Abelian Groups - László Fuchs [Springer]

1 Splitting Mixed Groups 577
so A may be thought of as the set of all .t; g/ 2 T ˚G with g D tCpT.ThenpA pT ˚pG A,
so A is quasi-splitting. If A were splitting, then there would exist a map W G ! T such that
g D g C pT for all g 2 G. ButG is cotorsion and reduced torsion, so it must be of bounded
order. Therefore, the image of G in T=pT would be finite, while G is infinite. Thus A cannot
split.
Next we prove a lemma in a more general setting than needed for the proof of
Theorem 1.10. LetC be a group and n >0an integer. Multiplication by n in C
factors as C !nC !C where W c 7! nc and is the inclusion map. Given the
bottom exact sequence, we build a commutative diagram with exact rows:
νμ: 0−−−−−→ A
∥
ν :0
α
−−−−−→ ′
B ′ β ′
⏐
↓
−−−−−→ C −−−−−→ 0
⏐
↓ μ
α
β
−−−−−→ A −−−−−→ nB + A −−−−−→ nC −−−−−→ 0
⏐
⏐
∥
↓
↓ν
:0 −−−−−→ A
α
−−−−−→
B
β
−−−−−→ C −−−−−→ 0.
Lemma 1.9 (C. Walker [1]). If e splits, then e represents an element of
Ext.C; A/Œn. IfCŒn D 0, then the converse is also true.
Proof. Observe that e D ne. Now,ife splits, then so does ne, and therefore e
belongs to Ext.C; A/Œn. Conversely,e 2 Ext.C; A/Œn means that e is contained in
the kernel of the endomorphism of Ext.C; A/ induced by multiplication by n in C.
Therefore, if CŒn D 0,theninviewofLemma5.1(iii) in Chapter 9 this is equivalent
to the splitting of e.
ut
The following theorem identifies the torsion subgroup of Ext.G; T/ as the set of
quasi-splitting extensions.
Theorem 1.10 (C. Walker [1]). A mixed group A is quasi-splitting if and only if
the exact sequence
0 ! T ! A ! G ! 0 (15.2)
.where T D t.A/ and G is torsion-free/ represents an element of finite order in
Ext.G; T/.
Proof. If this exact sequence represents an element of finite order in Ext.G; T/,then
the second part of the preceding lemma implies that the exact sequence
0 ! T ! nA C T ! .nA C T/=T ! 0 (15.3)
is splitting. Since nA nA C T A, this means that A is quasi-splitting.