Abelian Groups - László Fuchs [Springer]

532 14 Butler Groups
Theorem 1.7 (Butler [1]). A finite rank torsion-free group A is a Butler group if
and only if it satisfies the following conditions:
(a) Type.A/ is finite;
(b) for each critical type t,A ? .t/ has finite index in its purification A ? .t/ I
(c) for each critical type t, there is a direct decomposition
A.t/ D A t ˚ A ? .t/ (14.1)
where A t is a homogeneous completely decomposable group of type t.
Proof. First suppose A is a Butler group. (a) has been observed above. As a pure
subgroup of A; A.t/ is a Butler group, and hence A t is likewise Butler. Since a
homogeneous Butler group must be completely decomposable (see Corollary 1.5)(c)
follows in view of Lemma 3.6 in Chapter 12. Considering that A ? .t/ is a Butler
group, we have A ? .t/ D A 1 CCA k CA ? .t/ for certain rank one pure subgroups A i
of A. If none of A i can be dropped, then the A i are all of type t, and their intersections
with A ? .t/ are likewise of type t. Hence each A i \ A ? .t/ is of finite index in A i ,and
therefore A ? .t/ is of finite index in A ? .t/ .
Conversely, let A satisfy (a)–(c). (b) implies that A ? .t/ D F t C A ? . t/ for a
finitely generated free subgroup F t . Hence the subgroup A 0 D P t A t C P t F t is by
(a) a finite sum of rank one groups, so it is a Butler group. If AnA 0 is not empty, then
choose an element a 2 A n A 0 whose type s is maximal among the types of elements
in A n A 0 .Then
a 2 A.s/ D A s ˚ A ? .s/ D A s ˚ .F s C A ? .s//:
By the maximal choice of s,wehaveA ? .s/ A 0 ,soalsoA.s/ A 0 , in contradiction
to the choice of a.
ut
Bican’s Theorem The following remarkable characterization of Butler groups
relies on the partition of primes. If is a set of primes, then the localization Z of
Z at is the group of all rational numbers whose denominators are relatively prime
to every prime in . It should be observed that an immediate consequence of the
definition is that localizations of Butler groups are likewise Butler groups.
Theorem 1.8 (Bican [3]). A finite rank torsion-free group B is a Butler group if
and only if there is a partition
… D … 1 [[… k
of the set … of prime numbers such that for each `.`D 1;:::;k/, the tensor product
B ˝ Z …` .localization of B at …`/ is a completely decomposable group with totally
ordered typeset.