Abelian Groups - László Fuchs [Springer]

358 11 p-Groups with Elements of Infinite Height
We will refer to the next result in the proof of the main Theorem 3.6.
Lemma 3.4. Let A DhXI „i be a simply presented p-group, and Y a subset of X.
The subgroup N DhYi is a nice subgroup of A.
Proof. Let a 2 A n N, and write a D r 1 x 1 CC r k x k C s 1 y 1 CC s`y` as
in (11.4) wherex i and y j are distinct elements in X n Y and in Y, respectively; here
r i ; s j are positive integers < p. We claim that b D r 1 x 1 CCr k x k 2 a CN is proper
with respect to N. Pickanyc D t 1 y 0 1 CCt my 0 m 2 N, and calculate the height of
the element b C c 2 a C N. Evidently, we have h.b C c/ min i;j fh.x i /; h.y 0 j /g
min i fh.x i /gDh.b/.
ut
From the preceding proof it is clear that an x 2 X satisfying px 2hYi is proper
with respect to hYi.
Crawley–Hales Theorem We now have all the ingredients to establish the main
structure theorem. This remarkable result does no less than characterizes a large
class of p-groups in terms of cardinal invariants.
Theorem 3.5 (Crawley–Hales [1]). Two simply presented p-groups are isomorphic
if and only if their corresponding UK-invariants are identical.
Some advantage is to be gained by proving this result in the following strengthened
form, in order to be applicable to mixed groups as well.
Theorem 3.6 (Hill [11], Walker [3]). For a prime p, let G; H be nice subgroups
of A; C, respectively, such that A=G DhXI † X i and C=H DhYI † Y i are simply
presented p-groups of length . If
(i) there is a p-height-preserving isomorphism W G ! H; and
(ii) the Hill invariants for the prime p are equal:
f .A; G/ D f .C; H/ for every ;
then extends to an isomorphism W A ! C.
If only an inequality f .A; G/ f .C; H/ holds for every , then there is an
extension which is a homomorphism.
Proof. Since certain restraints are needed to control the step-wise extension of ,
we select, for each , an arbitrary, but fixed isomorphism
˛ W p AŒp=G./ ! p CŒp=H./:
Consider the set of all pairs .G ; / subject to the following conditions:
(a) G DhG; X i (for some subset X X) is a nice subgroup of A;
(b) is a height-preserving isomorphism of G with a nice subgroup H D
hH; Y i,whereY Y;
(c) G D ;
(d) for each , ˛ induces an isomorphism G ./=G./ ! H ./=H./.