Abelian Groups - László Fuchs [Springer]

100 3 Direct Sums of Cyclic Groups
As a first step, we reduce the proof of the theorem to the torsion case. This is one
of the rare situations when the discussion for torsion groups cannot be delegated to
p-groups.
Lemma 6.1 (Cohen–Gluck [1]). Let F be a free group and A D B ˚ C any group
with a free summand C. Given an epimorphism W F ! A, F admits a decomposition
F D F 1 ˚ F 2 such that .F 1 / D B and F 2 Š C.
Proof. Let W A ! C denote the projection along B. ThenF D F 1 ˚ F 2 with
F 1 D Ker and F 2 Š Im D C. The inclusion B F 1 cannot be proper. ut
Next, we reduce the proof to the countable case; this is a main step, supported
primarily by a straightforward back-and-forth argument.
Lemma 6.2 (Cohen–Gluck [1]). Suppose 0 ! H ! F !A ! 0 is an exact
sequence, and both F and A are direct sums of countable groups. Then there exist
.‘matching’/ direct decompositions
F D˚< F and A D˚< A (3.14)
for some ordinal such that, for each