Abelian Groups - László Fuchs [Springer]

574 15 Mixed Groups
Non-Splitting Mixed Groups We begin with examples of non-splitting mixed
groups. Our first example provides a stunningly simple method of manufacturing
non-splitting mixed groups, while the other two are key examples exhibiting nonsplitting
mixed groups of torsion-free rank 1.
Example 1.1. Let T be an unbounded reduced torsion group. Then Ext.Q=Z; T/ D T is a nonsplitting
reduced mixed group (the cotorsion hull of T) with T as torsion subgroup.
Example 1.2. Let p 1 ;:::;p n ;::: denote different primes, and define
T 1 D˚1nD1 ha ni with o.a n / D p n :
Thus T 1 is the torsion subgroup of the direct product B D Q 1
nD1 ha ni, a mixed group. Let b 0 D
.a 1 ;:::;a n ;:::/2 B.Fori ¤ n, the equation p n x D a i is uniquely solvable in ha i i, thus B contains
a unique element b n such that p n b n D .a 1 ;:::;a n 1 ;0;a nC1 ;:::/D b 0 a n : T 1 is the torsion part
of the group A 1 DhT 1 ; b 1 ;:::;b n ;:::i such that A 1 =T 1 is torsion-free of rank 1. We claim A 1 is not
splitting. Otherwise, we had A 1 D T 1 ˚ G for some G < A 1 ,sob n D t n C g n with t n 2 T 1 ; g n 2 G
for each n T. But this is impossible. ut
We stop for a moment to interpret this result as saying in effect that the cotorsion
hull of a reduced torsion group T is a largest mixed group with torsion part T that has
no non-trivial direct decomposition where T is contained in one of the summands.
Nunke Groups By making use of the generalized Prüfer p-groups H (which
will now be denoted as H .p/ to indicate dependence on p), we introduce nonsplitting
mixed groups of torsion-free rank 1, following Nunke [3]. The Nunke
group N .p/ of length is obtained as follows.