Abelian Groups - László Fuchs [Springer]

11 Summable p-Groups 401
(6) (Danchev) A reduced p-group is p !Cn -projective if and only if it has a large
subgroup that is p !Cn -projective.
11 Summable p-Groups
We continue the discussion of torsion groups with an interesting class which is
somewhat more general than direct sums of countable p-groups. The study was
initiated by Honda [3], and continued by Hill–Megibben [5] under the appropriate
name ‘summable groups.’
A p-group A is said to be summable if its socle S D AŒp is a free valuated vector
space. This means that S D˚ S where S is a subgroup satisfying p AŒp D
p C1 AŒp ˚ S . Hence every element ¤ 0 of S carries the same value .
Example 11.1. A †-cyclic p-group is summable, and so is the Prüfer group H !C1 whose socle is
isometric to ˚! S ; here, dim S D 1 for each !. Also, divisible p-groups are summable.
On the other hand, no unbounded torsion-complete p-group is summable.
We mention the following properties of summability.
(A) Countable p-groups are summable. Their socles as countably valuated vector
spaces are free Proposition 8.8 in Chapter 10.
(B) Direct sums of summable p-groups are summable. Hence direct sums of
countable p-groups are summable.
(C) Summands of summable p-groups are summable. The socle of a summable
p-group may be viewed as a direct sum of countable dimensional subspaces,
so by Theorem 8.10 in Chapter 10, every summand has a socle of the same
structure.
(D) Countable isotype subgroups of summable p-groups are summable.
The generalized Prüfer group H !1 D ˚sup n
satisfying pb D a. Thenb b 0 D .b b n / C .b n b n 1 / CC.b 1 b 0 / shows
that b b 0 must have infinitely many non-zero coordinate in the decomposition of
S, sinceb n b n 1 (of height n 1 ) is the only term that has non-zero coordinate in
S n 1
, except possibly for of b b n . However, this is impossible in a direct sum. ut
The preceding result is strengthened in (ii) of the following proposition.