ !, then G is a direct sum of cyclic groups, say,
Å iiÎ!Hi . Is there a recursive isomorphism G  ! Å iiÎ!Hi ?

4. Let
A = {(G, H, f ) : G is isomorphic to H and f is a monomorphism from a finite
subgroup of G to H }
Is B = {(G, H, f ) : f can be extended to an isomorphism from G to H } Borel
relative to A?

5. If G Å G is recursively isomorphic to H Å H, is it necessary that G is recursively
isomorphic to H?

6. Is there a Borel bijective map from the set of countable reduced 2-groups to the set of
countable reduced 3-groups that preserves embedding relation?

Uncountable reduced Borel p-groups

1. When does such a Borel group have a Borel generating tree (or a generating tree at all)?

2. What countable sequence of cardinals can be the Ulm invariant sequence of such a Borel group?

3. If two such Borel groups have the same Ulm invariants, must they be isomorphic?

4. If G Å G is Borel isomorphic to H Å H, is it necessary that G be isomorphic to H ? Can the isomorphism be Borel?

5. How complicated is the Poset of rank 1 Borel 2-groups (ordered  by rank preserving
monomorphisms)? What is the maximum size of an antichain?

6. Is there any universal rank 1 Borel 2-group?

7. Does every Borel subgroup of a rank 1 Borel 2-group with a Borel basis have a Borel basis? Or equivalently, are all minimal rank 1 uncountable Borel 2-groups Borel isomorphic?

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