Abelian Groups - László Fuchs [Springer]

7 Solid Chains 561
Lemma 7.6 (Bican–Fuchs [2]). Suppose 0 ! T ! G ! A ! 0 is a balancedexact
sequence where T is a torsion group. If A is a subgroup in a torsion-free
group B such that there is a solid chain leading from A up to B, then there exists a
commutative diagram with balanced-exact rows
0 −−−−→ T −−−−→ G −−−−→ A −−−−→ 0
⏐ ⏐
∥ ↓ ↓
0 −−−−→ T −−−−→ H −−−−→ B −−−−→ 0.
Proof. The claim is equivalent to the statement that the map Bext 1 .B; T/ !
Bext 1 .A; T/ induced by the inclusion A ! B is surjective. Consider a relative
balanced-projective resolution 0 ! K ! A ˚ C ! B ! 0; where C is completely
decomposable. The existence of a solid chain from A to B implies that K is a B 2 -
group. In the induced exact sequence
Bext 1 .B; T/ ! Bext 1 .A ˚ C; T/ D Bext 1 .A; T/ ! Bext 1 .K; T/
(for any torsion group T) the last term vanishes in view of Theorem 5.3. The
balancedness of the bottom sequence follows straightforwardly (it is easier if we
add ˚C to both G and A). Hence the claim follows.
ut
A sufficient criterion for a pure subgroup of a B 1 -group to be again such a group
can be given immediately.
Corollary 7.7. Let B be a B 1 -group, and A a pure subgroup of B. If there is a solid
chain from A to B, then A is likewise a B 1 -group.
Proof. Suppose there is a solid chain from A to B. Lemma 7.6 implies that any
balanced-exact sequence 0 ! T ! G ! A ! 0 can be embedded in a
commutative diagram like the one in Lemma 7.6 with balanced-exact bottom row.
Now, if B is a B 1 -group, then the bottom row splits, and hence so does the top row.
Consequently, A is a B 1 -group.
ut
Absolutely Solid Groups As a brief excursion, we consider the following
concept. Call a torsion-free group absolutely solid if it is solid in every torsionfree
group in which it is contained as a pure subgroup. The proofs of a few basic
properties of this notion are delegated to the exercises. Clearly, all countable torsionfree
groups are absolutely solid. We are interested in more exciting examples: in
B 2 -groups.
Theorem 7.8 (Rangaswamy [7], Bican–Fuchs [2]). B 2 -groups are absolutely
solid.
Proof. Let B be a B 2 -group, and G a torsion-free group containing B as a pure
subgroup. For the proof that B is solid in G, we may assume without loss of
generality that G=B is of rank 1, so there is a countable pure subgroup S of G such