Abelian Groups - László Fuchs [Springer]

562 14 Butler Groups
that G D B C S. By enlarging S if necessary, we may assume that S \ B is decent in
B; see Theorem 5.4. The isomorphism G=S Š B=.S \ B/ convinces us that G=S is a
B 2 -group, and as such it admits a solid chain. This chain lifts to a solid chain from S
to G,asS is countable, so solid in G. Thus, G admits a solid chain. By Theorem 7.3,
the kernel H in a balanced-projective resolution 0 ! H ! A ! B ! 0
of B (with completely decomposable A) isaB 2 -group; owing to Sect. 5(H), it is
also a TEP-subgroup in A. After choosing a relative balanced-projective resolution
0 ! K ! B ˚ C ! G ! 0 of B in G, we form the commutative diagram
H
⏐
↓
H
⏐
↓
0 −−−−→ L −−−−→ A ⊕ C −−−−→ G −−−−→ 0
⏐
⏐
↓
↓
∥
0 −−−−→ K −−−−→ B ⊕ C −−−−→ G −−−−→ 0,
where the vertical arrows are monic in the first set, and epic in the second set. Since
the bottom row and the middle column are balanced-exact, the middle row is also
balanced-exact, and hence it is a balanced-projective resolution of G. AsG admits
solid chains, L must be a B 2 -group. Therefore, K is a B 1 -group as the quotient of a
B 2 -group by a balanced TEP-subgroup (cp. Sect. 5(H)). Moreover, it is a B 2 -group,
as H is a B 2 -group (see Exercise 6). It remains to refer to Theorem 7.2 to conclude
that B is solid in G.
ut
F Notes. Theorem 7.8 was proved by Bican–Fuchs [2] for B 1 -groups in L, and the general
version by Rangaswamy [7].
Fuchs–Rangaswamy [3] show that the union of a countable chain of pure B 2 -subgroups is again
a B 2 -group. The same holds for smooth chains of length ! 1 provided the subgroups in the chain
are decent and of cardinalities @ 1 . Bican–Rangaswamy [1] extend the results to longer chains
under appropriate conditions on the factor groups. See also Bican–Rangaswamy–Vinsonhaler [1].
Exercises
(1) A torsion-free group of cardinality @ 1 admits a solid chain.
(2) The direct sum of groups with solid chains also has a solid chain.
(3) Furnish the details of proof in Lemma 7.5.
(4) If a torsion-free group has a solid chain, then it also has an H.@ 0 /-family of
solid subgroups.
(5) If the torsion-free group G contains a pure B 2 -subgroup B of infinite index ,
then it contains a pure subgroup S of cardinality such that G=S is a B 2 -group.
[Hint: Theorem 7.2.]
(6) Show that direct sums and summands of absolutely solid groups are likewise
absolutely solid.