Abelian Groups - László Fuchs [Springer]

6 Totally Projective p-Groups 371
(7) A balanced subgroup of a countable p-group is a summand. [Hint: balancedprojectivity.]
(8) Prove Lemma 5.7 by induction on . [Hint: it is trivial for finite , and easy
for limit ; for successor ordinals use y 2 p H Cn 1 with py D z and b 2 A
with pb D a; h.b/ 1, and extend to hyi !hbi.]
(9) (Warfield) A short exact sequence of p-groups relative to which the generalized
Prüfer groups have the projective property is balanced-exact.
(10) (E. Walker) In the next section we show that the balanced-projective p-groups
are exactly the simply presented ones. Using this, verify the following method
of getting a balanced-projective resolution 0 ! K ! G ! A ! 0 of a p-
group A. G is generated by x a , one generator for each a 2 A, subject to the
defining relations x 0 D 0, andpx a D x b if and only if pa D b holds in A. The
correspondence x a 7! a induces an epimorphism W G ! A, andK D Ker
is balanced in G.
(11) If G is a balanced subgroup of A, then Tor.G; X/ is balanced in Tor.A; X/ for
every group X. [Hint: Lemma 4.2 and Theorem 3.1 in Chapter 8.]
6 Totally Projective p-Groups
A most important type of p-group was discovered by Nunke [3] via homological
considerations. Later, it turned out that the class of p-groups with nice systems
introduced by Hill coincides with this class.
Total Projectivity Recall that †-cyclic groups can be characterized as groups A
satisfying Pext.A; C/ D 0 for all groups C. If we restrict our attention to p-groups,
and take into consideration that Pext is just the first Ulm subgroup of Ext, then we
can claim that a p-group A is †-cyclic exactly if p ! Ext.A; C/ D 0 for all groups C.
Nunke defined classes of p-groups A using this observation by replacing ! by an
arbitrary ordinal.
Let stand for an ordinal. A p-group A is called p -projective if
and totally projective if
p Ext.A; C/ D 0 for all groups C;
p Ext.A=p A; C/ D 0 for all ordinals and all groups C: (11.9)
Accordingly, A is totally projective if and only if the factor group A=p A is p -
projective for each .
Observe that if