Abelian Groups - László Fuchs [Springer]

7 Warfield Groups 599
Exercises
(1) Every simply presented mixed group admits an H.@ 0 /-family of nice subgroups.
(2) Determine the height-matrix of the global Nunke group N .
(3) (a) Find the height-matrices of the summands in Example 6.5.
(b) Check that H.a/ C H.b/ D H.c/ C H.d/.
7 Warfield Groups
The study of the so-called Warfield groups is one of the most challenging tasks in the
theory of mixed groups. They are slightly more general than the simply presented
groups, but their theory is less transparent.
Let A be a mixed group. The natural homomorphism ˛p W A ! A .p/ maps a 2 A
to a ˝ 1 2 A .p/ D A ˝ Z .p/ . The maps ˛p combined for all primes p define an
embedding ˛ W A ! Q p A .p/. IfF is a free subgroup of A such that A=F is torsion,
then ˛.A/=˛.F/ is the torsion subgroup of Q p A .p/=˛.F/. Hence we deduce that the
cokernel of ˛ is torsion-free and divisible.
(a) If the torsion-free rank of A is 1, and a 2 A is of infinite order such that A=hai
is a totally projective p-group, then for every b 2 A of infinite order, A=hbi is
also totally projective. This follows from the linear dependence of a and b, and
from the total projectivity of an extension of a finite group hai=hmai by a totally
projective p-group A=hai (Lemma 7.12 in Chapter 11).
(b) If A is of torsion-free rank 1, and a 2 A is of infinite order, then A=hai is totally
projective whenever the torsion subgroup tA is a totally projective p-group. In
fact, tA is isomorphic to a subgroup of countable index in A=hai,andap-group
is itself totally projective if it has a totally projective subgroup of countable
index (see Lemma 7.12 in Chapter 11).
Warfield Groups By a Warfield group is meant a summand of a simply
presented group. Thus simply presented torsion groups as well as completely
decomposable torsion-free groups are Warfield (but we usually think of mixed
groups when Warfield groups are mentioned). That not all mixed Warfield groups
are simply presented was first observed by Rotman–Yen [1] (in a different context);
we furnish the following counterexample due to Warfield [7].
Example 7.1. Consider the Z .p/ -module A generated by the symbols a n .n