Abelian Groups - László Fuchs [Springer]

1 The Ulm-Zippin Theory 345
Lemma 1.4. Assume A; C are p-groups, G is a subgroup of A, and H is a subgroup
of C such that f .A; G/ D f .C; H/ for some . There exists an a 2 AŒp of height
and proper with respect to G if and only if there is a c 2 CŒp of height and proper
with respect to H.
Proof. Observe that x 2 G./ means that x 2 AŒp is of height , and not proper
with respect to G, since—we repeat—x 2 p C1 A C G is another way of saying that
the coset x C G has a representative in p C1 A.
ut
Let A and C be reduced p-groups, and G A; H C subgroups. An
isomorphism W G ! H is called height-preserving if
h C .g/ D h A .g/ for every g 2 G:
It is important to keep in mind that heights are always computed in the large groups.
Manifestly, the restriction of an isomorphism of groups A ! C to subgroups
G ! H is always height-preserving.
Kaplansky–Mackey Lemma The next lemma is crucial in extending isomorphisms
between subgroups to larger subgroups. It is the most essential ingredient
in the proof of Ulm’s theorem below. The groups A; C in Lemma 1.5 need not be
p-groups, but the Hill invariants are computed for the chosen prime p.
Lemma 1.5 (Kaplansky–Mackey [1]). Let A and C be reduced groups, G a
subgroup of A, and H a subgroup of C such that A=G and C=H are p-groups.
Suppose W G ! H is a p-height-preserving isomorphism; furthermore, for all
, f .A; G/ f .C; H/ and
˛ W p AŒp=G./ ! p CŒp=H./ (11.1)
are monomorphisms. If a 2 A is proper with respect to G and pa 2 G, then can
be extended to a height-preserving isomorphism
for a suitable c 2 C such that
WhG; ai !hH; ci
˛.hG; ai./=G.// DhH; ci./=H./ for all :
Proof. Assume that a with h.a/ D has been chosen in its coset mod G, in addition
to being proper with respect to G, to satisfy also h.pa/ >C 1 whenever possible.
We distinguish two cases according as such a choice is possible or not.
Case I: h.pa/ >C 1. In this case pa D pb for some b 2 A of height C 1.
Then h.a b/ D , anda b is proper with respect to G, for otherwise there
would exist a g 2 G with h.a b C g/ > , leading to the contradiction
h.a C g/ >. By an earlier remark, a b … G./, thus there is a u 2 p CŒp