Abelian Groups - László Fuchs [Springer]

348 11 p-Groups with Elements of Infinite Height
Proof. The necessity of the conditions is immediate, the sole non-trivial argument
relies on the fact that only the last Ulm factor can be bounded.
For sufficiency, we first develop a sequence A . ˛/ of †-cyclic p-groups
whose UK-invariants are n !Cj for j 0,or˛ 2, and the existence has been established for shorter sequences.
The proof distinguishes several cases.
Case I. ˛ 1 exists and A˛ is a cyclic group, say, of order p`. Let C denote a
countable p-group with Ulm sequence A . < ˛ 1/; A 0˛ 1 where A0˛ 1 D
˚i p m .
By induction hypothesis, there is a countable p-group C with Ulm factors C D
hc i˚A 0 . < ˛/ where o.c / D p`C` .DefineA as the factor group of C
modulo the subgroup generated by all p`
c p`
c m .; < ˛/. Inthesame
way as in Case I, it follows that A will have the prescribed Ulm sequence.
Case V. ˛ is a limit ordinal and A˛ ¤ 0 is a †-cyclic group. This case can be
reduced to Case IV by imitating the method used in Case II.
ut
Example 1.10. Let G 0 ; G 1 be unbounded countable †-cyclic p-groups, and write G 0 D˚i