Abelian Groups - László Fuchs [Springer]

2 Prebalanced and Decent Subgroups 537
2 Prebalanced and Decent Subgroups
The further discussion of Butler groups is greatly enhanced by the introduction of
new concepts. We examine briefly three salient properties which prove relevant to
Butler groups both in the finite and in the infinite rank cases: prebalanced and decent
subgroups, as well as subgroups with the torsion extension property (TEP).
Balanced subgroups play a decisive role in the theory of torsion-free groups,
but the homological machinery they provide is unsatisfactory for the theory of
Butler groups. The reason is that finite rank Butler groups need not contain
any non-obvious balanced subgroup. However, there is a natural generalization
of balancedness (due to Richman [6]) that can furnish us with a sought-after
homological machinery. We also define the companion notion of decent subgroup
(Albrecht–Hill [1]).
Prebalanced Subgroups Let A be a subgroup of a (not necessarily torsion-free)
group G with torsion-free quotient G=A. A is said to be prebalanced in G if the
following condition is satisfied: for each rank 1 pure subgroup H=A of G=A, there
is a finite rank Butler subgroup B of G such that H D A C B.
Furthermore, A is called decent in G if the same conclusion holds whenever
H=A is any finite rank pure subgroup of G=A. Evidently, decent subgroups are
prebalanced, but balanced subgroups need not be decent as is shown by (c) in the
following example.
Example 2.1.
(a) Every pure subgroup in a finite rank Butler group B is decent, and hence prebalanced: if
C < C 0 are pure subgroups of B with finite rk C 0 =C, thenC 0 D C C C 0 with C 0 a finite rank
Butler group.
(b) The indecomposable group A in Example 4.3 in Chapter 12 has no proper balanced subgroups
¤ 0, but all of its proper pure subgroups are prebalanced and decent.
(c) Let C be any finite rank torsion-free group that is not Butler; e.g. of Pontryagin type
(Lemma 4.6 in Chapter 12). If 0 ! A ! G ! C ! 0 is a balanced-projective resolution of
C,thenA is (pre)balanced, but not decent in G.
It might be helpful if we insert the following remarks before proceeding with the
discussion.
(A) A pure-exact sequence 0 ! A ! G ! J ! 0 of torsion-free groups
with rk J D 1 is prebalanced exactly if there are a finite rank completely
decomposable group X D X 1 ˚ ˚ X n .rk X i D 1/ and an epimorphism
W X ! J that lifts to a homomorphism W X ! G. This follows at once from
the definition.
(B) A pure subgroup A of the torsion-free group G is prebalanced if and only if, for
each g 2 G there is a finite subset fa 1 ;:::;a n gA such that
hA; gi D A Chg C a 1 i CChg C a n i :
That is, G=A .g C A/ is of the same type as G .g C a 1 / __ G .g C a n /.As
G=A .g C A/ is the supremum of all G .g C a/ with a 2 A, we can increase