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The following results are
proved:
If A and B are abelian p-groups and the length of A is greater than the length of
B, then Tor(A,B) is a direct sum of countable groups if and only if (i) B is a
direct sum of countable groups and (ii) if the β-th Ulm invariant of B is
not zero, then every pβA-high subgroup of A is a direct sum of countable
groups.
If β is an ordinal, A is a p-group, and if one p6A-high subgroup of A is a direct
sum of countable groups then every pβA-high subgroup of A is a direct sum of
countable groups.
If A and B are p-groups of cardinality ≦ℵ1 without elements of infinite height,
then Tor(A,B) is a direct sum of cyclic groups.
For each n with 1 ≦ n < ω, there is a p-group G without elements of infinite
height such that G is not itself a direct sum of cyclic groups but every subgroup of G
having cardinality ≦ℵn is a direct sum of cyclic groups.
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