Abelian Groups - László Fuchs [Springer]

5 Pure-Complete and Quasi-Complete p-Groups 323
Quasi-Complete Groups Let A be a reduced p-group. It is called quasicomplete
(Head[1]) if the closure G (in the p-adic topology) of every pure
subgroup G is again pure in A. SinceG =G can be identified as the first Ulm
subgroup of A=G, the quasi-completeness of A means that, for any pure subgroup G
of A, the first Ulm subgroup .A=G/ 1 of A=G is pure in A=G, i.e. divisible.
We list a few trivialities for quasi-complete groups.
(A) Reduced quasi-complete p-groups are separable. This follows from 0 D A 1 .
(B) Torsion-complete groups are quasi-complete. This is a simple consequence of
Corollary 3.9.
(C) Closed pure subgroups in a quasi-complete group are quasi-complete.
(D) If G is a closed pure subgroup of a quasi-complete p-group A, then A=G is
likewise quasi-complete. For, if H=G is a pure subgroup of A=G,thenH is pure
in A,soŒ.A=G/=.H=G/ 1 Š .A=H/ 1 must be divisible.
Next we show that the property that we verified for torsion-complete groups in
the proof of Lemma 5.4 actually characterizes quasi-completeness.
Proposition 5.5 (Irwin–Richman–Walker [1], Koyama [1]). A reduced p-group
A is quasi-complete if and only if, for every pure subgroup H of A, and for every
subsocle S with HŒp S, there exists a pure subgroup G of A such that H G and
GŒp D S.
Proof. Let A be quasi-complete, H a pure subgroup of A such HŒp S AŒp; and
G a pure subgroup maximal with respect to the properties H G and GŒp S.To
prove GŒp D S, by way of contradiction assume that there is an x 2 S n G. Ifthe
coset x C G has finite height k in A=G, then write x C G D p k y C G with y 2 A.
Now hy C Gi is a summand of A=G, andG ˚hyi is a pure subgroup supported
by GŒp ˚hxi S, a contradiction. If x C G has infinite height in A=G, thenby
quasi-completeness it is contained in a subgroup G 0 =G Š Z.p 1 /. Clearly, G 0 is
pure in A and is supported by GŒp ˚hxi S, again a contradiction.
Conversely, let A have the stated property. First choosing S D A 1 Œp, we obtain
A 1 divisible, and hence 0. Next, for any pure subgroup H, choose G pure containing
H with socle S D H Œp. ThenG=H is pure in A=H with socle .A=H/ 1 Œp.
Consequently, G=H is divisible and equal to H =H,soA is quasi-complete. ut
Hence it follows at once:
Corollary 5.6. Quasi-complete p-groups are pure-complete.
ut
We isolate a preparatory result in the next lemma that will be needed in later
proofs.
Lemma 5.7 (Hill–Megibben [3]). Suppose A is a quasi-complete p-group, and B 0
is an unbounded summand of a basic subgroup B of A. Then B D A C B 0 .
Proof. Let B D B 0 ˚ B 00 ,and W B ! B 00 the projection map. Then .A/ is pure
in B 00 ,sinceB 00 .A/ and Coker. A/ is divisible. We want to prove that
Coker. A/ D 0 by showing that every b 2 B 00 Œp is in .A/.LetC denote a basic