Abelian Groups - László Fuchs [Springer]

8 Butler Groups of Uncountable Ranks 563
(7) A solid subgroup A of an absolutely solid group G is likewise absolutely solid.
[Hint: push-out for A ! G and 0 ! A ! H ! H=A ! 0.]
(8) Any extension of an absolutely solid group by a countable group is again
absolutely solid.
8 Butler Groups of Uncountable Ranks
We are at last on the final route toward a necessary and sufficient condition that
makes a B 1 -group into a B 2 -group. Armed with the arsenal of solid subgroups, the
only missing ingredient is the following key lemma which is a recast of Lemma 4.4
in Chapter 9.
Lemma 8.1 (Dugas–Hill–Rangaswamy [1]). Let be an uncountable regular
cardinal, and 0 D C 0 < C 1 < < C < ::: . < / a smooth chain of
pure subgroups of the torsion-free group C such that
(a) S < C D C;
(b) jC j