Abelian Groups - László Fuchs [Springer]

6 Solid Subgroups 551
understand the fine distinction between B 1 -andB 2 -groups of arbitrary cardinalities. It turns out
that the question whether or not the classes of B 1 -andB 2 -groups are identical has no definite
answer: it is undecidable in ZFC. We have shown above that B 2 -groups are always B 1 -groups,
and will provide a necessary and sufficient condition under which B 1 -groups are B 2 -groups (see
Theorem 8.2 below). Though it is of no practical use, still it will be discussed, because it is
instrumental in understanding why CH is a sufficient condition (Theorem 9.9). However, the proof
that it is undecidable in ZFC whether or not all B 1 -groups are B 2 -groups would require a significant
amount of preparations, which is beyond the scope of this book.
The equivalence problem of B 1 -andB 2 -groups is intrinsically tied to another, almost equally
important question as to when Bext 2 .G; T/ vanishes for all torsion-free G and all torsion T. To
answer this question, we will give a couple of equivalent conditions in Proposition 8.8. A third
relevant question that is concerned with the subgroup problem will be addressed briefly: for which
subgroups is the B 2 -property hereditary?
F Notes. It is still not known if Bext 1 .B; T/ D 0 for all direct sums T of finite cyclic groups
implies that B is a B 1 -group (problem raised by Arnold). To answer the question as to whether
or not Bext 1 .B; T/ D 0 for all countable T entails that B is a B 1 -group, we point out that this is
undecidable in ZFC (Dugas–Rangaswamy [1]). Indeed, choose B homogeneous of type Z, then
Bext 1 .B; T/ D Ext 1 .B; T/, soB is a Baer group, and for such groups the problem is undecidable
in ZFC, see Sect. 2 in Chapter 15.
Exercises
(1) In a B 2 -group, every countable subgroup is contained in a countable TEPsubgroup.
(2) Let 0 ! A ! B ! C ! 0 be a prebalanced-exact sequence of torsion-free
groups. A is a B 1 -group if both B and C are B 1 -groups. [Hint: (H).]
(3) (a) Prebalanced extension of a finitely Butler group by another such group is
finitely Butler.
(b) Let 0 D C 0 < C 1 < < C n < ::: .n