Abelian Groups - László Fuchs [Springer]

5 Pure-Complete and Quasi-Complete p-Groups 321
Corollary 4.6. The completion of a separable p-group in its large subgroup
topology is its torsion-completion.
Proof. Let A be a pure subgroup of the torsion-complete p-group B, andG the
completion of A in its large subgroup topology. The identity map of A extends to
a continuous map ı W G ! B which has to be surjective, because A is dense in B.
There is a similar map W B ! G, and the composite ı W B ! B must be the
identity.
ut
F Notes. Leptin [1] gave additional characterizations for torsion-completeness. In addition to
Theorem 3.10, he proved that if a basic subgroup B of a separable p-group A has the property that
every automorphism of B extends to an automorphism of A, then either A D B or A D B.
Crawley [1] solves one of the Kaplansky test problems for p-groups in the negative, by pointing
out that it is possible to have p-groups A; C satisfying A Š C ˚ C ˚ A,butA 6Š C ˚ A:
In the proceedings of the 1967 Montpellier conference, both Charles [3] and Cutler–Stringall
[1] pointed out the completeness of torsion-complete groups, using different approaches: the
former used the inductive topology arising from the topologies of the subgroups AŒp k , induced
by the p-adic topology, while the latter used the large subgroup topology.
Exercises
(1) Every subgroup C of a torsion-complete p-group A can be embedded in a
summand G of A such that jGj jCj @ 0
. [Hint: embed in pure subgroup.]
(2) A torsion-complete p-group that is contained in a †-cyclic p-group must be
bounded.
(3) A torsion-complete p-group A fails to have a proper pure subgroup Š A if and
only if its UK-invariants are finite.
(4) If C is a torsion-complete subgroup of a separable p-group A such that A=C is
bounded, then A is also torsion-complete. [Hint: apply Ext.Z.p 1 /; /.]
(5) (Koyama–Irwin) A separable p-group A is torsion-complete if and only if
every decomposition B D C 1 ˚ C 2 of its basic subgroup B induces a direct
decomposition A D C 1 ˚ C 2
.
(6) (Irwin–O’Neill) Suppose A D B ˚ C is a separable p-group that contains an
unbounded torsion-complete subgroup. Then either B or C has an unbounded
torsion-complete subgroup.
5 Pure-Complete and Quasi-Complete p-Groups
It does not seem unreasonable to anticipate that torsion-complete p-groups have
many more properties of interest. Some of these are shared by wider classes of
groups as well. In this section, we discuss briefly two such special properties. Both
are related to the existence of certain pure subgroups, motivated by the observations