The following results are
proved:
If A and B are abelian pgroups 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 βth Ulm invariant of B is
not zero, then every p^{β}Ahigh subgroup of A is a direct sum of countable
groups.
If β is an ordinal, A is a pgroup, and if one p^{6}Ahigh subgroup of A is a direct
sum of countable groups then every p^{β}Ahigh subgroup of A is a direct sum of
countable groups.
If A and B are pgroups of cardinality ≦ℵ_{1} 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 pgroup 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 ≦ℵ_{n} is a direct sum of cyclic groups.
