Abelian Groups - László Fuchs [Springer]

364 11 p-Groups with Elements of Infinite Height
(a) is a height-preserving isomorphism of G with a subgroup H of C;
(b) G D ,and is the restriction of 0 to G if < 0 ;
(c) for each , ˛ induces an isomorphism G ./=G./ ! H ./=H./.
Evidently, there is a maximal pair .G ; / in this set. (c) ensures that f .A; G /
f .C; H / for each . If < , then owing to (a), we are in the situation of
Lemma 1.5,so W G ! H can be extended to a height-preserving isomorphism
of G C1 with a subgroup of C such that (c) is not violated. This is a contradiction to
the choice of , so D ,andN D A, proving the assertion.
ut
We derive the following immediate corollary.
Corollary 4.6. The preceding lemma continues to hold if condition (ii) is dropped,
is assumed not to decrease heights, and the extension is required to be a nonheight-decreasing
homomorphism only.
Proof. We apply Lemma 4.5 to the groups A and A ˚ C with the nice subgroups
G and G ˚ H. The UK-invariants satisfy condition (ii), and the homomorphism
W G ! G ˚ H is height-preserving. Hence Lemma 4.5 guarantees that there is a
height-preserving monomorphism A ! A ˚ C. The projection to C is a map as
desired.
ut
F Notes. Groups with nice systems were studied by Hill [7]; he used the terminology “Axiom-
3 groups” (referring to their H.@ 0 /-family property). He proved the generalized Ulm theorem for
these groups—a major result. That it suffices to assume a priori the existence of nice composition
chains was observed in [IAG]. It turns out that this is not a weaker hypothesis after all; actually, we
do not need a deep result like Theorem 5.9 to show that a group with a nice composition chain has
a nice system: Hill has an ingenious direct proof in [16], based on his Theorem 5.5 in Chapter 1.
Exercises
(1) Let A be a p-group, and n >0an integer. p n A has a nice system if and only if A
has one.
(2) An unbounded torsion-complete p-group has no nice composition chain. [Hint:
it is enough to prove if basic subgroup is countable; in a nice composition chain
from a countable index on the cokernels are not nice subgroups.]
(3) The torsion subgroup of a direct product of infinitely many unbounded reduced
p-groups with nice composition chains never admits such a chain. [Hint: hint in
preceding exercise.]
(4) Let C be a p-group. A reduced p-group A with a nice composition chain is
isomorphic to an isotype subgroup (see next section) of C exactly if f .A/
f .C/ for all :
5 Isotypeness, Balancedness, and Balanced-Projectivity
We shall now turn our attention to subgroups that are of immense help in developing
the theory of totally projective p-groups. We start with an important device that