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**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 _{i}_{iÎ!}*H _{i}* . Is there a recursive isomorphism G ! Å

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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